text:
\section{Introduction}
This paper deals with Wavelets and nonlinear singular BVPs. Nonlinear BVPs are difficult to deal and if singularity is also present it becomes even more difficult. It is not easy to capture the behavior of the solutions near the point of singularity. If we apply suitable boundary conditions which forces the existence of unique continuous solutions and hence there is possibility of finding these solutions via numerical methods. Still since the coefficient blow up when near the singularity discretizing the differential equation is a challenge. Wavelets help us to treat this complicated situation in an easy way with less number of spatial points. To address both nonlinear BVPs and Wavelets we divide the introduction in two parts.
\subsection*{Nonlinear SBVPs arising in Exothermic Reactions}\label{P2_sec1} Here we propose a new class of nonlinear SBVP. For that let us consider the mathematical equation which governs, balance between heat generated and conducted away
\begin{equation}\label{P2_CHEM1}
\lambda\nabla^{2}T=-QW,
\end{equation}
where $T$ is gas temperature, $Q$ the heat of the reaction, $\lambda$ the thermal conductivity, $W$ the reaction velocity and $\nabla^{2}$ the Laplacian operator.

Chambre \cite{CHAMBRE1952} assumed that reaction is mono-molecular and velocity follows the Arrhenius law, given as
\begin{equation}\label{P2_CHEM2}
W=A\exp\left(\frac{-E}{RT}\right),
\end{equation}
and after some approximations and symmetry assumptions Chambre \cite{CHAMBRE1952} arrived at the following equation
\begin{equation}\label{P2_CHEM3}
Ly=-\delta \exp(t)
\end{equation}
where, $$L\equiv\frac{d^2}{dt^2}+\frac{k_g}{t}\frac{d}{dt},$$ $k_g$ depends on shape and size of the vessel and $\delta$ is a parameter.

Nakamura et al. \cite{NAKAMURA1989295}, while looking for an equation which can express the temperature dependence of the rate constant proposed the following
\begin{equation}\label{P2_CHEM4}
W=A\exp\left(\frac{-E}{R~\left(T_{0}^n+T^{n}\right)^{1/n}}\right),\quad n=1,2,3,\cdots
\end{equation}
where $R$ is a gas constant, $T$ is absolute temperature, $A$, $E_{0}$ and $T_{0}$ are parameters and $n$ is an integer.

Similar to analysis of Chambre \cite{CHAMBRE1952}, we arrive at the following differential equation
\begin{equation}\label{P2_CHEM5}
Ly=-B\exp\left(\frac{-A}{\left(c^n+y^{n}\right)^{1/n}}\right).
\end{equation}
There are several other examples which led us to consider the following class of nonlinear singular boundary value problem (SBVPs)
\begin{equation}\label{P2_1}
Ly+{f(t,y(t))}=0,\quad\quad\quad 0<t\le 1,
\end{equation}
subject to following the boundary conditions
\begin{subequations}
\begin{eqnarray}
\label{P2_1a}&& Case~(i)\quad ~~y'(0)=\alpha,\quad y(1)=\beta,\\
\label{P2_1b}&& Case~(ii)\quad ~y(0)=\alpha,\quad y(1)=\beta,\\
\label{P2_1c}&& Case~(iii)\quad y'(0)=\alpha,\quad ay(1)+by'(1)=\beta,
\end{eqnarray}
\end{subequations}
where $a,b,c$, $\alpha$, $\beta$ are real constants and ${f\left(t,y(t)\right)}$ is a real valued function. Boundary conditions at the singular end $t=0$ depend on the value of $k_g$.

There is huge literature on existence of solutions of such BVPs. Please refer \cite{PANDEY2008NARWA},  \cite{PANDEY2008JMAA}, \cite{PANDEY2009NATMA} and \cite{VERMA2011NATMA} and the references there in. Several numerical methods have also been proposed for solving these type of  non-linear singular boundary value problems (see \cite{Singh2016,MSAKVRP2019,VermaKayenat2018} and its references).

In \cite{RC2013} non linear singular Lane-Emden IVPs are solved with Haar Wavelet Quasilinearization approach, and nonlinearity is easily handled with quasilinearization. In \cite{SHI2017} Hermite wavelets operational matrix method is used to solve second order nonlinear singular initial value problems. In \cite{KAZE2013} Chebyshev wavelets operation matrices are used for solving nonlinear singular boundary value problems. In \cite{Zhou2016} method based on Laguerre wavelets is used to solve nonlinear singular boundary value problems. In \cite{MOH2011} method based on Legendre wavelet is proposed to solve singular boundary value problems. All these based on wavelets show high accuracy. In \cite{akvdt2018,rsg2019,rsgjshgag2019} Haar wavelets are used to solve SBVPs efficiently for higher resolution.

In this article we solve SBVP of type \eqref{P2_1} subject to boundary conditions \eqref{P2_1a},\eqref{P2_1b},\eqref{P2_1c} with help of Hermite Wavelet Newton Approach (HeWNA), Hermite Wavelet Quasilinearization Approach (HeWQA), Haar wavelet Newton Approach (HWNA) and Haar Wavelet Quasilinearization Approach (HWQA) and compare results to show accuracy of the method. Convergence of HeWNA method is also established. Novelty of this paper is that Newton Raphson has not been coupled with Haar wavelets and Hermite has not been used to solve Nonlinear Singular BVPs till now.

Some recent works on wavelets and their applications can be find in \cite{Hariharan2019,Assari2019,SCSS2019,HKMAST2019} and the references there in.

This paper is organized in the following manner. In section \ref{mra} we discuss MRA and Hermite wavelet is defined, in sub section \ref{P2_sec3} Haar wavelet is defined, in section \ref{P2_sec4} method of solution based on HeWQA, HeWNA, HWQA  and HWNA are proposed. In section \ref{P2_sec5}, convergence analysis of HWNA method is done and in the last section some numerical examples are presented to show accuracy of the method.
\begin{section}{Hermite and Haar Wavelet}\label{mra}
This section starts with MRA and definition of Hermite Wavelets.
\subsection{Wavelets and MRA}
The theory of Wavelets are developed by mathematicians as well engineers over the years. Wavelet word find its origin the name of French geophysicist Jean Morlet (1931--2007). He used wavelet to describe certain functions. Morlet and Croatian-French physicist Alex Grossman developed theory further which is used now a days \cite[p. 222]{MCPLAW2012}. Main issue that was being addressed in the process, was to overcome drawbacks of Fourier transforms. Wavelets are multi-indexed and it contains parameters which can be used to shift or dilate/contract the functions giving us basis functions. Thus computationally they are complex but they have better control and much better results are obtained at low resolution, i.e., less number of divisions are needed that we need in finite difference and all methods based on similar concepts. Here we consider methods based on wavelet transforms.

The following properties of wavelets enable us to choose them over other methods:

\begin{itemize}
\item Orthogonality
\item Compact Support
\item Density
\item Multiresolution Analysis (MRA)
\end{itemize}
\subsubsection{MRA}
Pereyra et al. \cite{MCPLAW2012} observe that an orthogonal multiresolution analysis (MRA) is a collection of closed subspaces of $L^2(\mathbb{R})$ which are nested, having trivial intersection, they exhaust the space, the subspaces are connected to each other by scaling property and finally there is a special function, the scaling function $\varphi$, whose integer translates form an orthonormal basis for
one of the subspaces. We give formal statement of MRA as defined in \cite{MCPLAW2012}.
\begin{definition}
An MRA with scaling function $\varphi$ is a collection of closed subspaces $V_j , j = \dots,—2,—1,0,1,2,\dots$ of $L^2(\mathbb{R})$ such that
\begin{enumerate}
\item $\mathbf{V}_{j} \subset \mathbf{V}_{j+1}$
\item $\overline{\bigcup\mathbf{V}_{j}}=\mathbf{L^2(\mathbb{R})}$
\item $ \bigcap \mathbf{V}_{j}={0}$
\item The function $f(x)$ belongs to $\mathbf{V}_{j}$ if and only if the function $f(2x) \in \mathbf{V}_{j+1}$.\
\item The function $\varphi$ belongs to $\mathbf{V}_{0}$, the set $\{\varphi(x-k), k \in \mathbf{\mathbb{Z}}\}$ is orthonormal basis for $\mathbf{V}_{0}$.
 \end{enumerate}
\end{definition}
The sequence of wavelet subspaces $W_j$ of $L^2(\mathbb{R})$, are such that $V_j\perp Wj$ , for all $j$ and $V_{j+1} = V_j\oplus W_j$. Closure of $\oplus_{j\in \mathbb{Z}} W_j$ is dense in $L^2(\mathbb{R})$ with respect to $L^2$ norm.

Now we state Mallat's theorem \cite{MALLAT1989} which guarantees that in presence of an orthogonal MRA, an orthonormal basis for $L^2(\mathbb{R})$ exists. These basis functions are fundamental functions in the theory of wavelets which helps us to develop advanced computational techniques.
\begin{theorem} {$\mathrm{(Mallat's~Theorem)}$}. Given an orthogonal MRA
with scaling function $\varphi$, there is a wavelet $\psi\in L^2(\mathbb{R})$ such that for
each $j\in \mathbb{Z}$, the family $\{\psi_{j,k}\}_{k\in\mathbb{Z}}$ is an orthonormal basis for $W_j$ .
Hence the family $\{\psi_{j,k}\}_{k\in\mathbb{Z}}$ is an orthonormal basis for $L^2(\mathbb{R})$. \label{Mallat}
\end{theorem}
\subsection{Hermite Wavelet (\cite{Saeed2014})}\label{P2_sec2}
Hermite Polynomials are defined on the interval $(-\infty,\infty)$ and can be defined with help of the recurrence formula:
\begin{eqnarray*}
&&H_{0}(t)=1\\
&&H_{1}(t)=2t\\
&&H_{m+1}(t)= 2tH_{m}(t)-2mH_{m-1}(t),~m=1,2,3,\cdots.
\end{eqnarray*}
Completeness and orthogonality (with respect to weight function $e^{-t^2}$) of Hermite polynomials enable us to treat them as wavelet (Theorem \ref{Mallat}).

Hermite wavelet on the interval $[0,1]$ are defined as
\begin{equation}
\label{P2_Def1}
\psi_{n,m}(t)=
2^{k/2}\frac{1}{\sqrt{n!2^{n}\sqrt{\pi}}}H_{m}(2^kt-\hat{n})\chi_{\left[\frac{\hat{n}-1}{2^k},\frac{\hat{n}+1}{2^k}\right)}
\end{equation}
where $k=1,2,\hdots$ is level of resolution, $n=1,2,\hdots, 2^{k-1},\hspace{0.25cm} \hat{n}=2n-1$ is translation parameter, $m=1,2,\hdots,M-1$ is order of Hermite polynomial.
\end{section}
\subsubsection{Approximation of Function with Hermite Wavelet}\label{P2_subsec2}
A function $f(t)$ defined on $L^2[0,1]$ can be approximated with Hermite wavelet in the following manner
\begin{equation}\label{P2_2}
f(t)= \sum^{\infty}_{n=1}\sum^{\infty}_{m=0}c_{nm}\psi_{nm}(t).
\end{equation}
For computation purpose we truncate \eqref{P2_2} and define,
\begin{equation}\label{P2_3}
f(t)\simeq \sum^{2^{k}-1}_{n=1}\sum^{M-1}_{m=0}c_{nm}\psi_{nm}(t)=c^{T}\psi(t)
\end{equation}
where  $\psi(t)$ is $2^{k-1}M\times 1$ matrix given as:
\begin{equation*}
\psi(t)= \left[\psi_{1,0}(t),\dots,\psi_{1,M-1}(t),\psi_{2,0}(t),\dots,\psi_{2,M-1}(t),\dots,\psi_{2^{k-1},0}(t),\dots,\psi_{2^{k-1},M-1}(t)\right]^T
\end{equation*}
$c$ is $2^{k-1}M\times 1$ matrix. Entries of $c$ can be computed as
\begin{equation}\label{P2_4}
c_{ij}=\int^{1}_{0}f(t)\psi_{ij}(t)dt
\end{equation}
with $i=1,2,\dots,2^k-1$ and $j=0,1,\dots,M-1$. Here $M$ is degree of Hermite polynomial.
\subsubsection{Integration of Hermite Wavelet}
As suggested in \cite{Gupta2015}, $\nu$-th order integration of $\psi(t)$ can also be approximated as
\begin{multline*}\label{P2_5}
\int^{t}_{0}\int^{t}_{0}\dots\int^{t}_{0}\psi(\tau)d\tau\simeq\\
\left[J^\nu\psi_{1,0}(t),\dots,J^\nu\psi_{1,M-1}(t),J^\nu\psi_{2,0}(t),\dots,J^\nu\psi_{2,M-1}(t),\dots,J^\nu\psi_{2^{k-1},0}(t),\dots,J^\nu\psi_{2^{k-1},M-1}(t)\right]^T
\end{multline*}
where
\begin{equation}
J^{\nu}\psi_{n,m}(t)= 2^{k/2}\frac{1}{\sqrt{n!2^{n}\sqrt{\pi}}}J^{\nu}H_{m}(2^kt-\hat{n})\chi_{\left[\frac{\hat{n}-1}{2^k},\frac{\hat{n}+1}{2^k}\right)},
\end{equation}
where $k=1,2,\hdots$ is level of resolution, $n=1,2,\hdots, 2^{k-1},\hspace{0.25cm} \hat{n}=2n-1$ is translation parameter, $m=1,2,\hdots,M-1$ is order of Hermite polynomial.
\begin{remark} Integral operator $J^{\nu}(\nu>0)$ of a function $f(t)$ is defined as
\begin{equation*}\label{P2_6}
J^{\nu}f(t)= \frac{1}{\nu!}\int_{0}^{t}(t-s)^{\nu-1}f(s)ds.
\end{equation*}
\end{remark}
\subsubsection{Hermite Wavelet Collocation Method}\label{P2_subsec3}
To apply Hermite wavelet on the ordinary differential equations, we need its discretized form. We use collocation method for discretization where mesh points are given by
\begin{eqnarray}
\label{P2_19}\bar x_{l} = l \Delta x,\quad\quad\quad l = 0,1,\cdots,M-1.
\end{eqnarray}
For the collocation points we define
\begin{eqnarray}
\label{P2_20}x_{l} = 0.5(\bar x_{l-1} + \bar x_{l}), \quad\quad\quad l = 0,1,\cdots,M-1.
\end{eqnarray}
For $k=1$, equation \eqref{P2_3} takes the form
\begin{equation}\label{P2_21}
f(t)\simeq \sum^{M-1}_{m=0}c_{1m}\psi_{1m}(t).
\end{equation}
We replace $t$ by $x_{l}$ in above equation and arrive at system of equations which can easily be solved to get the solution of the nonlinear SBVP.
\subsection{Haar Wavelet}\label{P2_sec3}
Let us assume that $x$ belongs to any interval $[P, Q]$, where $P$ and $Q$ are constant end points. Let us define $M = 2^{J}$, where $J$ is the maximal level of resolution. Devide $[P, Q]$ into $2M$ subintervals of equal length $\Delta x = (Q- P)/(2M)$. The wavelet number $i$ is defined as $i = m + k + 1$, where $j = 0, 1,\cdots,J$ and $k = 0, 1,\cdots,m - 1$ (here $m = 2j$).
The $i^{th}$ Haar wavelet is explained as
\begin{equation}
\label{P2_7}h_{i}(x)= \chi_{\left[{\eta }_{1}\left(i\right),{\eta }_{2}\left(i\right)\right)}-\chi_{\left[{\eta }_{2}\left(i\right),{\eta }_{3}\left(i\right)\right)}
\end{equation}
where
\begin{eqnarray}
\label{P2_8}\eta_{1} (i) = P + 2k\mu\Delta x,\quad
\eta_{2}(i) = P + (2k + 1)\mu\Delta x,\quad
\eta_{3}(i) = P + 2(k + 1)\mu\Delta x,\quad
\mu= M/m.
\end{eqnarray}
Above equations are valid for $i > 2$. For $i = 1$ case we have, $h_{i}(x) = \chi_{[P,Q]}$.

For $i = 2$ we have
\begin{eqnarray}
\label{P2_9}\eta_{1}(2) = P,\quad
\eta_{2}(2) = 0.5(2P + Q),\quad
\eta_{3}(2) = Q.
\end{eqnarray}
The thickness of the $i^{th}$ wavelet is
\begin{eqnarray}
\eta_{3}(i)- \eta_{1}(i) = 2\mu\Delta x = (Q- P)m^{-1} = (Q- P)2^{-j}.
\end{eqnarray}
If $J$ is fixed then by \eqref{P2_7}
\begin{eqnarray}
\label{P2_10}\int_{P}^{Q} h_{i}(x)h_{l}(x)dx=
\begin{cases}
(Q-P)2^{-j}, & l = i, \\
0, & l\neq i.
\end{cases}
\end{eqnarray}
The integrals $p_{\upsilon,i}(x)$ are defined as
\begin{eqnarray}
\label{P2_11} p_{\upsilon,i}(x)=\int_{P}^{x}\int_{P}^{x}\cdots\int_{P}^{x}h_{i}(t)dt^{v}=\frac{1}{(v-1)!}\int_{P}^{x}(x- t)^{(\upsilon-1)}h_{i}(t)dt,
\end{eqnarray}
where $\upsilon = 1,2,\cdots,n,\quad i = 1,2,\cdots,2M.$

Putting all values in the integral we get
\begin{multline}\label{P2_12}
p_{\upsilon , i}(x)=
\frac{1}{\upsilon!}[x-\eta_{1}(i)]^\upsilon\chi_{\left[{\eta}_{1}\left(i\right),{\eta}_{2}\left(i\right)\right)}+ \frac{1}{\upsilon!}\{[x-\eta_{1}(i)]^\upsilon-2[x-\eta_{2}(i)]^\upsilon\} \chi_{\left[{\eta }_{2}\left(i\right),{\eta }_{3}\left(i\right)\right]}\\
+\frac{1}{\upsilon!}\{[x-\eta_{1}(i)]^\upsilon-2[x-\eta_{2}(i)]^\upsilon+[x-\eta_{3}(i)]^\upsilon\}\chi_{(\eta_{3}(i),\infty)}
\end{multline}
for $i>1$ and for $i=1$ we have $\eta_{1}=P, \eta_{2} = \eta_{3}=Q$ and
\begin{equation}
\label{P2_13}p_{\upsilon,1}(x)=\frac{1}{\upsilon!}(x-P)^\upsilon.
\end{equation}

\subsubsection{Haar Wavelet Collocation Method}\label{P2_subsec3b}
Similar to case of previous section here again we define collocation points as follows
\begin{eqnarray}
&&\label{P2_14}\bar x_{t} = P + t \Delta x,\quad\quad\quad~ t = 0,1,\cdots,2M,\\
&&\label{P2_15}x_{t} = 0.5(\bar x_{t-1} + \bar x_{t}), \quad t = 0,1,\cdots,2M,
\end{eqnarray}
and replace $x \to x_{t}$ in \eqref{P2_7},\eqref{P2_8},\eqref{P2_9}. We define the Haar matrices $H, P_{1},P_{2},\cdots, P_\upsilon$ which are $2M \times 2M$ matrices. Entries of matrices are given by $H(i,t)=h_{i}(x_{t})$, $P_{v}(i,t)=P_{\upsilon, i}(x_{t})$, $\upsilon = 1,2,\cdots$.
Consider $P=0, Q=1,J=1$. Then $2M=4$, so $H,P_{1},P_{2}$ are defined as
\begin{eqnarray}
\label{P2_16}H=\left[
\begin{array}{cccc}
1& 1& 1& 1\\
1& 1& -1& -1\\
1& -1& 0& 0\\
0& 0& 1& -1
\end{array}\right],
\quad P_{1}=\frac{1}{8}
\left[
\begin{array}{cccc}
1& 3& 5& 7\\
1& 3& 3& 1\\
1& 1& 0& 0\\
0& 0& 1& 1
\end{array}\right],
\quad P_{2}=\frac{1}{128}\left[
\begin{array}{cccc}
1& 9& 25& 49\\
1& 9& 23& 31\\
1& 7& 8& 8\\
0& 0& 1& 7
\end{array}\right].
\end{eqnarray}
\subsubsection{Approximation of Function with Haar Wavelet}\label{P2_subsec3c}
A function $f(t)$ defined on $L^2[0,1]$ can be approximated by Haar wavelet basis in the following manner
\begin{equation}\label{P2_17}
f(t)= \sum^{\infty}_{i=0}a_{i}h_{i}(t).
\end{equation}
For computation purpose we truncate \eqref{P2_17} and define
\begin{equation}\label{P2_18}
f(t)\simeq \sum^{2M}_{i=0}a_{i}h_{i}(t),
\end{equation}
where $M$ is level of resolution.
\section{Numerical Methods Based on Hermite and Haar Wavelets}\label{P2_sec4}
In this section, solution methods based on Hermite wavelet and Haar wavelet are presented.
\subsection{Hermite Wavelet Quasilinearization Approach (HeWQA)}\label{P2_subsec4a}
In HeWQA we use quasilinearization to linearize SBVP then method of collocation for discretization and use Hermite wavelets for computation of solutions to nonlinear SBVP. We consider differential equation \eqref{P2_1} with boundary conditions \eqref{P2_1a}, \eqref{P2_1b} and \eqref{P2_1c}. Quasilinearizing this equation, we get the form
\begin{subequations}
\begin{eqnarray}
\label{P2_22a}Ly_{r+1}=-f(t,y_{r}(t))+\sum_{s=0}^{1}(y_{r+1}^{s}-y_{r}^{s})(-f_{y^{s}}(t,y_{r}(t)),
\end{eqnarray}
subject to linearized boundary conditions,
\begin{eqnarray}
\label{P2_22b}&&y'_{r+1}(0)=y_{r}(0),\quad{y_{r+1}(1)}={y_{r}(1)},\\
\label{P2_22c}&&y_{r+1}(0)=y_{r}(0),\quad{y_{r+1}(1)}={y_{r}(1)},\\
\label{P2_22d}&&y'_{r+1}(0)=y_{r}(0),\quad a{y_{r+1}(1)}+by'_{r+1}(1)=a{y_{r}(1)}+by'_{r}(1).
\end{eqnarray}
Here $s=0,1$, $f_{y^{s}}=\partial{f}/\partial{y^{s}}$ and $y_{r}^{0}(t)=y_{r}(t)$.

Thus we arrive at linearized form of given differential equation. Now we use Hermite wavelet method similar to described in \cite{CH1997}. Let us assume
\begin{eqnarray}
\label{P2_22e}y''_{r+1}(t)=\sum^{M-1}_{m=0}c_{1m}\psi_{1m}(t).
\end{eqnarray}
Then integrating twice we get the following two equations:
\begin{eqnarray}
&&\label{P2_22f}y'_{r+1}(t)=\sum^{M-1}_{m=0}c_{1m}J\psi_{1m}(t)+y'_{r+1}(0),\\
&&\label{P2_22g}y_{r+1}(t)=\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(t)+ty'_{r+1}(0)+y_{r+1}(0).
\end{eqnarray}
Here $J^{\nu}(\nu>0)$ is the integral operator defined previously.
\end{subequations}
\subsubsection{Treatement of the Boundary Value Problem}\label{P2_subsubsec4a}
Based on boundary conditions we will consider different cases and follow procedure similar to described in \cite{LU2008}.
\textbf{Case (i)}:\quad In equation \eqref{P2_1a} we have
$y'(0)=\alpha,y(1)=\beta$. So by linearization we have $y'_{r+1}(0)=\alpha,y_{r+1}(1)=\beta$. Now put $t=1$ in \eqref{P2_22g} we get
\begin{equation*}\label{P2_23}
y_{r+1}(1)=\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(1)+y'_{r+1}(0)+y_{r+1}(0),
\end{equation*}
so
\begin{equation}
\label{P2_24}y_{r+1}(0)=y_{r+1}(1)-\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(1)-y'_{r+1}(0).
\end{equation}
By using equation \eqref{P2_24} in \eqref{P2_22g} and simplifying we get
\begin{equation}\label{P2_27}
y_{r+1}(t)=(t-1)y'_{r+1}(0)+y_{r+1}(1)+\sum^{M-1}_{m=0}c_{1m}(J^2\psi_{1m}(t)-J^2\psi_{1m}(1)).
\end{equation}
Now putting values of $y'_{r+1}(0)$ and $y_{r+1}(1)$ in \eqref{P2_22f} and \eqref{P2_27} we get
\begin{eqnarray}
&&y'_{r+1}(t)=\alpha +\sum^{M-1}_{m=0}c_{1m}J\psi_{1m}(t),\\
&&\label{P2_28}
y_{r+1}(t)=(t-1)\alpha + \beta +\sum^{M-1}_{m=0}c_{1m}(J^2\psi_{1m}(t)-J^2\psi_{1m}(1)).
\end{eqnarray}
\textbf{Case (ii)}:\quad In equation \eqref{P2_1b} we have
$y(0)=\alpha,y(1)=\beta$. So by linearization we have $y_{r+1}(0)=\alpha,y_{r+1}(1)=\beta$. Now put $t=1$ in equation \eqref{P2_22g} we get
\begin{eqnarray}\label{P2_29}
y_{r+1}(1)=\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(1)+y'_{r+1}(0)+y_{r+1}(0),
\end{eqnarray}
so
\begin{eqnarray}\label{P2_30}
y'_{r+1}(0)=y_{r+1}(1)-\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(1)-y_{r+1}(0).
\end{eqnarray}
By putting these values in equation \eqref{P2_22f} and \eqref{P2_22g} we get
\begin{eqnarray*}
y'_{r+1}(t)=y_{r+1}(1)-y_{r+1}(0)+\sum^{M-1}_{m=0}c_{1m}(J\psi_{1m}(t)-J^2\psi_{1m}(1)),
\end{eqnarray*}
and
\begin{eqnarray*}\label{P2_31}
y_{r+1}(t)=(1-t)y_{r+1}(0)+ty_{r+1}(1)+\sum^{M-1}_{m=0}c_{1m}(J^2\psi_{1m}(t)-J^2\psi_{1m}(1)).
\end{eqnarray*}
Now we put values of $y_{r+1}(0)$ and $y_{r+1}(1)$ we get
\begin{eqnarray}\label{P2_32}
&&y'_{r+1}(t)=(\beta-\alpha)+\sum^{M-1}_{m=0}c_{1m}(J\psi_{1m}(t)-J^2\psi_{1m}(1)),\\
&&\label{P2_33}
y_{r+1}(t)=(1-t)\alpha + t\beta +\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(1)-y_{r+1}(0)).
\end{eqnarray}
\textbf{Case (iii)}:\quad In equation \eqref{P2_1c} we have
$y'(0)=\alpha, ay(1)+by'(1)=\beta$. So by linearization we have $y'_{r+1}(0)=\alpha,ay_{r+1}(1)+by'_{r+1}(1)=\beta$. Now put $t=1$ in equation \eqref{P2_22f} and \eqref{P2_22g} we get
\begin{eqnarray}\label{P2_34}
&&y'_{r+1}(1)=\sum^{M-1}_{m=0}c_{1m}J\psi_{1m}(1)+y'_{r+1}(0),\\
\label{P2_35}&&y_{r+1}(1)=\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(1)+y'_{r+1}(0)+y_{r+1}(0).
\end{eqnarray}
Putting these values in $ay_{r+1}(1)+by'_{r+1}(1)=\beta$ and solving for $y_{r+1}(0)$ we have
\begin{eqnarray*}\label{P2_37}
y_{r+1}(0)=\frac{1}{a}\left(\beta - ay'_{r+1}(0)-a\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(1)-b\left(\sum^{M-1}_{m=0}c_{1m}J\psi_{1m}(1)+y'_{r+1}(0)\right)\right).
\end{eqnarray*}
Hence from equation \eqref{P2_22g} we have
\begin{multline}\label{P2_38}
y_{r+1}(t)=\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(t)+ty'_{r+1}(0)\\
+\frac{1}{a}\left(\beta -ay'_{r+1}(0)-a\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(1)-b\left(\sum^{M-1}_{m=0}c_{1m}J\psi_{1m}(1)+y'_{r+1}(0)\right)\right).
\end{multline}
Now we put values of $y_{r+1}(0)$ and $y_{r+1}(1)$ in equation \eqref{P2_22f} and \eqref{P2_38} we get
\begin{eqnarray}
&&y'_{r+1}(t)=\alpha +\sum^{M-1}_{m=0}c_{1m}J\psi_{1m}(t),\\
&&\label{P2_39}y_{r+1}(t)=\frac{\beta}{a}+\left(t-1-\frac{b}{a}\right)\alpha+\sum^{M-1}_{m=0}c_{1m}\left(J^2\psi_{1m}(t)-J^2\psi_{1m}(1)-\frac{b}{a}J\psi_{1m}(1)\right).
\end{eqnarray}
Finally we put values of $y''_{r+1}$, $y'_{r+1}$ and $y_{r+1}$ for all these cases in the linearized differential equation \eqref{P2_22a}. Now we discritize the final equation with collocation method and then solve the resulting system assuming initial guess $y_{0}(t)$. We get required value of solution $y(t)$ of the nonlinear SBVPs at different collocation points.
\subsection{Hermite Wavelet Newton Approach (HeWNA)}\label{P2_subsec4b}
In this approach we use the method of collocation for discretization and then Hermite wavelet for approximation of the solutions. Finally Newton Raphson method is used to solve the resulting nonlinear system of equations.

We consider differential equation \eqref{P2_1} with boundary conditions \eqref{P2_1a}, \eqref{P2_1b} and \eqref{P2_1c}. Now we assume
\begin{eqnarray}
\label{P2_70}y''(t)=\sum^{M-1}_{m=0}c_{1m}\psi_{1m}(t).
\end{eqnarray}
Integrating twice we get the following two equations:
\begin{eqnarray}
&&\label{P2_71}y'(t)=\sum^{M-1}_{m=0}c_{1m}J\psi_{1m}(t)+y'(0),\\
&&\label{P2_72}y(t)=\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(t)+ty'(0)+y(0).
\end{eqnarray}
\subsubsection{Treatment of the Boundary Value Problem}\label{P2_subsubsec4b}
Based on boundary conditions we divide it in different cases.\\
\textbf{Case (i)}: In equation \eqref{P2_1a} we have
$y'(0)=\alpha, y(1)=\beta$. Now put $t=1$ in \eqref{P2_72} we get
\begin{eqnarray}\label{P2_40}
y(1)=\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(1)+y'(0)+y(0),
\end{eqnarray}
so
\begin{eqnarray}\label{P2_41}
y(0)=y(1)-\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(1)-y'(0).
\end{eqnarray}
By using equation \eqref{P2_41} in \eqref{P2_72} we get
\begin{eqnarray*}
\label{P2_42} &&y(t)=\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(t)+ty'(0)+y(1)-\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(1)-y'(0),\\
\label{P2_43}
&&y(t)=\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(t)+(t-1)y'(0)+y(1)-\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(1).
\end{eqnarray*}
Hence
\begin{eqnarray}\label{P2_44}
y(t)=(t-1)y'(0)+y(1)+\sum^{M-1}_{m=0}c_{1m}(J^2\psi_{1m}(t)-J^2\psi_{1m}(1)).
\end{eqnarray}
Now putting values of $y'(0)$ and $y(1)$ in \eqref{P2_71} and \eqref{P2_44} we get
\begin{eqnarray}
&&y(t)=\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(t)+ty'(0)+y(0),\\
&&\label{P2_45} y(t)=(t-1)\alpha + \beta +\sum^{M-1}_{m=0}c_{1m}(J^2\psi_{1m}(t)-J^2\psi_{1m}(1)).
\end{eqnarray}
\textbf{Case (ii)}: In equation \eqref{P2_1b} we have
$y(0)=\alpha,y(1)=\beta$. Now put $t=1$ in equation \eqref{P2_72} we get
\begin{eqnarray}\label{P2_46}
y(1)=\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(1)+y'(0)+y(0),
\end{eqnarray}
so
\begin{eqnarray*}\label{P2_47}
y'(0)=y(1)-\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(1)-y(0).
\end{eqnarray*}
Now using these values of $y'(0)$ and $y(1)$ in \eqref{P2_71} and \eqref{P2_72} and solving we get
\begin{eqnarray}
&&y'(t)=(\beta-\alpha)+\sum^{M-1}_{m=0}c_{1m}(J\psi_{1m}(t)-J^2\psi_{1m}(1)),\\
&&\label{P2_48}
y(t)=(1-t)\alpha + t\beta +\sum^{M-1}_{m=0}c_{1m}(J^2\psi_{1m}(t)-J^2\psi_{1m}(1)).
\end{eqnarray}
\textbf{Case (iii)}: In equation \eqref{P2_1c} we have
$y'(0)=\alpha, ay(1)+by'(1)=\beta$. Now put $t=1$ in equation \eqref{P2_71} and \eqref{P2_72} we get
\begin{eqnarray}\label{P2_49}
&&y'(1)=\sum^{M-1}_{m=0}c_{1m}J\psi_{1m}(1)+y'(0),\\
&&\label{P2_50}y(1)=\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(1)+y'(0)+y(0).
\end{eqnarray}
Putting these values in $ ay(1)+by'(1)=\beta$ and solving we will get value of $y(0)$, now put $y(0)$ and $y'(0)$ in \eqref{P2_72} we have
\begin{eqnarray}\label{P2_51}
y(t)=\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(t)+t y'(0)+\frac{1}{a}\left(\beta - ay'(0)-a\sum^{M-1}_{m=0}c_{1m}J^2\psi_{1m}(1)-b\left(\sum^{M-1}_{m=0}c_{1m}J\psi_{1m}(1)+y'(0)\right)\right).
\end{eqnarray}
Now by putting $y'(0)=\alpha$ in \eqref{P2_71} and \eqref{P2_51}, we have
\begin{eqnarray}
&&y'(t)=\alpha +\sum^{M-1}_{m=0}c_{1m}J\psi_{1m}(t),\\
&&\label{P2_52}
y(t)=\frac{\beta}{a}+\left(t-1-\frac{b}{a}\right)\alpha+\sum^{M-1}_{m=0}c_{1m}\left(J^2\psi_{1m}(t)-J^2\psi_{1m}(1)-\frac{b}{a}J\psi_{1m}(1)\right).
\end{eqnarray}
Now we put values of $y(t)$, $y'(t)$ and $y''(t)$ in \eqref{P2_1}. We discretize final equation with collocation method and solve the resulting nonlinear system with Newton Raphson method for $c_{1m}, m=0,1,\dots,M-1$. By substituting the values of $c_{1m}, m=0,1,\dots,M-1$, we get value of the solution of $y(t)$ of nonlinear SBVPs at different collocation points.
\subsection{Haar Wavelet Quasilinearization Approach (HWQA)}\label{P2_subsec4c}
As explained for HeWQA, we will follow same procedure in HWQA method. Here we are using Haar Wavelet in place of Hermite Wavelet. We consider differential equation \eqref{P2_1} with boundary conditions \eqref{P2_1a}, \eqref{P2_1b} or \eqref{P2_1c}. Applying method of quasilinearization, as we did in HeWQA,  we have equation \eqref{P2_22a} with linearized boundary conditions \eqref{P2_22b},\eqref{P2_22c} and \eqref{P2_22d}.

Let us assume
\begin{eqnarray}\label{P2_53e}
y''_{r+1}(t)=\sum_{i=0}^{2M}a_{i}h_{i}(t),
\end{eqnarray}
where $a_{i}$ are the wavelet coefficients. Then integrating twice we get following two equations:
\begin{eqnarray}\label{P2_53f}
&&y'_{r+1}(t)=\sum_{i=0}^{2M}a_{i}p_{1,i}(t)+y'_{r+1}(0),\\
&&\label{P2_53g}y_{r+1}(t)=\sum_{i=0}^{2M}a_{i}p_{2,i}(t)+ty'_{r+1}(0)+y_{r+1}(0).
\end{eqnarray}
\subsubsection{Treatement of the Boundary Value Problem}\label{P2_subsubsec4c}
Expressions for different boundary conditions in HWQA method are given below.\\
\noindent \textbf{Case (i)}: In equation \eqref{P2_1a} we have
$y'(0)=\alpha, y(1)=\beta$. Following same procedure as HeWQA
we have
\begin{eqnarray}
&&y'_{r+1}(t)=\alpha+\sum_{i=0}^{2M}a_{i}p_{1,i}(t),\\
&&\label{P2_54}y_{r+1}(t)=(t-1)\alpha + \beta +\sum_{i=0}^{2M}a_{i}(p_{2,i}(t)-p_{2,i}(1)).
\end{eqnarray}
\noindent \textbf{Case (ii)}: In equation \eqref{P2_1b} we have $y(0)=\alpha,y(1)=\beta$. So we have
\begin{eqnarray}
&&y'_{r+1}(t)=(\beta-\alpha)+\sum^{M-1}_{m=0}c_{1m}(p_{1,i}(t)-p_{2,i}(1)),\\
&&\label{P2_55}y_{r+1}(t)=(1-t)\alpha + t\beta +\sum_{i=0}^{2M}a_{i}(p_{2,i}(t)-tp_{2,i}(1)).
\end{eqnarray}
\textbf{Case (iii)}: In equation \eqref{P2_1c} we have $y'(0)=\alpha, ay(1)+by'(1)=\beta$. We finally have
\begin{eqnarray}
&&y'_{r+1}(t)=\alpha+\sum_{i=0}^{2M}a_{i}p_{1,i}(t),\\
&&\label{P2_57} y_{r+1}(t)=\frac{\beta}{a}+\left(t-1-\frac{b}{a}\right)\alpha+\sum_{i=0}^{2M}a_{i}\left(p_{2,i}(t)-p_{2,i}(1)-\frac{b}{a}p_{1,i}(1)\right).
\end{eqnarray}
\subsection{Haar Wavelet Newton Approach(HWNA)}\label{P2_subsec4d}
Here we use same procedure as that of HeWNA.
\subsubsection{Treatment of the Boundary Value Problem}\label{P2_subsubsec4d}
Expressions for different boundary conditions in HWNA method are given below.\\
\textbf{Case (i)}: In equation \eqref{P2_1a} we have
$y'(0)=\alpha,y(1)=\beta$. Following same procedure, final expression will take the form
\begin{eqnarray}
&&y'(t)=\alpha+\sum_{i=0}^{2M}a_{i}p_{1,i}(t),\\
&&\label{P2_58}y(t)=(t-1)\alpha + \beta +\sum^{2M}_{m=0}a_{i}(p_{2,i}(t)-p_{2,i}(1)).
\end{eqnarray}
\noindent\textbf{Case (ii)}: In equation \eqref{P2_1b} we have
$y(0)=\alpha,y(1)=\beta$. So by linearization we have $y(0)=\alpha,~y(1)=\beta$. Final expression is of the form
\begin{eqnarray}
&&y'(t)=(\beta-\alpha)+\sum^{M-1}_{m=0}c_{1m}(p_{1,i}(t)-p_{2,i}(1)),\\
&&\label{P2_59}
y(t)=(1-t)\alpha + t\beta +\sum^{2M}_{m=0}c_{1m}(p_{2,i}(t)-p_{2,i}(1)).
\end{eqnarray}
\textbf{Case (iii)}: In equation \eqref{P2_1c} we have
$y'(0)=\alpha,~ ay(1)+by'(1)=\beta$, so we have the following expression:
\begin{eqnarray}
&&y'(t)=\alpha+\sum_{i=0}^{2M}a_{i}p_{1,i}(t),\\
&&\label{P2_60} y(t)=\frac{\beta}{a}+\left(t-1-\frac{b}{a}\right)\beta+\sum^{2M}_{m=0}c_{1m}\left(p_{2,i}(t)-p_{2,i}(1)-\frac{b}{a}p_{1,i}(1)\right).
\end{eqnarray}
\section{Convergence}\label{P2_sec5}
Let us consider $2^{\mathrm{nd}}$ order ordinary differential equation in general form
\begin{equation*}
G(t,u,u',u'')=0.
\end{equation*}
We consider the HeWNA method. Let
\begin{equation}\label{P2_con1}
f(t)=u''(t)= \sum^{\infty}_{n=1}\sum^{\infty}_{m=0}c_{nm}\psi_{nm}(t).
\end{equation}
Integrating above equation two times we have
\begin{equation}\label{P2_con2}
u(t)= \sum^{\infty}_{n=1}\sum^{\infty}_{m=0}c_{nm}J^2\psi_{nm}(t)+B_{T}(t),
\end{equation}
where $B_{T}(t)$ stands for boundary term.
\begin{theorem}
Let us assume that, $f(t)=\frac{d^2u}{dt^2}\in L^2(\mathbb{R})$ is a continuous function defined on [0,1]. Let us consider $f(t)$ is bounded, i.e.,
\begin{equation}\label{P2_con3}
\forall  t \in [0,1] \quad \quad \exists \quad \eta :\left|\frac{d^2u}{dt^2}\right| \leq \eta.
\end{equation}
Then method based on Hermite Wavelet Newton Approach (HeWNA) converges.
\end{theorem}
\begin{proof}
In \eqref{P2_con2} by truncating expansion we have,
\begin{eqnarray}\label{P2_con4}
u^{k,M}(t)= \sum^{2^k-1}_{n=1}\sum^{M-1}_{m=0}c_{nm}J^2\psi_{nm}(t)+B_{T}(t)
\end{eqnarray}
So error $E_{k,M}$  can be expressed as
\begin{eqnarray}\label{P2_con5}
\|E_{k,M}\|_{2}=\|u(t)-u^{k,M}(t)\|_{2}=\left\|\sum^{\infty}_{n=2^k}\sum^{\infty}_{m=M}c_{nm}J^2\psi_{nm}(t)\right\|_{2}.
\end{eqnarray}
Expanding $L^{2}$ norm, we have
\begin{eqnarray}\label{P2_con6}
&&\|E_{k,M}\|_2^2=\int_{0}^{1}\left(\sum^{\infty}_{n=2^k}\sum^{\infty}_{m=M}c_{nm}J^2\psi_{nm}(t)\right)^2dt,\\
\label{P2_con8}
&&\|E_{k,M}\|_2^2=\sum^{\infty}_{n=2^k}\sum^{\infty}_{m=M}\sum^{\infty}_{s=2^k}\sum^{\infty}_{r=M}\int_{0}^{1}c_{nm}c_{sr} J^2\psi_{nm}(t)J^2\psi_{sr}(t)dt,\\
&&\label{P2_con9}
\|E_{k,M}\|_2^2\leq \sum^{\infty}_{n=2^k}\sum^{\infty}_{m=M}\sum^{\infty}_{s=2^k}\sum^{\infty}_{r=M}\int_{0}^{1}|c_{nm}|~|c_{sr}|~ |J^2\psi_{nm}(t)|~|J^2\psi_{sr}(t)|dt.
\end{eqnarray}
Now
\begin{eqnarray*}\label{P2_con7}
&&|J^{2}\psi_{nm}(t)|\leq \int_{0}^{t}\int_{0}^{t}|\psi_{nm}(t)|dtdt,\\
\label{P2_con10}
&&\leq \int_{0}^{t}\int_{0}^{1}|\psi_{nm}(t)|dtdt,
\end{eqnarray*}
since $t  \in [0,1]$. Now by \eqref{P2_Def1}, we have
\begin{eqnarray*}\label{P2_con11}
|J^{2}\psi_{nm}(t)|\leq  2^{k/2}\frac{1}{\sqrt{n!2^{n}\sqrt{\pi}}}\int_{0}^{t}\int_{\frac{\hat{n}-1}{2^k}}^{\frac{\hat{n}+1}{2^k}}|H_{m}(2^{k}t-\hat{n})|dtdt.
\end{eqnarray*}
By changing variable $2^kt-\hat{n}=y$, we get
\begin{eqnarray*}
|J^{2}\psi_{nm}(t)|&&\leq 2^{-k/2}\frac{1}{\sqrt{n!2^{n}\sqrt{\pi}}}\int_{0}^{t}\int_{-1}^{1}|H_{m}(y)|dydt,\\
&&\leq 2^{-k/2}\frac{1}{\sqrt{n!2^{n}\sqrt{\pi}}}\int_{0}^{t}\int_{-1}^{1}\left|\frac{H'_{m+1}(y)}{m+1}\right|dydt,\\
&&\leq 2^{-k/2}\frac{1}{(\sqrt{n!2^{n}\sqrt{\pi}})(m+1)}\int_{0}^{t}\int_{-1}^{1}|H'_{m+1}(y)|dydt.
\end{eqnarray*}
By putting $\int_{-1}^{1}|H'_{m+1}(y)|dy = h$, we get
\begin{eqnarray*}\label{P2_con12}
|J^{2}\psi_{nm}(t)|\leq 2^{-k/2}\frac{1}{(\sqrt{n!2^{n}\sqrt{\pi}})(m+1)}\int_{0}^{t}h dt.
\end{eqnarray*}
Hence
\begin{eqnarray}\label{P2_con13}
|J^{2}\psi_{nm}(t)|\leq 2^{-k/2}\frac{1}{(\sqrt{n!2^{n}\sqrt{\pi}})(m+1)}h,
\end{eqnarray}
since $t \in [0,1]$. Now for $|c_{nm}|$, we have
\begin{eqnarray}\label{P2_con14}
&&c_{nm}=\int_{0}^{1}f(t)\psi_{nm}(t)dt,\\
&&\label{P2_con15}
|c_{nm}| \leq \int_{0}^{1}|f(t)||\psi_{nm}(t)|dt.
\end{eqnarray}
Now using \eqref{P2_con3}, we have
\begin{eqnarray*}\label{P2_con20}
|c_{nm}| \leq \eta\int_{0}^{1}|\psi_{nm}(t)|dt.
\end{eqnarray*}
By \eqref{P2_Def1}, we have
\begin{eqnarray*}\label{P2_con16}
|c_{nm}|\leq  2^{k/2}\frac{\eta}{\sqrt{n!2^{n}\sqrt{\pi}}}\int_{\frac{\hat{n}-1}{2^k}}^{\frac{\hat{n}+1}{2^k}}|H_{m}(2^{k}t-\hat{n})|dt.
\end{eqnarray*}
Now by change of variable $2^kt-\hat{n}=y$, we get
\begin{eqnarray*}\label{P2_con17}
&&|c_{nm}|\leq  2^{-k/2}\frac{\eta}{\sqrt{n!2^{n}\sqrt{\pi}}}\int_{-1}^{1}|H_{m}y|dy,\\
&&\label{P2_con18} |c_{nm}|\leq  2^{-k/2}\frac{\eta}{\sqrt{n!2^{n}\sqrt{\pi}}}\int_{-1}^{1}\left|\frac{H'_{m+1}(y)}{m+1}\right|dy.
\end{eqnarray*}
By putting $\int_{-1}^{1}|H'_{m+1}(y)|dy = h$, we have
\begin{eqnarray}\label{P2_con19}
|c_{nm}|\leq 2^{-k/2}\frac{1}{(\sqrt{n!2^{n}\sqrt{\pi}})(m+1)}\eta h.
\end{eqnarray}
Now using equation \eqref{P2_con13} and \eqref{P2_con19} in \eqref{P2_con9}
\begin{eqnarray}\label{P2_con21}
&&\|E_{k,M}\|_2^2\leq 2^{-2k}\eta^2h^4\sum^{\infty}_{n=2^k}\sum^{\infty}_{m=M}\sum^{\infty}_{s=2^k}\sum^{\infty}_{r=M}\int_{0}^{1}\frac{1}{(\sqrt{n!2^{n}\sqrt{\pi}})^2(m+1)^2}\frac{1}{(\sqrt{s!2^{s}\sqrt{\pi}})^2(r+1)^2} dt,\\
&&\label{P2_con22}
\|E_{k,M}\|_2^2\leq 2^{-2k}\eta^2h^4\sum^{\infty}_{n=2^k}\frac{1}{n!2^{n}\sqrt{\pi}}\sum^{\infty}_{s=2^k}\frac{1}{s!2^{s}\sqrt{\pi}} \sum^{\infty}_{m=M}\frac{1}{(m+1)^2}\sum^{\infty}_{r=M}\frac{1}{(r+1)^2}.
\end{eqnarray}
Here all four series converges and
$\|E_{k,M}\| \longrightarrow 0$ as $k,M \rightarrow \infty$.
\end{proof}
\begin{remark}
The above theorem can easily be extended for the method HeWQA.
\end{remark}
\begin{theorem}
Let us assume that $f(t)=\frac{d^u}{dt^2}\in L^2(\mathbb{R})$ is a continuous function on $[0,1]$ and its first derivative is bounded for all $t\in [0,1]$ there exists $\eta$ such that $\left|\frac{df}{dt}\right|\leq \eta$, then the method based on HWQA and HWNA converges.
\end{theorem}
\begin{proof}
The proof is similar to the previous theorem.
\end{proof}
\section{Numerical Illustrations}\label{P2_sec6} In this section we apply HeWQA, HeWNA, HWQA and HWNA on proposed model which occurs in exothermic reaction \eqref{P2_CHEM5}. We also solve four other examples from real life and compare solutions with among these four methods and with exact solutions whenever available.

To examine the accuracy of methods we define maximum absolute error $L_\infty$ as
\begin{equation}
L_\infty=\max_{x \in [0,1]}|y(t)-y_w(t)|
\end{equation}
here $y(t)$ is exact solution and $y_w(t)$ is wavelet solution.
and the $L_2$-norm error as
\begin{equation}
L_2=\left(\sum_{j=o}^{2M}|y(x_j)-y_w(x_j)|^2\right)^{1/2}
\end{equation}
here $y(x_j)$ is exact solution and $y_w(x_j)$ is wavelet solution at the point $x_j$.

\subsection{Example 1 (Exothermic Reaction with Modified Arrehenius Law)}\label{P2_sec6a}
Consider the non-linear SBVP \eqref{P2_CHEM5} with given boundary condition
\begin{eqnarray}\label{P2_61}
Ly+B\exp\left(\frac{-A}{(c^n+y^{n})^{1/n}}\right)=0,\quad y'(0)=0,\quad y(1)=0,
\end{eqnarray}
we take some particular cases when $A=1,B=1,c=1$.

Comparison Graphs taking initial vector [$0,0,\hdots,0$] and $J=1$, $J=2$ with $n=1,~k_g=1$; $n=1,~k_g=2$; $n=2,~k_g=1$; $n=2,~k_g=2$;$n=3,~k_g=1$ and $n=3,~k_g=2$ are plotted in Figure \ref{exfig1}, Figure \ref{exfig2}, Figure \ref{exfig3}, Figure \ref{exfig4}, Figure \ref{exfig5} and Figure \ref{exfig6} respectively. Tables for solution is tabulated in table \ref{extab1}, \ref{extab2}, table \ref{extab3}, table \ref{extab4}, table \ref{extab5} and table \ref{extab6} respectively.
\begin{table}[H]
\caption{\small{ Comparison of HWQA, HeWNA, HWQA, HeWQA method solution for example \ref{P2_sec6a} with $n=1,~k_g=1$ taking $J=2$ :}}\label{extab1}
\begin{center}
\resizebox{10cm}{2cm}{\begin{tabular}{|c | l| l| l| l|}
\hline Grid Points	&	HWNA \cite{akvdt2018}	&	HeWNA	&	HWQA \cite{akvdt2018}	&	HeWQA\\
\hline
   0	&	0.098471606	&	0.098471868	&	0.099733232	&	0.098721649	\\
  1/16	&	0.098078784	&	0.09807895	&	0.099334324	&	0.098328004	\\
  3/16	&	0.094937743	&	0.094937908	&	0.096144758	&	0.095180981	\\
  5/16	&	0.08866797	&	0.08866813	&	0.089779277	&	0.088898418	\\
  7/16	&	0.079294153	&	0.079294305	&	0.080265493	&	0.079503596	\\
  9/16	&	0.066853514	&	0.066853646	&	0.067645598	&	0.06703223	\\
 11/16	&	0.051396022	&	0.051396117	&	0.051977236	&	0.051533416	\\
 13/16	&	0.032984684	&	0.032984721	&	0.033334595	&	0.033070901	\\
 15/16	&	0.011695904	&	0.011695851	&	0.011809628	&	0.01172465
\\\hline
\end{tabular}}
\end{center}
\end{table}

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.30]{arhinius1newton1haarJ1}\includegraphics[scale=0.30]{arhinius1newton1haarJ2}
\end{center}
\caption{Comparison plots of solution methods for $J=1,2$ for example \ref{P2_sec6a} with $n=1,~k_g=1$:}\label{exfig1}
\end{figure}

\begin{table}[H]
\caption{\small{ Comparison of HWQA, HeWNA, HWQA, HeWQA method solution for example \ref{P2_sec6a} with $n=1,~k_g=2$ taking $J=2$ :}}\label{extab2}														 \centering
\begin{center}
\resizebox{10cm}{2cm}{\begin{tabular}{|c | l| l| l| l|}
\hline Grid Points	&	   HWNA \cite{akvdt2018}	&	HeWNA	&	  HWQA \cite{akvdt2018}	&	HeWQA	\\
\hline0	&	0.063984889	&	0.063981865	&	0.064360996	&	0.064044224	\\
  1/16	&	0.063730596	&	0.063727533	&	0.064104699	&	0.063789741	\\
  3/16	&	0.061697078	&	0.061694111	&	0.062055233	&	0.06175505	\\
  5/16	&	0.057636632	&	0.057633858	&	0.057963614	&	0.057691954	\\
  7/16	&	0.05156246	&	0.051559975	&	0.051844545	&	0.051613138	\\
  9/16	&	0.043494423	&	0.04349232	&	0.043720272	&	0.043537903	\\
 11/16	&	0.0334591	&	0.033457472	&	0.033620792	&	0.033492504	\\
 13/16	&	0.02148988	&	0.021488815	&	0.021584087	&	0.021510584	\\
 15/16	&	0.007627057	&	0.007626644	&	0.007656348	&	0.007633719
\\\hline
\end{tabular}}
\end{center}
\end{table}

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.30]{arhinius1newtonhaarJ1}\includegraphics[scale=0.30]{arhinius1newtonhaarJ2}
\end{center}
\caption{Comparison plots of solution methods for $J=1,2$ for example \ref{P2_sec6a} with $n=1,~k_g=2$:}\label{exfig2}
\end{figure}

\begin{table}[H]
\caption{\small{Comparison of HWQA, HeWNA, HWQA, HeWQA method solution for example \ref{P2_sec6a} with $n=2,~k_g=1$ taking $J=2$ :}}\label{extab3}
\begin{center}
\resizebox{10cm}{2cm}{\begin{tabular}{|c | l| l| l| l|}
\hline
Grid Points	&	  HWNA \cite{akvdt2018}	&	HeWNA	&	HWQA \cite{akvdt2018}	&	HeWQA	\\
\hline
   0	&	0.092208342	&	0.092207929	&	0.092312756	&	0.092313207	\\
  1/16	&	0.091847576	&	0.091847154	&	0.091951987	&	0.091952438	\\
  3/16	&	0.088961585	&	0.088961165	&	0.089065836	&	0.089066284	\\
  5/16	&	0.083190705	&	0.08319029	&	0.083293585	&	0.083294024	\\
  7/16	&	0.07453702	&	0.074536618	&	0.074635444	&	0.074635864	\\
  9/16	&	0.063003362	&	0.06300299	&	0.063092011	&	0.063092397	\\
 11/16	&	0.048592958	&	0.048592641	&	0.048664618	&	0.048664943	\\
 13/16	&	0.031308953	&	0.031308728	&	0.031355798	&	0.031356022	\\
 15/16	&	0.011153812	&	0.011153735	&	0.01116988	&	0.011169952
\\\hline
\end{tabular}}
\end{center}
\end{table}

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.30]{arhinius1square1newtonhaarJ1}\includegraphics[scale=0.30]{arhinius1square1newtonhaarJ2}
\end{center}
\caption{Comparison plots of solution methods for example \ref{P2_sec6a} with $n=2,~k_g=1$:}\label{exfig3}
\end{figure}

\begin{table}[H]
\caption{\small{Comparison of HWQA, HeWNA, HWQA, HeWQA method solution for example \ref{P2_sec6a} with $n=2,~k_g=2$ taking $J=2$ :}}\label{extab4}							 \begin{center}
\resizebox{10cm}{2cm}{\begin{tabular}{|c | l| l| l| l|}
\hline
Grid Points	&	  HWNA \cite{akvdt2018}	&	HeWNA	&	HWQA \cite{akvdt2018}	&	HeWQA	\\
\hline
   0	&	0.061376073	&	0.061376325	&	0.061411277	&	0.061411345	\\
  1/16	&	0.061136122	&	0.061135877	&	0.061171325	&	0.061171393	\\
  3/16	&	0.059216558	&	0.05921632	&	0.05925171	&	0.059251777	\\
  5/16	&	0.055377823	&	0.055377598	&	0.055412494	&	0.055412559	\\
  7/16	&	0.049620654	&	0.049620448	&	0.04965375	&	0.049653812	\\
  9/16	&	0.041946044	&	0.041945866	&	0.0419757	&	0.041975756	\\
 11/16	&	0.032355109	&	0.032354967	&	0.032378845	&	0.032378891	\\
 13/16	&	0.020848901	&	0.020848809	&	0.020864147	&	0.020864177	\\
 15/16	&	0.007428191	&	0.007428161	&	0.007433258	&	0.007433263
\\\hline
\end{tabular}}
\end{center}
\end{table}

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.30]{arhinius1squarenewtonhaarJ1}\includegraphics[scale=0.30]{arhinius1squarenewtonhaarJ2}
\end{center}
\caption{Comparison plots of solution methods for $J=1,2$ for example \ref{P2_sec6a} with $n=2,~k_g=2$}\label{exfig4}
\end{figure}

\begin{table}[H]
\caption{\small{Comparison of HWQA, HeWNA, HWQA, HeWQA method solution for example \ref{P2_sec6a} with $n=3,~k_g=1$ taking $J=2$ :}}\label{extab5}
\begin{center}
\resizebox{10cm}{2cm}{
\begin{tabular}{|c | l| l| l| l| }
\hline
Grid Points	&	 HWNA \cite{akvdt2018}	&	 HeWNA	&	 HWQA \cite{akvdt2018}	&	 HeWQA	\\
\hline
   0	&	0.091982324	&	0.091982282	&	0.091998157	&	0.091998241	\\
  1/16	&	0.091622975	&	0.091622932	&	0.091622975	&	0.091638892	\\
  3/16	&	0.088748193	&	0.08874815	&	0.088748193	&	0.088764086	\\
  5/16	&	0.082998726	&	0.082998685	&	0.082998726	&	0.083014381	\\
  7/16	&	0.074374752	&	0.074374713	&	0.074374752	&	0.07438963	\\
  9/16	&	0.062876486	&	0.062876451	&	0.062876486	&	0.062889712	\\
 11/16	&	0.048504139	&	0.048504111	&	0.048504139	&	0.048514619	\\
 13/16	&	0.031257869	&	0.031257851	&	0.031257869	&	0.031264554	\\
 15/16	&	0.011137751	&	0.011137745	&	0.011137751	&	0.011139994	\\\hline
\end{tabular}}
\end{center}
\end{table}

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.30]{arhinius1cube1newtonhaarJ1}\includegraphics[scale=0.30]{arhinius1cube1newtonhaarJ2}
\end{center}
\caption{Comparison plots of solution methods for $J=1,2$ for example \ref{P2_sec6a} with $n=3,~k_g=1$}\label{exfig5}
\end{figure}

\begin{table}[H]
\caption{\small{Comparison of HWQA, HeWNA, HWQA, HeWQA method solution for example \ref{P2_sec6a} with $n=3,~k_g=2$ taking $J=2$:}}\label{extab6}
\begin{center}
\resizebox{10cm}{2cm}{
\begin{tabular}{|c | l| l| l| l| }
\hline
Grid Points	&	 HWNA \cite{akvdt2018}	&	 HeWNA	&	 HWQA \cite{akvdt2018}	&	 HeWQA	\\
\hline
   0	&	0.061315351	&	0.061315829	&	0.061318845	&	0.061321705	\\
  1/16	&	0.061075828	&	0.061075814	&	0.061079322	&	0.061083071	\\
  3/16	&	0.059159646	&	0.059159634	&	0.059163135	&	0.059166226	\\
  5/16	&	0.055327307	&	0.055327295	&	0.055330739	&	0.055333256	\\
  7/16	&	0.049578851	&	0.04957884	&	0.049582101	&	0.049584212	\\
  9/16	&	0.041914328	&	0.041914319	&	0.041917195	&	0.041918897	\\
 11/16	&	0.032333785	&	0.032333778	&	0.032336026	&	0.032337266	\\
 13/16	&	0.020837254	&	0.02083725	&	0.020838652	&	0.020839592	\\
 15/16	&	0.007424747	&	0.007424746	&	0.007425199	&	0.007424945
\\\hline
\end{tabular}}
\end{center}
\end{table}

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.30]{arhinius1cubenewtonhaarJ1}\includegraphics[scale=0.30]{arhinius1cubenewtonhaarJ2}
\end{center}
\caption{Comparison plots of solution methods for $J=1,2$ for example \ref{P2_sec6a} with $n=3,~k_g=2$}\label{exfig6}
\end{figure}
The example defined by equation \eqref{P2_61} is new and does not exist in literature. So we are not in a situation to compare the results. We have considered $n=1,2,3$ and $k_g=1,2$. Tables \ref{extab1}, \ref{extab2}, \ref{extab3}, \ref{extab4}, \ref{extab5} , \ref{extab6} and figures \ref{exfig1},  \ref{exfig2}, \ref{exfig3},  \ref{exfig4}, \ref{exfig5}, \ref{exfig6} demonstrate the behaviour of the solution for $J=1,2$. HWNA, HeWNA, HWQA and HeWQA all give numerics which are very well comparable and shows that our proposed techniques are working well.

We also observed for small changes in initial vector, for example taking $[0.1,0.1,\hdots,0.1]$ or $[0.2,0.2,\hdots,0.2]$ doesn't significantly change the solution in any case.

\subsection{Example 2 (Stellar Structure)}\label{P2_sec6b}
Consider the non-linear SBVP:
\begin{eqnarray}\label{P2_62}
Ly(t)+y^{5}(t)=0,\quad y'(0)=0,\quad y(1)=\sqrt{\frac{3}{4}},\quad k_g=2,
\end{eqnarray}
Chandrasekhar (\cite{CS1967}, p88 ) derived above two point nonlinear SBVP. This equation arise in study of stellar structure. Its exact solution is $y(t)=\sqrt{\frac{3}{3+x^2}}$.

Comparison Graphs taking initial vector [$\sqrt{\frac{3}{4}},\sqrt{\frac{3}{4}},\hdots, \sqrt{\frac{3}{4}}$] and $J=1$, $J=2$ are plotted in Figure \ref{exfig7}. Tables for solution and error are tabulated in table \ref{extab7} and table \ref{exerr1}.
\begin{table}[H]
\caption{\small{Comparison of HWNA, HeWNA, HWQA, HeWQA methods solution with analytical solution for example \ref{P2_sec6b} taking $J=2$ :}}\label{extab7}											 \centering
\begin{center}
\resizebox{12cm}{2cm}{
\begin{tabular}	{|c | l|  l|  l| l| l| }
\hline
Grid Points	&	  HWNA \cite{akvdt2018}	&	HeWNA	&	  HWQA \cite{akvdt2018}	&	HeWQA	&	   Exact\\	\hline
0	&	1.00023666	&	0.999999992	&	1.00023666	&	0.999999992	&	1	\\
  1/16	&	0.999586961	&	0.99934958	&	0.999586961	&	0.99934958	&	 0.999349593	\\
  3/16	&	0.994419294	&	0.994191616	&	0.994419294	&	0.994191616	&	 0.994191626	\\
  5/16	&	0.984319576	&	0.984110835	&	0.984319576	&	0.984110835	&	 0.984110842	\\
  7/16	&	0.969730094	&	0.96954859	&	0.969730094	&	0.96954859	&	 0.969548596	\\
  9/16	&	0.9512486	&	0.951101273	&	0.9512486	&	0.951101273	&	 0.951101277	\\
 11/16	&	0.92956584	&	0.92945791	&	0.92956584	&	0.92945791	&	 0.929457914	\\
 13/16	&	0.905403371	&	0.905338132	&	0.905403371	&	0.905338132	&	 0.905338136	\\
 15/16	&	0.879460746	&	0.879439538	&	0.879460746	&	0.879439538	&	 0.879439536
	\\\hline
\end{tabular}}
\end{center}
\end{table}
\begin{table}[H]
\caption{\small{Comparison of error of HWNA, HeWNA, HWQA, HeWQA methods for example \ref{P2_sec6b} taking $J=2$ :}}\label{exerr1}											 \centering
\begin{center}
	\resizebox{8cm}{0.8cm}{
		\begin{tabular}	{|c | l|  l|  l| l| }
			\hline      																Error	&	  HWNA \cite{akvdt2018}	&	HeWNA	&	  HWQA \cite{akvdt2018}	&	HeWQA\\ \hline
			$L_\infty$	&	0.000237368	& 2.49669$\times 10^{-9}$  	&	0.000237368	&	2.49669$\times 10^{-9}$\\
			$L_2$	&	0.000471959	&	1.97638$\times 10^{-8}$	&  0.000471959		&		1.97638$\times 10^{-8}$ \\\hline
	\end{tabular}}
\end{center}
\end{table}
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.30]{ex13newtonhaarJ1}\includegraphics[scale=0.30]{ex13newtonhaarJ1Error}
\end{center}
\begin{center}
\includegraphics[scale=0.30]{ex13newtonhaarJ2}\includegraphics[scale=0.30]{ex13newtonhaarJ2Error}
\end{center}
\caption{Comparison plots and error plots of solution methods for $J=1,2$ for example \ref{P2_sec6b}}\label{exfig7}
\end{figure}
In this test case since exact solution of the SBVP governed by \eqref{P2_62} exists, We have compared our solutions with exact solution in table \ref{extab7} and figure \ref{exfig7}. Numerics again prove that method gives results with best accuracy for $J=1$ and $J=2$.

We also observed for small changes in initial vector, for example taking $[0.8,0.8,\hdots,0.8]$ or $[0.7,0.7,\hdots,0.7]$ doesn't significantly change the solution.

\subsection{Example 3 (Thermal Explosion)}\label{P2_sec6c}
Consider the non linear SBVP:
\begin{eqnarray}\label{P2_63}
Ly(t)+e^{y(t)}=0,\quad y'(0)=0,\quad y(1)=0, ~k_g=1.
\end{eqnarray}
Above nonlinear SBVP is derived by Chamber \cite{CHAMBRE1952}. This equation arises in the thermal explosion in cylindrical vessel. The exact solution of this equation is $y(x) = 2 \ln{\frac{4-2\sqrt{2}}{(3-2\sqrt{2})x^2+1}}$.

Comparison Graphs taking initial vector [$0,0,\hdots, 0$] and $J=1$, $J=2$ are plotted in Figure \ref{exfig8}. Tables for solution and error are tabulated in table \ref{extab8} and table \ref{exerr2}.
\begin{table}[H]
\caption{\small{Comparison of HWNA, HeWNA, HWQA, HeWQA methods solution with analytical solution for example \ref{P2_sec6c} taking $J=2$:}}\label{extab8}			 \centering
\begin{center}
\resizebox{12cm}{2cm}{
\begin{tabular}	{|c | l|  l|  l| l| l| }
\hline

Grid Points	&	   HWNA \cite{akvdt2018}	&	  HeWNA	&	 HWQA \cite{akvdt2018}	&	HeWQA	&	Exact	\\  \hline
0	&	0.316727578	&	0.316694368	&	0.316727578	&	0.316694368	&	0.316694368	 \\
  1/16	&	0.315388914	&	0.315354403	&	0.315388914	&	0.315354403	&	 0.315354404	\\
  3/16	&	0.304700946	&	0.304666887	&	0.304700946	&	0.304666887	&	 0.304666888	\\
  5/16	&	0.283494667	&	0.283461679	&	0.283494667	&	0.283461679	&	 0.283461679	\\
  7/16	&	0.252100547	&	0.252069555	&	0.252100547	&	0.252069555	&	 0.252069555	\\
  9/16	&	0.210993138	&	0.210965461	&	0.210993138	&	0.210965461	&	 0.210965462	\\
 11/16	&	0.160768168	&	0.16074555	&	0.160768168	&	0.16074555	&	 0.16074555	\\
 13/16	&	0.102115684	&	0.102100258	&	0.102115684	&	0.102100258	&	 0.102100258	\\
 15/16	&	0.035791587	&	0.035785793	&	0.035791587	&	0.035785793	&	 0.035785793
\\\hline
\end{tabular}}
\end{center}
\label{Table2}
\end{table}
\begin{table}[H]
	\caption{\small{Comparison of error of HWNA, HeWNA, HWQA, HeWQA methods for example \ref{P2_sec6c} taking $J=2$ :}}\label{exerr2}											 \centering
	\begin{center}
		\resizebox{8cm}{0.8cm}{
			\begin{tabular}	{|c | l|  l|  l| l| }
				\hline      																Error	&	  HWNA \cite{akvdt2018}	&	HeWNA	&	  HWQA \cite{akvdt2018}	&	HeWQA\\ \hline
				$L_\infty$	&	0.0000345103	& 1.07541$\times 10^{-10}$  	&	0.0000345103	&	1.07541$\times 10^{-10}$\\
				$L_2$	&	0.0000771278	&	4.99369$\times 10^{-10}$	&  0.0000771278	&		4.99369$\times 10^{-10}$ \\\hline
		\end{tabular}}
	\end{center}
\end{table}

\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.30]{ex2newtonhaarJ1}\includegraphics[scale=0.21]{ex2newtonhaarJ1Error}
\end{center}
\begin{center}
\includegraphics[scale=0.30]{ex2newtonhaarJ2}\includegraphics[scale=0.30]{ex2newtonhaarJ2Error}
\end{center}
\caption{Comparison plot and error plots of solution methods for $J=1,2$ for example \ref{P2_sec6c}}\label{exfig8}
\end{figure}
This is test case derived by Chambre  \cite{CHAMBRE1952} long back again exact solution is available. Table \ref{extab8} and figure \ref{exfig8} show that numerics are in good agreement with exact solutions or $J=1$ and $J=2$.

We also observed for small changes in initial vector, for example taking $[0.1,0.1,\hdots,0.1]$ or $[0.2,0.2,\hdots,0.2]$ doesn't significantly change the solution.
\subsection{Example 4 (Rotationally Symmetric Shallow Membrane Caps)}\label{P2_sec6d}
Consider the non linear SBVP:
\begin{eqnarray}\label{P2_65}
Ly(t)+\left(\frac{1}{8y^{2}(t)}-\frac{1}{2}\right)=0,\quad y'(0)=0,\quad y(1)=1,~k_g=1.
\end{eqnarray}
Above nonlinear SBVP is studied in papers \cite{RW1989,JV1998}. Exact solution of this problem is not known.

Comparison Graphs taking initial vector [$1,1,\hdots, 1$] and $J=1$, $J=2$ are plotted in Figure \ref{exfig9}. Tables for solution is tabulated in table \ref{extab9}.
\begin{table}[H]
\caption{\small{ Comparison of HWQA, HeWNA, HWQA, HeWQA method solution for example \ref{P2_sec6d} taking $J=2$:}}	\label{extab9}													 \centering
\begin{center}
\resizebox{10cm}{2cm}{\begin{tabular}{|c |l| l| l| l|}
\hline
Grid Points	&	HWNA \cite{akvdt2018}	&	HeWNA	&	HWQA \cite{akvdt2018}	&	HeWQA	\\	\hline
   0	&	0.954137376	&	0.954135008	&	0.954137376	&	0.954135008	\\
  1/16	&	0.954314498	&	0.954311604	&	0.954314498	&	0.954311604	\\
  3/16	&	0.95573187	&	0.95572956	&	0.95573187	&	0.95572956	\\
  5/16	&	0.958569785	&	0.958567713	&	0.958569785	&	0.958567713	\\
  7/16	&	0.962834546	&	0.962832683	&	0.962834546	&	0.962832683	\\
  9/16	&	0.968535496	&	0.968533886	&	0.968535496	&	0.968533886	\\
 11/16	&	0.975684891	&	0.975683641	&	0.975684891	&	0.975683641	\\
 13/16	&	0.984297738	&	0.984296771	&	0.984297738	&	0.984296771	\\
 15/16	&	0.994391588	&	0.994391728	&	0.994391588	&	0.994391728
\\\hline
\end{tabular}}
\end{center}
\end{table}
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.30]{ex5newtonhaarJ1}\includegraphics[scale=0.30]{ex5newtonhaarJ2}
\end{center}
\caption{Comparison plots of solution methods for $J=1,2$ for example \ref{P2_sec6d}}\label{exfig9}
\end{figure}
In this real life example again exact solution is not known so comparison is not done with exact solution. Table \ref{extab9} and figure \ref{exfig9} show that computed results are comparable for $J=1,2$.

We also observed for small changes in initial vector, for example taking $[0.9,0.9,\hdots,0.9]$ or $[0.8,0.8,\hdots,0.8]$ doesn't significantly change the solution.

\subsection{Example 5 (Thermal Distribution in Human Head)}\label{P2_sec6e}
Consider the non linear SBVP:
\begin{eqnarray}\label{P2_66}
Ly(t)+e^{-y(t)}=0,\quad y'(0)=0,\quad 2y(1)+y'(1)=0,~k_g=2.
\end{eqnarray}
This SBVP is derived by Duggan and Goodman \cite{RA1986}. Exact solution of this problem is not known to the best of our knowledge.

Comparison Graphs taking initial vector [$0,0,\hdots, 0$] and $J=1$, $J=2$ are plotted in Figure \ref{exfig10}. Tables for solution is tabulated in table \ref{extab10}.
\begin{table}[H]
\caption{\small{Comparison of HWQA, HeWNA, HWQA, HeWQA method solution for example \ref{P2_sec6e} taking $J=2$:}}\label{extab10}														 \centering
\begin{center}
\resizebox{10cm}{2cm}{
\begin{tabular}	{|c | l| l| l| l| }
\hline
Grid Points	&	  HWNA \cite{akvdt2018}	&	  HeWNA	&	  HWQA \cite{akvdt2018}	&	HeWQA	\\	\hline
0	&	0.269855704	&	0.269948774	&	0.272263769	&	0.272366612	\\
  1/16	&	0.269358573	&	0.269451863	&	0.27176762	&	0.271870738	\\
  3/16	&	0.265377954	&	0.265471233	&	0.267793921	&	0.267896983	\\
  5/16	&	0.257388082	&	0.257481347	&	0.259810468	&	0.259913411	\\
  7/16	&	0.245331028	&	0.245424295	&	0.247745058	&	0.247847809	\\
  9/16	&	0.229118226	&	0.229211536	&	0.231489202	&	0.231591678	\\
 11/16	&	0.208628362	&	0.2087218	&	0.210897975	&	0.211000089	\\
 13/16	&	0.183704413	&	0.183798121	&	0.18579005	&	0.18589165	\\
 15/16	&	0.154149664	&	0.154243862	&	0.155947881	&	0.156048741	\\
1	&	0.13756259	&	0.137656718	&	0.139174003	&	0.139274111
\\\hline
\end{tabular}}
\end{center}
\end{table}
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.30]{ex7newtonhaarJ1}\includegraphics[scale=0.30]{ex7newtonhaarJ2}
\end{center}
\caption{Comparison plots of solution methods for $J=1,2$ for example \ref{P2_sec6e}}\label{exfig10}
\end{figure}
In absence of exact solution the comparison has not been made with exact solution. But comparison of all four methods for in the given problem due to Duggan and Goodman \cite{RA1986}, in table \ref{extab10} and figure \ref{exfig10} shows accuracy of the present method.

We also observed for small changes in initial vector, for example taking $[0.1,0.1,\hdots,0.1]$ or $[0.2,0.2,\hdots,0.2]$ doesn't significantly change the solution in any case.

\section{Conclusions}\label{P2_conclusions} In this research article, we have proposed a new model governing exothermic reactions and four different numerical methods based on wavelets, namely HWQA, HWNA, HeWQA, HeWNA for solving these nonlinear SBVPs arising in different branches of science and engineering (cf. \cite{RW1989, JV1998, RA1986, CHAMBRE1952, CS1967}). We have applied these methods in five real life examples [see equations (\ref{P2_61}), (\ref{P2_62}), (\ref{P2_63}), (\ref{P2_65}) and (\ref{P2_66})]. Singularity of differential equations can also be very well handled with help of these four proposed methods based. Difficulty arise due to non-linearity of differential equations is dealt with the help of quasilinearization in HWQA and HeWQA method. In the other two proposed method, HWNA and HeWNA, we will solve the resulting non-linear system with help of Newton-Raphson method. Boundary conditions are also handled well by the proposed methods. Main advantage of proposed methods is that solutions with high accuracy are obtained using a few iterations. We also observe that small perturbation in initial vector does not significantly change the solution. Which shows that our method is numerically stable.

Our convergence analysis shows that that $||E_{k,M}||$ tends to zero as $M$ tends to infinite. Which shows that accuracy of solution increases as $J$ increases.

Computational work illustrate the validity and accuracy of the procedure. Our computations are based on a higher resolution and codes developed can easily be used for even further resolutions.

redpajama_set_name: RedPajamaArXiv


text:
Q: Javascript File Reader I may just not be understanding the file reader api, but
When I try to run:
for (var i = 0, f1, f2; f1 = files[sorted_index[i]], f2 = files[sorted_index[i+1]]; i=i+2) {
    var file_one;
    var file_two;

    if(f1.size < f2.size){
        file_one = f1;
        file_two = f2;
    } else {
        file_one = f2;
        file_two = f1;
    }

    var file_one_contents;
    var file_two_contents;

  var reader = new FileReader();
  reader.readAsText(file_one);
  reader.onload = readSuccess;

  function readSuccess(evt){
    file_one_contents = evt.target.result;
  }
  var reader = new FileReader();
    reader.readAsText(file_two);
    reader.onload = readSuccess2;

  function readSuccess2(evt2){
    file_two_contents = evt2.target.result;
  }

    console.log(file_one_contents);
    console.log(file_two_contents);

The console log only contains undefined in it.
The goal of the script it two read in two CSVs and take the data from the pair of files and do some computations.
Thanks!

A: The API is asynchronous. The "success" functions are called when the operation completes, and that won't be immediate.
Move your console.log() calls to inside the handler functions.
edit — If you need to wait to start doing stuff until both files are ready, you can do something like this:
  var countdown = 2;

  var reader = new FileReader();
  reader.readAsText(file_one);
  reader.onload = readSuccess;

  function readSuccess(evt){
    file_one_contents = evt.target.result;
    countdown--;
    if (countdown === 0) go();
  }
  var reader = new FileReader();
    reader.readAsText(file_two);
    reader.onload = readSuccess2;

  function readSuccess2(evt2){
    file_two_contents = evt2.target.result;
    countdown--;
    if (countdown === 0) go();
  }

There are more sophisticated ways to do it, of course, but that simple trick just waits until the counter is zero before calling "go()", which represents the function that'd work on processing the files.

A: I had a similar problem which solved the file read waiting by using ".onloadend" instead of "onload". In the code below x is bound to a "div" element
reader.onloadend = function (evt) {
    x.innerHTML = evt.target.result;
}

With "onload" it was all undefined.

redpajama_set_name: RedPajamaStackExchange


text:
2008 Dodge Challenger SRT8 Test Drive: On the Streets, 425-hp Hemi Muscles With Modern Civility
We'd already driven a covered version of the 2008 Challenger on a rainy test track before its official unveil in February, but now it's time for the real rubber roastin' to begin. We can safely say that the Challenger ain't what it used to be—it's a whole lot better.
By Basem Wasef Published: Apr 29, 2008
LOS ANGELES — Decades before paddle shifters and traction control, muscle cars like the 1970 Dodge Challenger brought raw V8 power to the masses. After selling over 83,000 Challengers in the first year alone, Dodge knew it had struck muscle-car gold. And like that—poof—it disappeared. Thirty-eight years later, in spite of pricey fuel and economic uncertainty, Dodge is at it again. We'd already driven a covered version of the 2008 Challenger on a rainy test track before its official unveil in February, but now it's time for the real rubber roastin' to begin.
The nouveau Challenger strikes a faultless visual balance between nostalgia and clean modernity. It hits all the right notes, from its set-in headlamps and grille to its metal fuel door and flat-black spoiler lip. In person, the Challenger is imposing, making modern Mustangs look slight by comparison. Coupes just don't have brawny dimensions like this anymore—but they should.
Unfortunately, the concept car's retro-cool interior was ditched at the 11th hour, but what remains is a spacious cocoon that's typical Dodge: stark, no-nonsense, functional. The seats are nicely bolstered with ultra-suede inserts and offer an excellent combination of support and comfort. The trapezoidal door panel shapes hint at Mopars of yore. The only available options on the $37,995 Challenger SRT8 are a MyGIG multimedia system with navigation, a power sunroof and sticky Goodyear F1 Supercar tires.
Press the red Start button to fire up the 6.1-liter, 425-hp V8, and you'll hear an exhaust note that's not nearly as imposing as the Challenger's gruff exterior. Throughout our day piloting a Hemi Orange SRT8 around Southern California, we were on alert for auditory evidence of the beast under the hood. But that exhaust is surprisingly mellow. A bass-inflected song becomes more evident above 4000 rpm, and microseconds between shifts were greeted with a short bark. For the most part, however, this V8 is surprisingly discreet—nothing a cat-back exhaust system can't fix.
The V8 does its duty with a pancake-flat powerband, pulling steadily to its 6400-rpm redline. Smoky burnouts come easily, as long as the driver's footwork is quick—an electronic throttle-control system is designed to cut the gas when both pedals are pressed simultaneously for more than two seconds.
On the serpentine roads that cut through the Angeles Crest Forest, the Challenger maintained a level of composure that suggested the stiff chassis and Bilstein coil-overs were well-sorted. Make no mistake: The Challenger weighs in at 4140 pounds, so it's not exactly a Boxster in the canyons. But the Challenger exhibits unperturbed grip on wide, high-speed sweepers.
Later that day, we tackled Willow Springs Raceway, and the Challenger rose to the occasion again. It's still a bulky car with a bit of uncertainty at turn-in and more steering boost than we'd like, but when shifted manually with the Autostick feature, it offers a satisfying combination of grip and power. A degree of dialed-in understeer prevents the Challenger from getting too twitchy, but heavy throttle on stretches like Turn 3—an uphill left-hander—will gently break the fat rear Goodyear F1s loose. Huge 14.2-in. front and 13.8-in. rear slotted Brembo brakes work better than they ought to on a 2-ton-plus ride like this. We found ourselves braking later and later on the track, and were never without enough stopping power. A new knock-back mitigation system resets the calipers when lateral acceleration exceeds 0.6 g's, improving pedal feel and brake response.
This Hemi is most definitely not shy or watered down. But what surprised us the most about the Challenger SRT8 was its civility. Power is abundant, but never to the point that we couldn't rein her in. Our two-hour highway drive home from Willow Springs was quiet and smooth—characteristics more akin to a grand touring car than a muscle car. The only things we miss are a more vocal exhaust note and a manual gearbox, the latter of which will be available in the fall of 2009.
After flogging the 2008 SRT8 on the track and driving home in insulated comfort, we can safely say that the Challenger ain't what it used to be—it's a whole lot better.
Save Time Scraping Your Windshield With a De-Icer
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redpajama_set_name: RedPajamaCommonCrawl


1. Extracts by exhibition curator Tse Hao Guang, taken from a write-up on the poem written by Chow Teck Seng supplied to him.
2. A recording of Chow Teck Seng reading his own poem, ""我们对着鱼缸说国语".
3. Chua Mia Tee, National Language Class, Collection of National Gallery Singapore.


text:
Aterpnodes geminipuncta is een vlinder uit de familie van de spanners (Geometridae). De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1900 door Warren.

geminipuncta

redpajama_set_name: RedPajamaWikipedia


Trump and Putin will meet July 16 in Helsinki, Finland, for their first official summit.
"You appear to be hell-bent on pushing for your European army, hell-bent for a foreign policy without veto, and hell-bent to be on a direct collision course with President Trump, and with NATO", Farage said on Tuesday while blasting the EU leadership during a debate at the European Parliament.
Russia's President Vladimir Putin talks to U.S. President Donald Trump during their bilateral meeting at the G20 summit in Hamburg, Germany, July 7, 2017.
The prospect of Trump and Putin meeting alone has raised concern among members of the US national security community, some of whom have fretted that Putin, who has served as either president or prime minister since 2000, will use the meeting to gain leverage over the U.S.
"The president believes a better relationship with Russia would be good for both America and Russia, but the ball really is in Russia's court and the president will continue to hold Russia accountable for its malign activity", Jon Huntsman, U.S. ambassador to Russia, told reporters on a conference call.
"The president has determined that now is the time for direct communication between himself and president Putin and it is in the interests of the United States, the interests of Russian Federation and the interests of peace and security around the world", the official said.
President Donald Trump has declared that his incessant jibes at USA allies to spend more on defense has made NATO Secretary General Jens Stoltenberg "Trump's biggest fan", while Russian Federation must regret ever liking the ex-Apprentice host. At Trump's rally, the president suggested he had done enough to allay fears among Americans that he did not view US allies favorably.
The summit also will include a bilateral meeting and a working lunch between Mr. Trump and Mr. Putin. The president will arrive in Brussels the evening of July 10 and meet with other North Atlantic Treaty Organisation leaders the following morning for a discussion of "defense and deterrence issues".
Trump has expressed scorn for the alliance and is expected to upbraid European NATO members for not spending enough on their own defense.
Speaking of the possible topics of the upcoming meeting, Huntsman said "some of them might be around strategic stability and arms control".
Her own interior minister, Horst Seehofer , has threatened to resign. "Both the chancellor and (Bavaria's Horst) Seehofer are damaged".
Centers for Disease Control and Prevention, the epidemic caused more than 42,000 deaths from opioid overdoses in the United States in 2016.
In Portland, Oregon, for example, several stay-at-home moms are organizing their first rally while caring for young kids. Julie Hoplamazian, an Episcopal priest marching in Brooklyn.
Argentina are expected to make several changes to their team for the match at the Saint Petersburg Stadium. Somehow, Argentina are through to the last-16 of the World Cup .
Similar to her visit to Tucson, the first lady took part in a round table where she heard from officials at the Phoenix facility. Last week , her critics blasted her trip to visit immigrant children that the government had separated from their parents.
Twitter users are now calling her "Permit Patty". "I have no problem with enterprising young women", she told the publication. She also said the incident had nothing to do with race; it was only about the volume of their salesmanship.


Fantastic! Only know their Rough Trade stuff. Another band I discovered recently is Galaxie 500, who sound quite like Field Mice, St Christopher, etc.
What Rough Trade stuff do you mean? AFAIK, they were only ever signed to Sarah Records.
Ooops, Sarah Records, yes, but they came to Germany via Rough Trade Records (which I dcan't expect anybody to know. My bad.) The Rough trade sticker was physically huge on the vinyl sleeves, that's why i have this mnenonic confusion.


text:
Der Autonome Bezirk Qianxinan der Bouyei und Miao () liegt im Südwesten der chinesischen Provinz Guizhou. Qianxinan hat eine Fläche von 16.817 km² und 3.015.112 Einwohner (Stand: Zensus 2020).

Administrative Gliederung
Auf Kreisebene setzt sich Qianxinan aus zwei kreisfreien Städten und sechs Kreisen zusammen. Diese sind (Stand: Ende 2018):
 Stadt Xingyi (兴义市), 2.913 km², 833.000 Einwohner, Bezirkshauptstadt, Verkehrs-, Verwaltungs- und Handelszentrum von Qianxinan;
 Stadt Xingren (兴仁市), 1.783 km², 423.400 Einwohner;
 Kreis Pu'an (普安县), 1.446 km², 260.500 Einwohner, Hauptort: Großgemeinde Panshui (盘水镇);
 Kreis Qinglong (晴隆县), 1.301 km², 248.500 Einwohner, Hauptort: Großgemeinde Liancheng (莲城镇);
 Kreis Zhenfeng (贞丰县), 1.502 km², 313.500 Einwohner, Hauptort: Großgemeinde Mingu (珉谷镇);
 Kreis Wangmo (望谟县), 3.021 km², 240.900 Einwohner, Hauptort: Großgemeinde Fuxing (复兴镇);
 Kreis Ceheng (册亨县), 2.608 km², 186.900 Einwohner, Hauptort: Großgemeinde Zhelou (者楼镇);
 Kreis Anlong (安龙县), 2.244 km², 365.000 Einwohner, Hauptort: Großgemeinde Xin'an (新安镇).

Ethnische Gliederung der Bevölkerung (2000)
Beim Zensus im Jahr 2000 hatte Qianxinan 2.864.920 Einwohner (Bevölkerungsdichte: 170,49 Einw./km²).

 Anmerkung: Bei den "Li" in Guizhou handelt es sich offensichtlich nicht um die Li-Nationalität der Inselprovinz Hainan. Immer wieder gelingt es einigen Gruppen in Guizhou, deren Selbstbezeichnung offenbar "Li" ist, deren Nationalität aber (noch) nicht offiziell "anerkannt" wurde, sich bei den Volkszählungen (1982, 1990, 2000) in der Kategorie "Lizu" (Li-Nationalität) klassifizieren zu lassen. Das führt zu dem irrigen Eindruck, es handle sich um Li der Insel Hainan.

Siehe auch
 Administrative Gliederung der Volksrepublik China
 Autonome Verwaltungseinheiten Chinas

Einzelnachweise

Ort in Guizhou
Autonomer Bezirk (China)


text:
MVA Foundation/GCCDC Awarded 1st Neighborhood Partnership Program in the Mid Mon Valley
6-Year plan approved by Governor Wolf to invest $1.5 million for community improvement projects and services
Front Left to Right: Ian Luluicci, KeyBank; Renee Bonari, KeyBank; Nathan B. Sommer, KeyBank; Dana Scarpino, UMPC Health Plan; Ali Doyle, the Governor's Regional Representative; John LaCarte, LaCarte Enterprises; Jamie Colecchi, MVA; Bud Cook, PA State Representative
Charleroi, PA, November 14, 2022 – The Mon Valley Alliance Foundation, the Greater Charleroi Community Development Corporation, and several nonprofits and business partners today announced details and funding for the first Neighborhood Partnership Program in the Mid Mon Valley, a six-year plan that will drive $1.5 million into Charleroi for community improvement projects and social services.
The Charleroi NPP proposal was approved as part of $35.9 million in tax credits awarded under the state Neighborhood Assistance Program through which private companies receive state tax credits for investing in non-profit projects. It was one of two new NPPs awarded in the 12-county Southwestern Pennsylvania region, bringing the total locally to 25, according to Johnna A. Pro, Regional Director for the state's Depar


Beauty is found in life's moments. Take time with the ones you love, and never take things for granted. It's the little things that you will cherish for the rest of your life! Those little moments, like stopping to smell flowers, is what inspired this design. Hang this beautiful print to remind yourself that you don't need to wait for what others will do- the choice for change is yours! 18" x 24" Frame not included.


text:
Home » Football News
Benjamin Mendy fighting to save his £475k Lamborghini from being crushed
Elijah Odetokun Posted On December 4, 2020
Benjamin Mendy is racing to save his £475k Lamborghini Aventador from being crushed. The Man City ace had his supercar seized after driving without a valid licence or insurance.
20 days after the car was taken French defender Mendy has yet to come up with any documentation. Police usually keep a car for 14 days before selling it at auction or crushing it.
A source said: "Ben is a great footballer but he's not great on bureaucratic details like paperwork.
"He didn't realise he wasn't insured so it's a monumental blunder on his part.
"He's desperately trying to correct it but the fact it's now 20 days since his car was seized, he's at risk of it being destroyed.
"The police know he wants it back and are happy to return it but only once he's provided the necessary paperwork.
"It's worrying that after almost three weeks he still hasn't managed to sort it out. It's a beautiful car and he just wants it back in one piece."
Mendy, 26, was pulled over at 5am on November 15 in Prestbury, Cheshire.
The motor, with a French number plate, was taken to Macclesfield police station and is now at a secure pound.
Mendy, who joined City in 2017, bought the car in France and was pictured driving it in Cannes this summer.
The SVJ, which can do 0-60mph in three seconds, is an improved version of the Aventador S, with only 900 made.
The interior of the car.
Front view of the car
Mendy seen driving his Lamborghini
Mendy pictured with his car in France
Mendy pictured with his car
Rear view of the car
According to the Cheshire Police website, any motorist who needs to reclaim a vehicle has just 14 days to do so.
The force charges £150 to take a car and £20 a day for storage. A spokesman said: "Arrangements are being made for the vehicle to be recovered."
Last year 137,410 vehicles were seized for no insurance, according to the Motor Insurers' Bureau. Uninsured drivers face their cars being crushed, a £300 fine and six penalty points.
In June, Man United star and international teammate of Mendy, Paul Pogba had his French-registered Rolls-Royce seized in Manchester.
Foreign vehicles that have been in Britain for more than six months in a year must be registered with the DVLA and fitted with UK plates.
But if the owner is a UK resident they must register their car within two weeks of import.
Source – The Sun Football
Aguero adds to Guardiola's squad concern, to self-isolate
Manchester United semifinal jinx under Solskjær continues
Manchester City confirm three new cases of coronavirus
Taiwo Awoniyi scores but ends up on the losing side in Berlin derby


text:
It's Pretty Easy Being Green
Since legendary LSU landscape designer Steele Burden first began planting live oaks, crepe myrtles and azaleas in 1932, the iconic campus greenery has become as recognizable as the Quad's arches.
The LSU campus is now one of the most beautiful in the nation, having been listed among the 20 best campuses in America in Thomas Gaines' The Campus as a Work of Art, and thrice named a Tree Campus USA by the Arbor Day Foundation. The prestigious reputation is maintained by LSU Landscape Services, which annually partners with LSU Campus Life to give the grounds a fresh face.
Spring Greening Day is an annual half-day service project to beautify the campus in preparation for commencement, as well as an opportunity for students, faculty and staff to learn about sustainability and personal practices to support the environment. Materials are purchased through the LSU Foundation's Campus Beautification Fund.
"LSU Spring Greening Day is a wonderful opportunity for students and other LSU family to work and see the immediate gratification of plants going in the ground," said Fred Fellner, assistant director of landscape services with LSU Facility Services. "We are focused mostly on sustainable planting that frames and enhances the campus's largest and most prevalent green plants, the storied live oaks. To this end, we plant ground cover, place sod or spread mulch under the trees to beautify the grounds, control erosion and improve the ecosystem."
Photo by Eddy Perez
April 24 marked the sixth year Tigers have donned gardening gloves and picked up shovels to nurse the campus they love. Despite a rainy forecast, 103 volunteers planted more than 4,700 plants and flowers, including an LSU AgCenter "Super Plant" variety called Serena angelonia.
"I was very impressed with the turn-out we had, given how bad the weather was in the beginning," said Taylor Delatte, one of the event's student planners. "I have never helped organize and event like this before so I was excited to be a part of it."
Consistently sponsored by Albemarle, the event supports the Green Tiger Project, which aims to integrate campus beautification with campus sustainability, and supports the Campus Beautification Fund. The fund offers alumni, friends and fans a way to contribute to the university's campus enhancement efforts, like Spring Greening Day and the living LSU Christmas Tree, a Little Gem Magnolia planted in 2014.
"Spring Greening Day gives students a chance to participate in beautification by planting all throughout campus," explained petroleum engineering student Yaser Suleiman, who also helped plan the beautification project. "It's a really fun way of getting students out and involved. They have fund by getting their hands dirty and being a part of making their campus better. Spring Greening Day is important because it gets students to have fun while being involved making their surroundings more beautiful. It made me have a deeper appreciation for the campus and all the hard work it takes to keep LSU one of the most beautiful campuses in America."


text:
Robert Miller Congratulates Future Team on Winning Abracon's 2018 Demand Creation Award
July 11, 2019 Technology Press Release No comments
Montreal, Canada (bizpr) July 9, 2019 – Future Electronics, a ****** leading distributor of electronic components, recently won Abracon's Demand Creation Distributor of the Year Award for 2018.
Robert Miller, President of Future Electronics, thanked everyone who contributed to Abracon's excellent performance in the previous year.
The award demonstrates Future's close alignment with Abracon's values of customer support, technical expertise and integrity, and recognizes Future's industry-leading achievement among Abracon's distributors.
"We appreciate the recognition from Abracon of our demand creation efforts," said Mark Teskey, Supplier Development Manager at Future Electronics. "It really shows our commitment to the strong partnership between our two organizations."
Robert Miller Congratulates Future Team on Winning Abracon's 2018 Demand Creation Award full post
(280 words, estimated 1:07 mins reading time)
Future Electronics Wins 2018 Demand Creation Distributor of the Year Award from Abracon
June 26, 2019 Technology Press Release No comments
Montreal, Canada (bizpr) June 21, 2019 – Future Electronics, a ****** leading distributor of electronic components, recently won Abracon's 2018 Demand Creation Distributor of the Year Award at EDS in Las Vegas.
"We appreciate the recognition from Abracon of our demand creation efforts" said Mark Teskey, Supplier Development Manager at Future Electronics. "It demonstrates our commitment to the strong partnership between our organizations."
Abracon (http://bit.ly/2XSfl8I) is a leading ****** manufacturer of frequency control, timing, synchronization, RF, connectivity and power components. The award recognizes Future's industry-leading achievement among Abracon's distributors, and demonstrates Future's close alignment with Abracon's values of customer support, technical expertise and integrity.
Future Electronics Wins 2018 Demand Creation Distributor of the Year Award from Abracon full post
(237 words, estimated 57 secs reading time)
Robert Miller Congratulates Future Team for Wakefield-Vette Highest POS Revenue Growth Award Win
Montreal, Canada (bizpr) June 18, 2019 – Future Electronics, a ****** leading distributor of electronic components, recently won Wakefield-Vette's 2018 Highest POS Revenue Growth Award.
Wakefield-Vette, a ****** leader in thermal management solutions, presented Future Electronics with the award in recognition of their outstanding point-of-sales growth, which led all other distributors in the electronic components industry.
Robert Miller, President of Future Electronics, thanked everyone on the Future team who contributed to Wakefield-Vette's excellent sales performance.
Robert Miller Congratulates Future Team for Wakefield-Vette Highest POS Revenue Growth Award Win full post
Future Electronics Wins Wakefield-Vette's Highest POS Revenue Growth Award for Second Straight Year
June 5, 2019 Technology Press Release No comments
Montreal, Canada (bizpr) June 4, 2019 – Future Electronics, a ****** leading distributor of electronic components, recently won Wakefield-Vette's 2018 Highest POS Revenue Growth Award at EDS in Las Vegas.
Wakefield-Vette ( http://bit.ly/2K7i0ah ), a ****** leader in thermal management solutions, presented Future Electronics with the award in recognition of Future's outstanding point-of-sales growth, which led all other distributors in the electronic components industry.
"Future Electronics is very excited to receive the award for Highest POS Revenue Growth from Wakefield-Vette for the second consecutive year," said Kathryn Sartain, Product Marketing Manager at Future Electronics. "We thank them for their ongoing support, and we look forward to continuing our success in 2019."
Future Electronics Wins Wakefield-Vette's Highest POS Revenue Growth Award for Second Straight Year full post
Future Electronics Wins Dialight's 2018 Distributor of the Year Award
Montreal, Canada (bizpr) May 30, 2019 – Future Electronics, a ****** leading distributor of electronic components, recently won Dialight's Distributor of the Year Award for 2018 at EDS in Las Vegas.
Dialight ( http://bit.ly/2YYklc7 ), the ****** leader in industrial LED lighting technology, awarded Future Electronics in recognition for their outstanding results in 2018, where they outpaced all other distributors.
"It's a great honor to be recognized by Dialight, a world class optoelectronics company with a long history of delivering high-quality LED solutions to the market," said Jaysen Jones, Corporate Product Marketing Manager at Future Electronics. "We value our 17 years of partnership, and we thank them for this prestigious award."
Future Electronics Wins Dialight's 2018 Distributor of the Year Award full post
Future Electronics Wins TE's 2018 ICT Americas Distributor of the Year Award
Montreal, Canada (bizpr) May 28, 2019 – Future Electronics, a ****** leading distributor of electronic components, recently won TE Connectivity's ICT Americas Distributor of the Year Award for 2018 at TE's annual Regional Summit in San Diego.
TE Connectivity, a ****** leader in connectivity and sensors, presented the award to Future Electronics in recognition for their exceptional performance in generating new business among industrial and commercial transportation customers.
"It is a great honor to be recognized by the Industrial and Commercial Transportation Business Unit as their Distributor of the Year in the Americas," said Jason Manzano, Product Marketing Manager at Future Electronics. "It speaks volumes to the great collaboration between Future and TE."
Future Electronics Wins TE's 2018 ICT Americas Distributor of the Year Award full post
Future Electronics Wins Altech's 2018 Sales Achievement Award for Second Straight Year
May 31, 2019 Technology Press Release No comments
Montreal, Canada (bizpr) May 25, 2019 – Future Electronics, a ****** leading distributor of electronic components, recently won Altech's 2018 Sales Achievement Award at EDS in Las Vegas.
Altech Corporation, a leading supplier of interconnect and electromechanical components, presented Future Electronics with the award in recognition for Future's sales growth, which outpaced all other distributors in the electronic components industry and culminated in another record year of sales for the partnership.
"We are pleased that Future Electronics was recognized by Altech for our outstanding performance for the second consecutive year," said Jason Lauder, Product Marketing Manager at Future Electronics.
Future Electronics Wins Altech's 2018 Sales Achievement Award for Second Straight Year full post
Robert Miller Congratulates Future Team on Winning ON Semi's 2018 Top Distributor for the Americas
April 26, 2019 Technology Press Release No comments
Montreal, Canada (bizpr) April 23, 2019 – Future Electronics, a ****** leading distributor of electronic components, was recently awarded 2018 Top Distribution Partner for the Americas by ON Semiconductor, a leading supplier of semiconductor-based solutions.
Robert Miller, President of Future Electronics, thanked everyone who contributed to the company's industry-leading performance on behalf of ON Semiconductor in 2018.
ON Semiconductor has enjoyed significant growth in distribution sales, which have accounted for over half of the company's revenue since 2015, and over 60% of revenue in 2018.
Robert Miller Congratulates Future Team on Winning ON Semi's 2018 Top Distributor for the Americas full post
Future Electronics Named ON Semiconductor's 2018 Top Distribution Partner for the Americas
April 8, 2019 Technology Press Release No comments
Montreal, Canada (bizpr) April 8, 2019 – Future Electronics, a ****** leading distributor of electronic components, was recently awarded 2018 Top Distribution Partner for the Americas by ON Semiconductor, a leading supplier of semiconductor-based solutions.
"Future Electronics is being recognized in the Americas region for successfully growing product sales and generating significant new business," said Jeff Thomson, ON Semiconductor's Vice President of ****** Channel Sales. "Future effectively supported both our customers' needs and our company initiatives for operational excellence."
ON Semiconductor has enjoyed significant growth in distribution sales, which have accounted for over half of the company's revenue since 2015, and over 60% of revenue in 2018. In the third quarter of 2018, ON Semiconductor reached a tremendous milestone, breaking $1 billion in distribution resales. "The support of our worldwide distribution partners is fundamental to the success of our company," Thomson said.
"Future Electronics is honored to receive this prestigious award from ON Semiconductor, our valued partner and an important industry leader," said Karim Yasmine, Future Electronics Corporate Vice President. "Future Electronics' mission is to delight our customers and suppliers, and we are very grateful to be recognized for our efforts."
For more information about Future Electronics, visit www.FutureElectronics.com.
About Future Electronics
Future Electronics is a ****** leader in electronics distribution, recognized for providing customers with ****** supply chain solutions, custom-tailored engineering services and a comprehensive suite of passives and semiconductor products. Founded by Robert Miller in 1968, Future Electronics has over 5,500 employees and operates in 169 offices in 44 countries around the world. Future Electronics is globally integrated, with a unified IT infrastructure that delivers real-time inventory availability and access to customers. With the highest level of service, the most advanced engineering capabilities, and the largest available-to-sell inventory in the world, Future's mission is always to Delight the Customer®. For more information, visit www.FutureElectronics.com.
Martin H. Gordon
FUTURE ELECTRONICS
www.FutureElectronics.com
514-694-7710 (ext. 2236)
martin.gordon@FutureElectronics.com
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Robert Miller Congratulates Future Team on Winning Nexperia's 2018 Americas Distributor of the Year
Montreal, Canada (bizpr) April 2, 2019 – Future Electronics, a ****** leading distributor of electronic components, has been awarded 2018 Distributor of the Year for the Americas by Nexperia, a ****** leader in Discretes, Logic and MOSFET devices.
Robert Miller, President of Future Electronics, thanked his team for their hard work and dedication, and congratulated everyone who contributed to the company's excellent performance. Future achieved record revenue on behalf of Nexperia in 2018, and was both their largest and fastest-growing channel partner.
"In just two years as a stand-alone company, Nexperia once again significantly outperformed the market in 2018," said Conrad Rodriguez, VP Americas for Nexperia. "Our distribution partners played a vital role in achieving this success, reaching a record $1B in resales through the channel. While all partners contributed to the record growth in the Americas, one star was brighter than the rest!"
Future Electronics also posted the largest market-share gain amongst all distribution partners, and brought in the highest number of new customers.
Robert Miller founded Future Electronics in 1968, and has always believed that his employees are the company's greatest asset.
Future Electronics is a ****** leader in electronics distribution, recognized for providing customers with ****** supply chain solutions, custom-tailored engineering services and a comprehensive suite of passives and semiconductor products. Founded by Robert Miller in 1968, Future Electronics has over 5,500 employees and operates in 169 offices in 44 countries around the world. Future Electronics is globally integrated, with a unified IT infrastructure that delivers real-time inventory availability and access to customers. With the highest level of service, the most advanced engineering capabilities, and the largest available-to-sell inventory in the world, Future's mission is always to Delight the Customer®. For more information, visit: www.FutureElectronics.com.
Robert Miller Thanks Future Electronics Suppliers and Team for Making AEU '18 a Success
November 29, 2018 Technology Press Release No comments
Montreal, Canada (bizpr) November 28, 2018 – Future Electronics, a ****** leading distributor of electronic components, recently held AEU '18 at their corporate headquarters in Montreal, Canada.
Advanced Engineering University (AEU) is Future's premier event for training the members of its Advanced Engineering Group on the latest technologies and applications from their valued partner suppliers. Robert Miller, President of Future Electronics, thanked the company's Advanced Engineering Group and all participating suppliers for making AEU '18 a success.
Robert Miller Thanks Future Electronics Suppliers and Team for Making AEU '18 a Success full post


Aoife Cunningham is a blogger and third level student from Dublin who is currently studying English and German. Her thoughts can be read here. She hopes to venture into the world of journalism or teaching, but her ultimate goal is to one day be published. She also planning on getting more involved with charity work, having previously spent a year as a voluntary youth worker, but for now is concentrating on balancing her academic life with work and leisure.
Because I am a Girl I ask …. why gender inequality still exists.
Because I am a Girl I believe …. Everyone should be equal.
Because I am a Girl I hope …. That the media will no longer heavily influence society (females particularly) to look a certain way. I hope that all sizes, choices and appearances are accepted, rather than judged or influenced.
Because I am a Girl I wonder …. What the Suffragettes would think of today's society.
Because I am a Girl I dream …. Of being successful, and perhaps inspiring someone, someday.
Because I am a Girl I remember …. Playing around with nail varnish and experimenting with different colors – nothing has changed!
Because I am a Girl I like …. Makeup, glitter and the occasional soppy film.
Because I am a Girl I dislike …. Bad hair days!
Because I am a Girl I feel …. That there's a long way to go before all countries and cultures expel gender inequality, or the notion of viewing males as being more powerful. I also feel proud that female authors are receiving the same amount of recognition as male authors, I can't help but wonder what some deceased authors would make of their increasing popularity!


Eldercare is one of the UK's leading providers of technology enabled care services, empowering more people to fulfil their dreams and stay in their homes for longer.
With most companies looking and feeling the same, Eldercare drew on our expertise at getting personal with consumers to completely transform their brand and set a new standard in their sector.
We understood that with over 1/3 of the British population now over 75, home care brands were failing to keep pace with the changing needs and attitudes around this deeply emotive issue.
A new approach was needed, one that was truly inclusive, emotionally warm and sincere.
That's why we spent time understanding the clients business and their customers to develop a brand that put the user and their families at the forefront of all communication and demonstrated 'care' in everything that it did, from the choice of images, to how it spoke.


Top level staff at the region's mental health trust are moving to new positions within Norfolk's health service.
Dawn Collins and Julie Cave, both Norfolk and Suffolk Foundation Trust (NSFT) board members will take up secondment positions at the county's community healthcare trust and sustainability and transformation partnership (STP) respectively.
While Rebecca Driver, who headed up communications for the STP and was director of communications at Great Yarmouth and Waveney Clinical Commissioning Group, will move to NSFT.
It comes as the latest inspection report into NSFT is due to be published soon, but a spokesman insisted this was not connected to that, or to the early departure of the trust's chair this week.
Mrs Cave, managing director and former interim chief executive at NSFT, will take up her long-term secondment at the STP next month, as interim chief operating officer.
Mrs Cave initially applied for the permanent chief executive role at NSFT but confirmed in March she had pulled out of the process and Antek Lejk was appointed to the position.
Mrs Collins, deputy chief nurse at NSFT, has joined Norfolk Community Health and Care (NCHC) on a part-time secondment.
She will work there until Spring, spending up to two days a week to provide additional capacity to enable Anna Morgan, director of nursing and quality, to carry out work for the STP.
Ms Driver started at NSFT on November 20 and will be in the post of interim director of corporate affairs and communications for 12 months.
On Tuesday it was announced NSFT's chairman Gary Page leaving his post with immediate effect.
Mr Page was due to leave early next year when he term in office comes to an end but he left early to make way for a new leadership team.


we made it! our first meal was at mcdonalds. is that crazy or what? we never eat there, but it is different here . . . they have McCafe. similar to starbucks!
we are staying in a dorm with a lady who is there all day cooking and cleaning, so we still get to practice our spanish. the pillows are made out of cement! i swear! but the place is really nice and spacious. two girls are there also (english speaking).
today we started school. we each have our own teacher. it is good but very hard.
well, adios for now. i will write more later.


We love the products we make. We love the positive reaction people have to our products. We don't make traditional furniture. We make furniture that is unique, safe, washable, durable, and fun.
The most durable seamless soft seating. Non-porous and waterproof flexible thermoplastic finish with a flexible foam core. No hidden pockets or stitched seams where bacteria could hide and multiply. Surface has great stain resistance and can be easily cleaned and disinfected. A solid base with fixed legs, or caster wheels for mobility. Light texture with matte finish. Passes CAL133 Fire Test. Perfect for high traffic spaces like schools, libraries, healthcare, museums, retail, and more.
Utilizing BIONIC® Polymer, Tenjam has made a giant leap forward in recycled materials and proven that Recovered Coastal Plastic (RCP) can be economically recovered, recycled, and turned into high performance products suitable for indoor and outdoor use. Tenjam is possibly the first furniture manufacturer in the world to incorporate plastic marine debris into commercial quality furniture and the first to incorporate BIONIC® Polymer into any type of product. BIONIC® Polymer is made with plastic recovered from shorelines, waterways, and coastal communities. The result is our most durable and lowest cost products.
All products in this category incorporate Recovered Coastal Plastic (RCP) and are suitable for indoor and outdoor use. These products have a hard outer shell with supreme durability and stain resistance. Bring on the saltwater, sand, sun, rain, and snow—these products can take it. We run the products through multiple tests including UV color stability, impact resistance, weight capacity, and stain resistance tests. You can be confident that these products will hold up and look great for many years in high traffic spaces.


text:
require 'optparse'

module VagrantPlugins
  module CommandPush
    class Command < Vagrant.plugin("2", :command)
      def self.synopsis
        "deploys code in this environment to a configured destination"
      end

      # @todo support multiple strategies if requested by the community
      def execute
        opts = OptionParser.new do |o|
          o.banner = "Usage: vagrant push [strategy] [options]"
        end

        # Parse the options
        argv = parse_options(opts)
        return if !argv

        name = validate_pushes!(@env.pushes, argv[0])

        @logger.debug("'push' environment with strategy: `#{name}'")
        @env.push(name)

        0
      end

      # Validate that the given list of names corresponds to valid pushes.
      #
      # @raise Vagrant::Errors::PushesNotDefined
      #   if there are no pushes defined
      # @raise Vagrant::Errors::PushStrategyNotProvided
      #   if there are multiple push strategies defined and none were specified
      # @raise Vagrant::Errors::PushStrategyNotDefined
      #   if the given push name do not correspond to a push strategy
      #
      # @param [Array<Symbol>] pushes
      #   the list of pushes defined by the environment
      # @param [String] name
      #   the name provided by the user on the command line
      #
      # @return [Symbol]
      #   the compiled list of pushes
      #
      def validate_pushes!(pushes, name = nil)
        if pushes.nil? || pushes.empty?
          raise Vagrant::Errors::PushesNotDefined
        end

        if name.nil?
          if pushes.length == 1
            return pushes.first.to_sym
          else
            raise Vagrant::Errors::PushStrategyNotProvided, pushes: pushes
          end
        end

        if !pushes.include?(name)
          raise Vagrant::Errors::PushStrategyNotDefined,
            name: name,
            pushes: pushes
        end

        return name.to_sym
      end
    end
  end
end

redpajama_set_name: RedPajamaGithub