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Wiley Open Research
Mathematical Methods in the Applied Sciences Open Research
Public Documents (326)
Under Review (202)
The well-posedness and long-time behavior of the nonlocal diffusion porous medium equ...
Chang Zhang
In this paper, we mainly consider the well-posedness and long-time behavior of solutions for the nonlocal diffusion porous medium equations with nonlinear term. Firstly, we obtain the well-posedness of the solutions in L1(Ω) for the equations. Secondly, we prove the existence of a global attractor by proving there exists a compact absorbing set. Finally, we apply index theory to consider the dimension of the attractor and prove that there exists an infinite dimensional attractor of the equations under proper conditions.
Tur´an type inequalities for the supertrigonometric functions
Lu-Lu Geng
This paper is devoted to the study of Tur\'{an} type inequalities for some well-known special functions such as supersine and supercosine which are derived by using a new form of the Cauchy-Bunyakovsky-Schwarz inequality.
INTEGRAL REPRESENTATIONS AND INEQUALITIES OF EXTENDED CENTRAL BINOMIAL COEFFICIENTS
Chunfu Wei
In the paper, the author presents three integral representations of extended central binomial coefficient, proves decreasing and increasing properties of two power-exponential functions involving extended (central) binomial coefficients, derives several double inequalities for bounding extended (central) binomial coefficient, and compares with known results.
Dynamic transitions and bifurcations of 1D reaction-diffusion equations: The Self-adj...
Taylan Sengul
This paper deals with the classification of transition phenomena in the most basic dissipative system possible, namely the 1D reaction diffusion equation. The emphasis is on the relation between the linear and nonlinear terms and the effect of the boundaries which influence the first transitions. We consider the cases where the linear part is self-adjoint with 2nd order and 4th order derivatives which is the case which most often arises in applications. We assume that the nonlinear term depends on the function and its first derivative which is basically the semilinear case for the second order reaction-diffusion system. As for the boundary conditions, we consider the typical Dirichlet, Neumann and periodic boundary settings. In all the cases, the equations admit a trivial steady state which loses stability at a critical parameter. We aim to classify all possible transitions and bifurcations that take place. Our analysis shows that these systems display all three types of transitions: continuous, jump and mixed and display transcritical, supercritical bifurcations with bifurcated states such as finite equilibria, circle of equilibria, and slowly rotating limit cycle. Many applications found in the literature are basically corollaries of our main results. We apply our results to classify the first transitions of the Chaffee-Infante equation, the Fisher-KPP equation, the Kuramoto Sivashinsky equation and the Swift-Hohenberg equation.
Special decompositions and linear superpositions of nonlinear systems: BKP and disper...
Xiazhi Hao
The existence of decomposition solutions of the well-known nonlinear BKP hierarchy is analyzed. It is shown that these decompositions provide simple and interesting relationships between classical integrable systems and the BKP hierarchy. Further, some special decomposition solutions display a rare property: they can be linearly superposed. With the emphasis on the case of the fifth BKP equation, the structure characteristic having linear superposition solutions is analyzed. Finally, we obtain similar superposed solutions in the dispersionless BKP hierarchy.
Local and global existence in L^{p} for the inhomogeneous nonlinear Schrödinger equat...
deng Wang
This paper investigates the local and global existence for the inhomogeneous nonlinear Schrödinger equation with the nonlinearity λ|x|^{-b}|u|^{β}u. It is show that a global solution exists in the mass-subcritical for large data in the spaces L^{p}, p < 2 under some suitable conditions on b,β and p. The solution is established using a data-decomposition argument, two kinds of generalized Strichartz estimates in Lorentz spaces and a interpolation theorem.
Sparse Representations of Random Signals
Tao Qian
Sparse (fast) representations of deterministic signals have been well studied. Among other types there exists one called adaptive Fourier decomposition (AFD) for functions in analytic Hardy spaces. Through the Hardy space decomposition of the $L^2$ space the AFD algorithm also gives rise to sparse representations of signals of finite energy. To deal with multivariate signals the general Hilbert space context comes into play. The multivariate counterpart of AFD in general Hilbert spaces with a dictionary has been named pre-orthogonal AFD (POAFD). In the present study we generalize AFD and POAFD to random analytic signals through formulating stochastic analytic Hardy spaces and stochastic Hilbert spaces. To analyze random analytic signals we work on two models, both being called stochastic AFD, or SAFD in brief. The two models are respectively made for (i) those expressible as the sum of a deterministic signal and an error term (SAFDI); and for (ii) those from different sources obeying certain distributive law (SAFDII). In the later part of the paper we drop off the analyticity assumption and generalize the SAFDI and SAFDII to what we call stochastic Hilbert spaces with a dictionary. The generalized methods are named as stochastic pre-orthogonal adaptive Fourier decompositions, SPOAFDI and SPOAFDII. Like AFDs and POAFDs for deterministic signals, the developed stochastic POAFD algorithms offer powerful tools to approximate and thus to analyze random signals.
Mathematical modeling of bulk and directional crystallization with the moving phase t...
Liubov Toropova
This paper is devoted to the mathematical modeling of a combined effect of directional and bulk crystallization in a phase transition layer with allowance for nucleation and evolution of newly born particles. We consider two models with and without fluctuations in crystal growth velocities, which are analytically solved using the saddle-point technique. The particle-size distribution function, solid-phase fraction in a supercooled two-phase layer, its thickness and permeability, solidification velocity, and desupercooling kinetics are defined. This solution enables us to characterize the mushy layer composition. We show that the region adjacent to the liquid phase is almost free of crystals and has a constant temperature gradient. Crystals undergo intense growth leading to fast mushy layer desupercooling in the middle of a two-phase region. The mushy region adjacent to the solid material is filled with the growing solid phase structures and is almost desupercooled.
Speed determinacy of the traveling waves for a three species time-periodic Lotka-Volt...
Qiong Wu
In this paper, speed selection of the time periodic traveling waves for a three species time-periodic Lotka-Volterra competition system is studied via the upper-lower solution method as well as the comparison principle. Through constructing specific types of upper and lower solutions to the system, the speed selection of the minimal wave speed can be determined under some sets of sufficient conditions composed of the parameters in the system.
Blow up for the solutions of the pressureless Euler-Poisson equations with time-depen...
Jianli Liu
The Euler-Poisson equations can be used to describe the important physical phenomena in many areas, such as semiconductor modeling and plasma physics. In this paper, we show the singularity formation mechanism for the solutions of the pressureless Euler-Poisson equations with time-dependent damping for the attractive forces in R^n (n ≧1) and the repulsive forces in R. We obtain the blow up of the derivative of the velocity under the appropriate assumptions.
A complete analytical solution to the integro-differential model describing the nucle...
Margarita Nikishina
In this paper, a complete analytical solution to the integro-differential model describing the nucleation and growth of ellipsoidal crystals in a supersaturated solution is obtained. The asymptotic solution of the model equations is constructed using the saddle-point method to evaluate the Laplace-type integral. Numerical simulations carried out for physical parameters of real solutions show that the first four terms of the asymptotic series give a convergent solution. The developed theory was compared with the experimental data on desupersaturation kinetics in proteins. It is shown that the theory and experiments are in good agreement.
MULTIPLICITY OF SOLUTIONS TO CLASS OF NONLOCAL ELLIPTIC PROBLEMS WITH CRITICAL EXPONE...
Elard Hurtado
In this paper, we establish existence of infinitely many weak solutions for a class of quasilinear stationary Kirchhoff type equations, which involves a general variable exponent elliptic operator with critical growth. Precisely, we study the following nonlocal problem \begin{equation*} \begin{cases} -\displaystyle{M}(\mathscr{A}(u))\operatorname{div}\Bigl(a(|\nabla u|^{p(x)})|\nabla u|^{p(x)-2}\nabla u\Bigl) = \lambda f(x,u)+ |u|^{s(x)-2}u \text{ in }\Omega, \\ u = 0 \text{ on } \partial \Omega, \end{cases} \end{equation*} where $\Omega$ is a bounded smooth domain of $\mathbb{R}^N,$ with homogeneous Dirichlet boundary conditions on $\partial \Omega,$ the nonlinearity $f:\overline{\Omega}\times \mathbb{R}\to \mathbb{R}$ is a continuous function, $a:\mathbb{R}^{+}\to\mathbb{R}^{+}$ is a function of the class $C^{1},$ $M:\mathbb{R}^{+}_{0}\to\mathbb{R}^{+}$ is a continuous function, whose properties will be introduced later, $\lambda$ is a positive parameter and $p,s\in C(\overline{\Omega})$. We assume that $\mathscr{C}=\{x\in \Omega: s(x)=\gamma^{*}(x)\}\neq \emptyset,$ where $\gamma^{*}(x)=N\gamma(x)/(N-\gamma(x))$ is the critical Sobolev exponent. We will prove that the problem has infinitely many solutions and also we obtain the asymptotic behavior of the solution as $\lambda\to 0^{+}$. Furthermore, we emphasize that a difference with previous researches is that the conditions on $a(\cdot)$ are general overall enough to incorporate some interesting differential operators. Our work covers a feature of the Kirchhoff's problems, that is, the fact that the Kirchhoff's function $M$ in zero is different from zero, it also covers a wide class of nonlocal problems for $p(x)>1,$ for all $x\in \overline{\Omega}.$ The main tool to find critical points of the Euler Lagrange functional associated with this problem is through a suitable truncation argument, concentration-compactness principle for variable exponent found in \cite{bonder}, and the genus theory introduced by Krasnoselskii.
On the planar Schrödinger-Poisson system with zero mass potential
Fangfang Liao
In this paper, we prove that the following planar Schrödinger-Poisson system with zero mass -Δu+φu=f(u), x∈R^2, Δφ= 2πu^2, x∈R^2, admits a nontrivial radially symmetric solution under weaker assumptions on f by using some new analytical approaches.
Calculation of Retention Indices of Essential Oils with the aid of the Van den Dool a...
Rifat BATTALOGLU
The aim of this article is to study the relationships and models among the Van den Dool and Kratz equation, the gas chromatography, and the Bézier curves constructed by aid of the Bernstein polynomials. Another aim of this article is to introduce open problems that contribute to real-world problems involving mathematics, chemistry, and plant biology, including the Van den Dool and Kratz equation, the gas chromatography, and Bézier curves. Searching for the solutions of these problems may have qualities that will create the potential that can enter the field of study of many researchers. As a result of these goals, the usability of Bézier curves was investigated while determining the chemical composition of essential oil obtained from P. Aladaghensis Leblebici. By applying the retention index from the Van den Dool and Kratz equation and evaluating chemical compositions of the essential oil are characterized by gas chromatography-mass spectrometry (GC-MS). The Van den Dool and Kratz equation results have the potential to be used not only in the chemical compositions of the oils, but also in applied mathematics and other fields. Moreover, we construct a new special finite sum. A lower bound and inequality are also given for the finite special sum involving the dead time associated with the isocratic step. Some applications and criticisms are given that include this lower bound and inequality for these sums and its effects on the chemical compositions of essential oil and the Van den Dool and Kratz equation.
On multilinear distorted multiplier estimate and its applications
kailong yang
In this article, we investigate the multilinear distorted multiplier estimate (Coifman-Meyer type theorem) associated with the Schr\"{o}dinger operator $H=-\Delta + V$ in the framework of the corresponding distorted Fourier transform. Our result is the "distorted" analog of the multilinear Coifman-Meyer multiplier operator theorem in \cite{CM1}, which extends the bilinear estimates of Germain, Hani and Walsh's in \cite{PZS} to multilinear case for all dimensions. As applications, we give the estimate of Leibniz's law of integer order derivations for the multilinear distorted multiplier for the first time and we obtain small data scattering for a kind of generalized mass-critical NLS with good potential in low dimensions $d=1,2$.
On vanishing limits of the shear viscosity and Hall coefficients for the planar compr...
Xia Ye
This paper deals with an initial-boundary value problem of the planar compressible Hall-magnetohydrodynamic (for short, Hall-MHD) equations. For the fixed shear viscosity and Hall coefficients, it is shown that the strong solutions of Hall-MHD equations and corresponding MHD equations are global. As both the shear viscosity and the Hall coefficients tend to zero, the convergence rate for the solutions from Hall-MHD equations to MHD equations is given. The thickness of boundary layer is discussed by spatially weighted estimation and the characteristic of boundary layer is described by constructing a boundary layer function.
Global dynamics and bifurcation analysis of a fractional-order SEIR epidemic model wi...
Parvaiz Ahmad Naik
The present paper studies a fractional-order SEIR epidemic model for the transmission dynamics of infectious diseases such as HIV and HBV that spreads in the host population. The total host population is considered bounded, and Holling type-II saturation incidence rate is involved as the infection term. Using the proposed SEIR epidemic model, the threshold quantity, namely basic reproduction number R0, is obtained that determines the status of the disease, whether it dies out or persists in the whole population. The model's analysis shows that two equilibria exist, namely, disease-free equilibrium (DFE) and endemic equilibrium (EE). The global stability of the equilibria is determined using a Lyapunov functional approach. The disease status can be verified based on obtained threshold quantity R0. If R0 < 1, then DFE is globally stable, leading to eradicating the population's disease. If R0 > 1, a unique EE exists, and that is globally stable under certain conditions in the feasible region. The Caputo type fractional derivative is taken as the fractional operator. The bifurcation and sensitivity analyses are also performed for the proposed model that determines the relative importance of the parameters into disease transmission. The numerical solution of the model is obtained by the generalized Adams- Bashforth-Moulton method. Finally, numerical simulations are performed to illustrate and verify the analytical results.
On singularities of solution of the elasticity system in a bounded domain with angula...
Yasir Nadeem
This paper aims to give a mathematically rigorous description of the corner singularities of the weak solutions for the plane linearized elasticity system in a bounded planar domain with angular corner points on the boundary. The qualitative properties of the solution including its regularity depend crucially on these corner points or such types of boundary conditions. In particular, the resulting expansion of the solutions of the underlying problem involves singular vector functions, inlines, depending on a certain parameter ξμ. We derive the transcendental equations for all ten possible cases of combinations of the boundary conditions generated by the basic four ones in classical elasticity proposing in the two natural directions of the boundary, i.e., tangential and normal direction, respectively, which depends on ξμ . So, a MATLAB program is developed whereby ξμ can be computed, and figures showing their distributions are presented. The leading singular exponents are computed for these combinations of the boundary conditions, wherein critical angles ω_ critical are listed such that for interior angles ω