A textbook or reference for applied physicists or mathematicians; geophysicists; or civil, mechanical, or electrical engineers. It assumes the usual undergraduate sequence of mathematics in engineering or the sciences, the traditional calculus, differential equations, and Fourier and Laplace transforms. It explains how to use those and the Hankel transforms to solve linear partial differential equations that are encountered in engineering and sciences. No date is noted for the first edition; the second includes numerical methods and asymptotic techniques for inverting particularly complicated transforms.
Transform Methods for Solving Partial Differential Equations 2 and (1.1.2) Equation 1.1.2 is the Fourier transform of f(t) while Equation 1.1.1 is the inverse Fourier transform, which converts a Fourier transform back to f(t). If, following Hamming,1 we imagine that f(t) is a light beam, then the Fourier transform, like a glass prism, breaks up the function into its component frequencies ω , each of intensity F(ω). In optics, the various frequencies are called colors; by analogy the Fourier transform gives us the color spectrum of a function. On the other hand, the inverse Fourier transform blends a function's spectrum to give back the original function. If f(t) is an even function, we can replace Equation 1.1.1 with the Fourier cosine transform (1.1.3) with (1.1.4) On the other hand, if f(t) is an odd function, then Equation 1.1.1 can be replaced with the Fourier sine transform (1.1.5) with (1.1.6) Fourier cosine and sine transforms are also useful when f(t) is only defined on the semi-infinite interval [0, ). Clearly, for a Fourier transform to exist, then the integral in Equation 1.1.2 must also exist. A sufficient condition is that f(t) is absolutely integrable on (- , ), or (1.1.7) Some of the simplest functions, such as f(t)=sin(at) and f(t)=cos(at), violate this condition and appear not to have a Fourier transform. Actually, they do exist but it requires the use of the Dirac delta function to express them. To avoid the use of generalized functions, some investigators argue that all physical processes suffer a certain amount of dissipation. For that 1Hamming, R.W., 1977: Digital Filters. Prentice-Hall, p. 136.
The Fundamentals 3 reason, the Fourier transform should include a damping factor e - t so that the definition of the Fourier transform becomes (1.1.8) where ω′=ω-i , >0. In this modified form, the inverse Fourier transform is (1.1.9) • Example 1.1.1 Let us find the Fourier transform of (1.1.10) when we include a small amount of damping >0. From the definition of the Fourier transform, (1.1.11) Direct integration yields (1.1.12) For those familiar with Laplace transforms, this is the same answer as the Laplace transform of cos(bt) with s replaced by iω′. For this reason, Van der Pol and Bremmer2 have called the transform pair Equation 1.1.8 and Equation 1.1.9, a two-sided Laplace or bilateral transform. On the other hand, from the definition of the inverse Fourier transform (1.1.13) In principle, we can compute any Fourier transform from the definition. However, it is far more efficient to derive some simple relationships that 2Van der Pol, B., and H.Bremmer, 1955: Operational Calculus Based on the Two-Sided Laplace Integral. Cambridge University Press, 415 pp. See Equation 11 in Chapter 2.
Transform Methods for Solving Partial Differential Equations 6 The Fourier transform of f(t)*g(t) is (1.1.27) (1.1.28) (1.1.29) and the convolution theorem holds. • Example 1.1.4 For our final example,3 let us find the inverse f(x) of the Fourier transform (1.1.30) using the convolution theorem where c, t and are positive and real. We begin by noting that (1.1.31) and (1.1.32) From the convolution theorem, (1.1.33) if . Then, if x>ct, (1.1.34) For |x|≤ct, (1.1.35) 3Taken from Tanaka, K., and T.Kurokawa, 1973: Viscous property of steel and its effect on strain wave front. Bull. JSME, 16, 188--193.
The Fundamentals 7 Finally, if x<-ct, (1.1.36) In Section 3.1 and Section 5.1 we will discuss the inversion of Fourier transforms by complex variables. In this section, we have given a quick overview of Fourier transforms. For greater detail, as well as drill exercises, the reader is referred to Chapter 5 of the author 's Advanced Engineering Mathematics with MATLAB.4 1.2 LAPLACE TRANSFORMS Consider a function f(t) such that f(t)=0 for t<0. Then the Laplace integral (1.2.1) defines the Laplace transform of f(t), which we write [f(t)] or F(s). The Laplace transform of f(t) exists, for sufficiently large s, provided f(t) satisfies the following conditions: 4Duffy, D.G., 2003: Advanced Engineering Mathematics with MATLAB. Chapman & Hall/ CRC, 818 pp. • Example 1.2.1 Let us find the Laplace transform for the Heaviside step function: (1.2.2) The Heaviside step function is essentially a bookkeeping device that gives us the ability to "switch on" and "switch off" a given function. For example, if we want a function f(t) to become nonzero at time t=a, we represent this process by the product f(t)H(t-a). From the definition of the Laplace transform, (1.2.3) • f (t)=0 for t<0, • f (t) is continuous or piece-wise continuous is every interval, • tn|f(t)|< as t→0 for some number n, where n<1, • e-sot|f(t)| < as t→ , for some number s0. The quantity s0 is called the abscissa of convergence.
Transform Methods for Solving Partial Differential Equations 8 • Example 1.2.2 The Dirac delta function or impulse function, often defined for computational purposes by (1.2.4) plays an especially important role in transform methods because its Laplace transform is (1.2.5) (1.2.6) (1.2.7) (1.2.8) A special case is [δ(t)]=1. The Fourier transform of δ(t-a) is similar, namely e-iaω. • Example 1.2.3 Although we could compute Equation 1.2.1 for every function that has a Laplace transform, these results have already been tabulated and are given in many excellent tables.5 However, there are four basic transforms that the reader should memorize. They are (1.2.9) (1.2.10) (1.1.11) (1.2.12) 5 The most complete set is given by Erdélyi, A., W.Magnus, F.Oberhettinger, and F. G.Tricomi, 1954: Tables of Integral Transforms, Vol I. McGraw-Hill Co., 391 pp.
Transform Methods for Solving Partial Differential Equations 10 by integration by parts. If f(t) is of exponential order, 6 e-st f(t) tends to zero as t→ , for large enough s, so that (1.2.18) Similarly, if f(t) and f′(t) are continuous, f′′(t) is piece-wise continuous, and all three functions are of exponential order, then (1.2.19) In general, (1.2.20) on the assumption that f(t) and its first n-1 derivatives are continuous, f(n)(t) is piece-wise continuous, and all are of exponential order so that the Laplace transform exists. Consider now the transform of the function e-at f(t), where a is any real number. Then, by definition, (1.2.21) or (1.2.22) Equation 1.2.22 is known as the first shifting theorem and states that if F(s) is the transform of f(t) and a is a constant, then F(s+a) is the transform of e-at f(t). • Example 1.2.4 Let us find the Laplace transform of f(t)=e-at sin(bt). Because the Laplace transform of sin(bt) is b/(s2+b2), (1.2.23) where we have simply replaced s by s+a in the transform for sin(bt). • Example 1.2.5 Let us find the inverse of the Laplace transform (1.2.24) 6 By exponential order we mean that there exist some constants, M and k, for which |f(t)|≤Me kt for all t>0.
Transform Methods for Solving Partial Differential Equations 12 • Example 1.2.6 Let us find the convolution between cos(t) and sin(t). (1.2.37) (1.2.38) (1.2.39) (1.2.40) The reason why we introduced convolution derives from the following fundamental theorem (often called Borel's theorem7). If (1.2.41) then (1.2.42) In other words, we can invert a complicated transform by convoluting the inverses to two simpler functions. • Example 1.2.7 Let us find the inverse of the transform (1.2.43) (1.2.44) Therefore, (1.2.45) (1.2.46) (1.2.47) (1.2.48) 7Borel, É., 1901: Leçons sur les séries divergentes. Gauthier-Villars, p. 104.
The Fundamentals 13 In this section, we have given a quick overview of Laplace transforms. For greater detail, as well as drill exercises, the reader is referred to Chapter 6 of the author's Advanced Engineering Mathematics with MATLAB.8 1.3 LINEAR ORDINARY DIFFERENTIAL EQUATIONS Most analytic techniques for solving a partial differential equation involve reducing it down to an ordinary differential equation or a set of ordinary differential equations that is hopefully easier to solve than the original partial differential equation. From the vast number of possible ordinary differential equations, we focus on second-order equations. All of the following techniques extend to higher-order equations. Consider the ordinary differential equation (1.3.1) where a, b and c are real. For the moment let us take f(x)=0. Assuming a solution of the form y(x)=Aemx and substituting into Equation 1.3.1, (1.3.2) This purely algebraic equation is the characteristic or auxiliary equation. Because Equation 1.3.2 is quadratic, there are either two real roots, or else a repeated real root, or else conjugate complex roots. At this point, let us consider each case separately and state the solution. Any undergraduate book on ordinary differential equations will provide the details for obtaining these general solutions. Case I: Two distinct real roots m1 and m 2, (1.3.3) Case II: A repeated real root m1, (1.3.4) Case III: Conjugate complex roots m1=p+qi and m2=p-qi, (1.3.5) 8Duffy, op. cit.
The Fundamentals 15 because cos2(x)=[1+cos(2x)]/2. Substituting Equation 1.3.12 into Equation 1.3.11 and equating coefficients for the constant, cosine and sine terms, we find that A=1/2, B=-3/50 and C=2/25. The remaining task is to compute the arbitrary constants in the homogeneous solution. In this book we always have conditions at both ends of a given domain, even if one of these points is at infinity. We now illustrate the procedure used in solving these boundary-value problems. • Example 1.3.3 Solve the boundary-value problem (1.3.13) where s>0. The general solution to Equation 1.3.13 is (1.3.14) We have chosen to use hyperbolic functions because the domain lies between x=0 and x=1. Now, (1.3.15) and (1.3.16) Solving for A and B, (1.3.17) Therefore, (1.3.18) Problems 1. Solve the boundary-value problem where a and s are real and positive.
Transform Methods for Solving Partial Differential Equations 16 1.4 COMPLEX VARIABLES Complex variables provide analytic tools for the evaluation of integrals with an ease that rarely occurs with real functions. The power of integration on the complex plane has its roots in the basic three C's: the Cauchy- Riemann equations, the Cauchy-Goursat theorem and Cauchy's residue theorem. The Cauchy-Riemann equations have their origin in the definition of the derivative in the complex plane. Just as we have the concept of the function in real variables, where for a given value of x we can compute a corresponding value of y=f(x), we can define a complex function w=f(z) where for a given value of we may compute w=f(z)=u(x, y)+iv(x,y). In order for f'(z) to exist in some region, u(x, y) and v(x, y) must satisfy the Cauchy-Riemann equations: (1.4.1) If ux , uy, vx and vy are continuous in some region surrounding a point z0 and satisfy Equation 1.4.1 there, then f(z) is analytic there. If a function is analytic everywhere in the complex plane, then it is an entire function. Alternatively, if the function is analytic everywhere except at some isolated singularities, then it is meromorphic. Note that f(z) must satisfy the CauchyRiemann equations in a region and not just at a point. For example, f(z)=|z| satisfies the Cauchy-Riemann equations at z=0 and nowhere else. Consequently, this function is not analytic anywhere on the complex plane. Integration on the complex plane is more involved than in real, single variables because dz=dx+i dy. We must specify a path or contour as we integrate from one point to another. Of all of the possible contour integrals, a closed contour is the best. To see why, we introduce the following results: Cauchy-Goursat theorem:9 If f(z) is an analytic function at each point within and on a closed contour C, then . This theorem leads immediately to The principle of deformation of contours: The value of a line integral of an analytic function around any simple closed contour remains unchanged if we deform the contour in such a manner that we do not pass over a point where f(z) is not analytic. Consequently we can evaluate difficult integrals by deforming the contour so that the actual evaluation is along a simpler contour or the computations 9 See Goursat, E., 1900: Sur la définition générale des fonctions analytiques, d'après Cauchy. Trans. Am. Math. Soc., 1, 14--16. are made easier. See Example 1.4.1.
The Fundamentals 17 Most integrations on the complex plane, however, deal with meromorphic functions. Our next theorem involves these functions; it is Cauchy's residue theorem: 10 If f(z) is analytic inside a closed contour C (taken in the positive sense) except at points z1, z2, ..., zn where f(z) has singularities, then (1.4.2) where Res[f(z); zj] denotes the residue of f(z) for the singularity located at The question now turns to what is a residue and how do we compute it. The answer involves the nature of the singularity and an extension of the Taylor expansion, called a Laurent expansion: (1.4.3) for 0<|z -zj|<a. The first summation is merely the familiar Taylor expansion; the second summation involves negative powers of z--zj and gives the behavior at singularity. The residue equals the coefficient a -1. The construction of a Laurent expansion for a given singularity has two practical purposes: (1) it gives the nature of the singularity and (2) we will occasionally use it to give the actual value of the residue. Turning to the nature of the singularity, there are three types: • Essential Singularity: Consider the function f(z)=cos(1/z). Using the expansion for cosine, (1.4.4) for 0<|z|< . Note that this series never truncates in the inverse powers of z. Essential singularities have Laurent expansions which have an infinite number of inverse powers of z--zj. The value of the residue for this essential singularity at z=0 is zero. • Removable Singularity: Consider the function f(z)=sin(z)/z. This function appears, at first blush, to have a singularity at z=0. Upon applying the expansion for sine, we see that (1.4.5) 10 See Mitrinovic, D.S., and J.K. , 1984: The Cauchy Method of Residues. D. Reidel Publishing Co., 361 pp. Section 10.3 gives the historical development of the residue theorem.
The Fundamentals 19 The desirability of dealing with closed contour integrals should be clear by now. This is true to such an extent that mathematicians have devised several theorems that allow us to change a line integral into a closed one by adding an arc at infinity. The one of greatest relevance to us is by C.Jordan:11 Jordan's lemma: Suppose that, on a circular arc CR with radius R and center at the origin, f(z)→0 uniformly as R→ . Then (1.4.14) if CR is in the first and/or second quadrant; (1.4.15) if CR is in the third and/or fourth quadrant; (1.4.16) if CR is in the second and/or third quadrant; and (1.4.17) if CR is in the first and/or fourth quadrant. Technically, only Equation 1.4.14 is actually Jordan's lemma while the remaining points are variations. Proof: We shall prove the first part; the remaining portions follow by analog. We begin by noting that (1.4.18) Now, (1.4.19) (1.4.20) Therefore, (1.4.21) 11 Jordan, C., 1894: Cours D'Analyse de l'École Polytechnique. Vol. 2. GauthierVillars, pp. 285--286. See also Whittaker, E.T., and G.N.Watson, 1963: A Course of Modern Analysis. Cambridge University Press, p. 115.
Transform Methods for Solving Partial Differential Equations 20 where 0≤θ0<θ1≤ π. Because the integrand is positive, the right side of Equation 1.4.21 is largest if we take θ0=0 and θ1=π. Then (1.4.22) We cannot evaluate the integrals in Equation 1.4.22 as they stand. However, because sin(θ)≥2θ/π, we can bound the value of the integral by (1.4.23) If m>0, |IR| tends to zero with MR as R→ . • Example 1.4.1 To illustrate how useful distorting the original contour may be in evaluating an integral, consider 12 (1.4.24) where the contour C is the circle |z|<-ln(a) and 1=n. The integrand has an infinite number of simple poles at zm=-ln(a)+2mπi with m=0,±1,±2,... (which lie outside the original contour) and a (n +1)th-order pole at z=0. Because the straightforward evaluation of this integral by the residue theorem would require differentiating the denominator n times, we choose to evaluate Equation 1.4.24 by expanding the contour so that it is a circle of infinite radius with a cut that excludes the simple poles at zm. See Figure 1.4.1. Then, by the residue theorem, (1.4.25) where I is the contribution from the circle at infinity. Because the residue off(z)atzmis--zm --1--n, (1.4.26) 12 Based upon Götze, F., and H.Friedrich, 1980: Berechnungs- und Abschätzungsformeln für verallgemeinerte geometrische Reihen. Z. Angew. Math. Mech., 60, 737--739.
Transform Methods for Solving Partial Differential Equations 22 • Example 1.4.2 Let us evaluate where a and k are real and positive. First, 1.4.32 We close the line integral along the real axis by introducing an infinite semicircle in the upper half-plane as dictated by Jordan's lemma. Therefore, (1.4.33) (1.4.34) • Example 1.4.3 When the definite integral involves hyperbolic functions, a rectangular closed contour is generally the best one to use. For example, consider the contour integral13 (1.4.35) where C is the closed rectangular contour ABCD shown in Figure 1.4.2. Along AD as R→ , (1.4.36) Similarly, along BC (1.4.37) Along AB (1.4.38) 13 Taken from Hawthorne, W.R., 1954: The secondary flow about struts and airfoils. J. Aeronaut. Sci., 21, 588--608.
Transform Methods for Solving Partial Differential Equations 24 The only poles located inside the closed contour occur at z±=± πi[1-1/ (2 )]. To compute their residues, we note that (1.4.44) (1.4.45) because cos[π/(2 )]=-cosh(z±) and =z-z± . Therefore, the poles are second order and the residues equal Hence the sum of residues is Substituting this sum and Equation 1.4.43 into Equation 1.4.41, we finally have In this section, we have given a quick overview of complex variables as it applies to single-valued functions. For greater detail, as well as drill exercises, the reader is referred to Chapter 1 of the author's Advanced Engineering Mathematics with MATLAB.14 In those instances where there are multivalued functions due to the presence of z raised to some rational power, inverse functions or logarithms, we must make them single-valued. This is the subject of the next two sections. 1.5 MULTIVALUED FUNCTIONS, BRANCH POINTS, BRANCH CUTS AND RIEMANN SURFACES In this section, we introduce functions that yield several different values of w for a given z, i.e., multivalued functions. We must make these functions single- valued so that we can apply the techniques from the previous section. Furthermore, this condition is also necessary for a well-posed physical problem. Consider the complex function w=z1/2. For each value of z there are two possible values of w. For example, if z=-i, then w equals either or . The points w1 and w2 are (1.4.46) 14 Duffy, op. cit.
The Fundamentals 25 distinct members from two branches of the same function that "branch off' from the same point, z=0. The number of branches depends upon the nature of f(z). For example, log(z) has an infinite number of branches, namely, ln(r)+ i+nπi, n=0,1, 2,.... In the case of real variables, the branches easily separate. For example, the square root of a positive, real number has two distinct branches: a and -a, where a is a real, positive number. However, in complex functions the two branches are hardly distinguishable because they do not separate at all. Therefore, if we wish to keep them separate, we must do it artificially. Consider again the complex function .Ifwemove around a closed path that does not encircle the origin, the values of r and vary continuously. At the end, the final value of equals our initial value, 0. Consequently, we can say that all of the values of w along this contour belong to the same branch, . Let our closed contour now enclose the origin. Because the final value of = 0+2π, the final value of w now equals .We have reached the other branch of w in a continuous manner. Consequently, z=0 appears to be a special point; one branch ties to the other there because they have the same value. We reach each branch from the other after a complete turn around the origin. A branch point is any point having this property. In this example, infinity is also a branch point. We can show this by the substitution z=1/z ′ and an examination of the transformed function about z′=0. In our example with w=z1/2, we reached the other branch by completing a closed contour around the origin. However, we cannot say exactly when, in our journey, we crossed the boundary from w1 and w2. For example, it might have been when we crossed the positive real axis or when we crossed the negative real axis. The conclusions are the same either way. This ambiguity leads to the concept of the branch cut. The reason why we must define a branch cut between branch points lies in the fact that we cannot make a multivalued function single-valued by excluding the branch point and a small neighborhood around it. Its multivaluedness does not depend on the mere existence of the branch point itself, but on the possibility of encircling it. In summary, a branch cut is a line that we choose that connects two branch points. Furthermore, it defines the separation between the branches. For this reason the branch cut is a barrier that we may not cross.15 A geometrical interpretation of this process is to limit each branch to a particular Riemann surface. We can view each Riemann surface as a floor in a large department store. On each floor (Riemann surface) you can only obtain one type of branch (for example, square roots with positive real parts). However, we can reach other floors (surfaces), if desired, through a 15 The Russian mathematical term for the edge along a branch cut is bereg, which commonly means "the bank of a river."
Transform Methods for Solving Partial Differential Equations 26 set of escalators, located at the branch cut, that take you up to the next higher Riemann surface (if it exists) or down to the next lower Riemann surface (if it exists). These (very thin) escalators extend between the branch points. As the architect of your Riemann surface, there is great flexibility in choosing where your branch cuts lie. Figure 1.5.1 and Figure 1.5.2 show some of the more popular choices for and , respectively. The advantage of introducing a Riemann surface is that every continuous curve on the z-plane maps the multivalued function into a continuous curve on the w-plane. This relationship between the Riemann surfaces allows us to apply to multivalued analytic functions all of the techniques of integration, analytic continuation and so forth, which depend on a continuous path being drawn from one point to another. In the next section, we illustrate the mechanics of integration involving multivalued functions. 1.6 SOME EXAMPLES OF INTEGRATION THAT INVOLVE MULTIVALUED FUNCTIONS In this section we perform contour integration with multivalued functions. Essentially, the introduction of branch points and cuts requires careful bookkeeping of phases (or arguments). Once we set up our bookkeeping, the evaluation follows directly. Figure 1.5.1: Some popular branch cut configurations for . The branch cuts are denoted by wavy lines.
The Fundamentals 27 • Example 1.6.1 Let us evaluate the integral16 (1.6.1) where -a<b<c<d. Figure 1.6.1 shows the contour The integrand is a multivalued function with branch points at z=-a and z=c and simple poles at z=b and z=d. We begin by expanding our original contour C without bound so that it includes all of the singularities. Figure 1.6.1 shows this enlarged contour . Next we employ the following theorem: Theorem:17 If zf(z) tends uniformly to a limit k as |z| increases indefinitely, the value of , taken around a very large circle, center the origin, tends toward 2 ik. Figure 1.5.2: Some popular branch cut configurations for . The branch cuts are denoted by wavy lines. 16 This example is taken from Glagolev, N.I., 1945: Resistance of cylindrical bodies in rolling (in Russian). Prikl. Mat. Mek., 9, 318--333. 17 Forsyth, A.R., 1965: Theory of Functions of a Complex Variable. Dover Publications, Inc., p. 41.
The Fundamentals 31 z+a=(x +a)e2πi along C2, C3 and C4, and z+a=(x+a)e0i along C6, C7 and C8 with -a<x< c. After substituting into Equation 1.6.17, (1.6.18) or (1.6.19) • Example 1.6.2 Let us evaluate (1.6.20) where -a<x<a and 0<μ<1. Figure 1.6.3 shows that the contour runs just below the real axis from-a to a and then back to -a just above the real axis.18 The integrand is a multivalued function with three branch points at z=a, x and-a. The branch cut runs along the real axis from -a to a. The value of Equation 1.6.20 follows from the limit of zf(z) as |z|→ . (See the previous example.) If the argument of z-a, z+a and z-x lies between 0 and 2π, this limit equals one and (1.6.21) Figure 1.6.3: The contour used to evaluate Equation 1.6.20. 18 Reprinted from J. Appl. Math. Mech., 23, V.Kh.Arutiunian, The plane contact problem of the theory of creep, 1283--1313, ©1959, with kind permission from Pergamon Press Ltd., Headington Hill Hall, Oxford OX3 0BX, UK.
The Fundamentals 33 and (1.6.30) Substituting these integrals into Equation 1.6.22 and simplifying, (1.6.31) • Example 1.6.3 Let us simplify the integral with a singular kernel19 (1.6.32) by evaluating the complex integral (1.6.33) where and are nonintegers. The contour C is a circle of infinite radius with appropriate branch cuts as shown in Figure 1.6.4. From the residue Figure 1.6.4: The contour used to evaluate Equation 1.6.34. 19 Taken from Liu, P.L.-F., 1986: Hydrodynamic pressures on rigid dams during earthquakes. J. Fluid Mech., 165, 131--145. Reprinted with the permission of Cambridge University Press.
The Fundamentals 35 Although we must still evaluate an integral numerically, the integrand is no longer singular. Let us now redo this integral with the contour shown in Figure 1.6.5. The phase of both z and z-1 now run from - to . The contributions from CR, C2 and C10 are zero. However, for the other contours, (1.6.43) (1.6.44) (1.6.45) (1.6.46) (1.6.47) (1.6.48) Figure 1.6.5: The contour used in redoing Equation 1.6.34.
The Fundamentals 41 5. Evaluate around the illustrated contour and verify that 6. Use the illustrated contour and two different complex functions to show that and Problem 3 Problem 4 Problem 5 Problem 6 Hint: The square root will have different signs at the two poles.
The Fundamentals 43 10. Show22 that where 0≤ n, is real and noninteger and C is a dumbbell-shaped contour lying along the negative real axis from z=-1 and z=0. 11. Evaluate around the illustrated contour and show23 12. Evaluate Problem 11 22 Taken from Jury, E.I., and C.A.Galtieri, 1961: A note on the inverse z transformation. IRE Trans. Circuit Theory, CT-8, 371--374. ©1961 IEEE. 23 Taken from Ahmadi, A.R., and S.E.Widnall, 1994: Energetics of oscillating lifting surfaces by the use of integral conservation laws. J. Fluid Mech., 266, 347--370. Reprinted with the permission of Cambridge University Press.
Transform Methods for Solving Partial Differential Equations 44 13. Evaluate around the contour used in Problem 4, where p= /[2 -2 arccos( )] with the branch 0≤arccos( )<2 . Show25 that Hint: 14. By first showing that around the illustrated contour and show 24 Problem 12 24 Ibid. 25 Taken from Greenwell, R.N., and C.Y.Wang, 1980: Fluid flow through a partially filled cylinder. Appl. Sci. Res., 36, 61--75. Reprinted by permission of Kluwer Academic Publishers.
The Fundamentals 45 15. Show26 that where -1<q<1 and the plus sign applies when p>q while the negative sign is for p<q. Step 1: Consider first the case p=1. From the contours shown in Figure 1.6.8, show that Figure 1.6.8: Contour used in solving Problem 15. Redrawn from Lewis, P.A., and G.R.Wickham, Philos. Trans. R. Soc. London, Ser. A, 340, 503--529 (1992). and C2 is the contour shown in Figure 1.6.6, prove that 26 Taken from Lewis, P.A., and G.R.Wickham, 1992: The diffraction of SH waves by an arbitrary shaped crack in two dimensions. Phil. Trans. R. Soc. London, Ser. A, 340, 503--529.
Transform Methods for Solving Partial Differential Equations 46 Step 2: By parameterizing each contour as follows: where 0<ρ< and tan(θ)=ρ/2, and where -1<ρ< 1, show that and Note that the simple pole at z=q makes no contribution. Step 3: Evaluate the integral for by noting that (1+q)+(1-q)=2, replacing cosine and sine by their exponential equivalents, and letting 1- e2θi= 2 . You should find that Finally, by matching imaginary parts, show that Step 4- Now redo the first three steps with p=-1 and show that Then, use analytic continuity to argue for the more general result. 16. Evaluate27 27 Taken from Chen, C.F., 1962: Linearized theory for supercavitating hydrofoils with spoiler-flaps. J. Ship Res., 6, No. 3, 1--9. Reprinted with the permission of the Society of Naval Architects and Marine Engineers (SNAME). Material originally appearing in SNAME publications cannot be reprinted without written permission from the Society, 601 Pavonia Ave., Jersey City, NJ 07306.
The Fundamentals 49 Step 8: Conclude the problem by showing that 1.7 BESSEL FUNCTIONS In Section 1.3 we dealt only with ordinary differential equations that have constant coefficients. In problems involving cylindrical coordinates, we will solve the equation (1.7.1) commonly known as Bessel's equation of order n with a parameter . The general solution to Equation 1.7.1 is (1.7.2) where Jn(.) and Yn(.) are nth order Bessel functions of the first and second kind, respectively. Bessel functions have been exhaustively studied and a vast literature now exists on them.28 The Bessel function Jn(z) is an entire function, has no complex zeros, and has an infinite number of real zeros symmetrically located with respect to the point z=0, which is itself a zero if n>0. All of the zeros are simple, except the point z=0, which is a zero of order n if n>0. On the other hand, Yn(z) is analytic in the complex plane with a branch cut along the segment (- , 0] and becomes infinite as z→0. Considerable insight into the nature of Bessel functions is gained from their asymptotic expansions. These expansions are (1.7.3) and (1.7.4) where denotes an arbitrarily small positive number. Therefore, Bessel functions are sinusoidal in nature and decay as z-1/2 . 28 The standard reference is Watson, G.N., 1966: A Treatise on the Theory of Bessel Functions. Cambridge University Press, 804 pp.
Transform Methods for Solving Partial Differential Equations 52 where (1.7.15) Consider now the special case where f(x, y) is only a function of so that f(x, y)=g(r). Then, changing to polar coordinat es through the substitution x=r cos(θ), y=r sin(θ), k=ρ cos(φ) and =ρ sin(φ), we have that (1.7.16) (1.7.17) Therefore, the integral in Equation 1.7.15 becomes (1.7.18) (1.7.19) If we introduce =θ-φ, the integral inside the square brackets can be evaluated as follows: and Equation 1.7.21 is equivalent to Equation 1.7.20 because the integral of a periodic function over one full period is the same regardless of where the integration begins. Equation 1.7.22 follows from the integral definition of the Bessel function.29 Therefore, (1.7.23) Finally, because Equation 1.7.23 is clearly a function of F(k, )=G( ) and (1.7.24) (1.7.20) (1.7.21) (1.7.22) 29 Ibid., Section 2.2, Equation 5.
The Fundamentals 55 Modern research,31 however, has shown that during the early nineteenth century both English and French mathematicians developed both symbolic calculus and operational methods. For example, Cauchy used operational methods, applying the Fourier transform to solve the wave equation. Later in the nineteenth century, this knowledge was apparently forgotten until Heaviside rediscovered the Laplace transform. 32 However, because of Heaviside's lack of mathematical rigor, a controversy of legendary proportions33 developed between him and the mathematical "establishment." It remained for the English mathematician T.J.I'a.Bromwich34 and the German electrical engineer K.W.Wagner 35 to justify Heaviside's work. Although each was ignorant of the other 's work, both used function theory. Wagner concentrated on the expansion formula, whereas Bromwich gave a broader explanation of the operational calculus. Having given a brief history of transform methods, let us explore the relationship of transform methods to the traditional methods of separation of variables and numerical methods by studying the sound waves that arise when a sphere of radius a begins to pulsate at time t=0. The symmetric wave equation in spherical coordinates is (1.8.1) where c is the speed of sound, u(r, t) is the velocity potential and - ∂u/ ∂r gives the velocity of the parcel of air. At the surface of the sphere r=a, the radial velocity must equal the velocity of the pulsating sphere (1.8.2) where (t), the displacement of the surface of the pulsating sphere, equals B sin(ωt)H(t). The air is initially at rest. 31 Cooper, J.L.B., 1952: Heaviside and the operational calculus. Math. Gaz., 36, 5--19; Lützen, J., 1979: Heaviside's operational calculus and the attempts to rigorise it. Arch. Hist. Exact Sci., 21, 161--200; Deakin, M.A.B., 1981: The development of the Laplace transform, 1737--1937: I.Euler to Spitzer, 1737--1880. Arch. Hist. Exact Sci., 25, 343--390; Deakin, M.A.B., 1982: The development of the Laplace transform, 1737--1937: II. Poincaré to Doetsch, 1880-- 1937. Arch. Hist. Exact Sci., 26, 351--381; Petrova, S.S., 1986: Heaviside and the development of the symbolic calculus. Arch. Hist. Exact Sci., 37, 1--23; Deakin, M.A.B., 1992: The ascendancy of the Laplace transform and how it came about. Arch. Hist. Exact Sci., 44, 265--286. 32 Heaviside, O., 1893: On operators in physical mathematics. Proc. R. Soc. London, Ser. A, 52, 504--529. 33 Nahin, P., 1988: Oliver Heaviside: Sage in Solitude. IEEE Press, Chapter 10. 34 Bromwich, T.J.I'a., 1916: Normal coordinates in dynamical systems. Proc. London Math. Soc., Ser. 2, 15, 401--448. 35 Wagner, K.W., 1915/16: Über eine Formel von Heaviside zur Berechnung von Einschaltvorgängen. Arch. Electrotechnik, 4, 159--193.
Transform Methods for Solving Partial Differential Equations 60 differential equations to find the Laplace transform U(r, s). Finally, we can use the same inversion techniques as those employed in single variable problems since the independent variable r acts as a parameter. With the straightforward nature of this procedure, a natural question is "Why isn't this technique used for all linear partial differential equations?" There are essentially three difficulties. First, taking the Laplace transform of the partial differential equation may be difficult. For example, for most partial differential equations with nonconstant coefficients, we do not know how to take their transform. Second, having taken the Laplace transform of the partial differential equation, we may be unable to solve the resulting ordinary differential equation. Finally, even if we can solve the differential equation, we may be unable to invert the transform analytically. Of these three difficulties, inversion seems the greatest stumbling block, probably because transform methods would be immediately abandoned if we could not find analytically the transform of the dependent variable. For this reason, we will spend considerable time on various inversion techniques. First, we will apply integration techniques on the complex plane to evaluate the inversion integral. Indeed, we shall do this so often that this book will read like a book on applied complex variables.36 When this technique fails, we still have the option of inverting the transform using asymptotic or numerical methods. Using numerical methods for only the inversion is preferable to using numerical methods to solve the entire problem since we still have an analytic solution in the other independent variable. In addition to Laplace transforms, we may also use Fourier transforms to solve partial differential equations. Again, the most difficult aspect of the analysis is the inversion and, again, we may apply complex variables and numerical methods to find its inverse. We illustrate this in Chapter 3 for single-valued transforms while we treat multivalued transforms in Chapter 5. 36 "Had Heaviside been able to make full use of Cauchy's method of complex integration, then (to quote a well-known saying) 'we should have learned something'." Quote taken from Bromwich, T.J.I'a., 1928: Note on Prof. Carslaw paper. Math. Gaz., 14, p. 227.
61 Chapter 2 Methods Involving Single-Valued Laplace Transforms Having given an overview of transform techniques, we begin with this chapter to examine in greater detail how Laplace transforms may be used to solve linear partial differential equations. We limit ourselves to the situation where the transform does not contain a branch point or cut. We address that question in Chapter 4. 2.1 INVERSION OF LAPLACE TRANSFORMS BY CONTOUR INTEGRATION Laplace transforms are a popular tool for solving initial-value problems. In most undergraduate courses, the use of tables, special theorems, partial fractions and convolution are the methods taught for finding the inverse. In most problems involving partial differential equations, these techniques fail us. In 1916, Bromwich1 showed that we can express the inverse of a Laplace transform (Bromwich's integral) as the contour integral (2.1.1) 1 Bromwich, T.J. I'a., 1916: Normal coordinates in dynamical systems. Proc. London Math. Soc., Ser. 2, 15, 401--448.
Methods Involving Single-Valued Laplace Transforms 63 where a is real. From Bromwich's integral, (2.1.8) Here c is greater than the real part of any of the singularities in Equation 2.1.8. Our first task in the inversion of F(s) is the classification of the singularities of the integrand of Equation 2.1.8. Using the infinite product for the hyperbolic sine,2 (2.1.9) Thus, we have a second-order pole at z=0 and simple poles at zn=±n i/a, where n=1, 2, 3,.... Figure 2.1.1: An outstanding mathematician at Cambridge University at the turn of the twentieth century, Thomas John I'Anson Bromwich (1875--1929) came to Heaviside's operational calculus through his interest in divergent series. Beginning a correspondence with Heaviside, Bromwich was able to justify operational calculus through the use of contour integrals by 1915. After his premature death, individuals such as J.R.Carson and Sir H. Jeffreys brought Laplace transforms to the increasing attention of scientists and engineers. (Portrait courtesy of ©The Royal Society.) 2Gradshteyn, I.S., and I.M.Ryzhik, 1965: Table of Integrals, Series and Products. Academic Press, Section 1.431, Formula 2.
Transform Methods for Solving Partial Differential Equations 64 We can convert the line integral in Equation 2.1.8, with the Bromwich contour lying parallel and slightly to the right of the imaginary axis, into a closed contour using Jordan's lemma through the addition of an infinite semicircle joining i to -i as shown in Figure 2.1.2. We now apply the residue theorem. For the second-order pole at z=0, (2.1.10) (2.1.11) (2.1.12) (2.1.13) after using sinh(az)=az+O(z3). For the simple poles zn=±nπi/a, (2.1.14) (2.1.15) (2.1.16) Figure 2.1.2: Contours used in the inversion of Laplace transform given by Equation 2.1.7.
Methods Involving Single-Valued Laplace Transforms 65 because cosh(±n i)=cos(n )=(-1)n. Thus, summing the residues gives (2.1.17) (2.1.18) Figure 2.1.3 illustrates this inverse at various times t. • Example 2.1.2 For our second example, we invert (2.1.19) where q=s1/2/a, and the constants a, L and x are real. One immediate concern is the presence of s1/2 because this is a multivalued function. However, when we replace the hyperbolic cosine and sine functions with their Taylor expansions, F(s) contains only powers of s and is, in fact, single-valued. From Bromwich's integral, (2.1.20) where q=z1/2/a. Using the Taylor expansions for the hyperbolic cosine and sine, we find that z=0 is a second-order pole. The remaining poles are located where sinh(qL)=-i sin(iqL)=0. Therefore, or zn=- n2 2a2/L2, where n=1, 2, 3,.... We have chosen the positive sign because Figure 2.1.3: The inverse of the Laplace transform given by Equation 2.1.7.
Transform Methods for Solving Partial Differential Equations 66 z1/2 must be single-valued; a negative sign would lead to the same result. Further analysis reveals that these poles are simple. Having classified the poles, we now close the line contour which lies slightly to the right of the imaginary axis with an infinite semicircle in the left half-plane and use the residue theorem. See Figure 2.1.4. The values of the residues are (2.1.21) (2.1.22) (2.1.23) (2.1.24) (2.1.25) and (2.1.26) Figure 2.1.4: Contour used in the inversion of Laplace transform given by Equation 2.1.19.
Methods Involving Single-Valued Laplace Transforms 67 (2.1.27) (2.1.28) (2.1.29) Summing the residues, (2.1.30) Figure 2.1.5 illustrates this inverse at various times t and values of x. • Example 2.1.3 To illustrate inversion by Bromwich's integral when Bessel functions are present, consider the transform 3 (2.1.31) The power series representations for I0(z) and I1(z) are (2.1.32) Figure 2.1.5: The inverse of the Laplace transform given by Equation 2.1.19. 3 Taken from Raval, U., 1972: Quasi-static transient response of a covered permeable inhomogeneous cylinder to a line current source. Pure Appl. Geophys., 96, 140--156. Published by Birkhäuser Verlag, Basel, Switzerland.
Methods Involving Single-Valued Laplace Transforms 69 where a and r are real, positive. One of the integral representations of K0(.) is (2.1.42) Therefore, (2.1.43) (2.1.44) (2.1.45) (2.1.46) because . • Example 2.1.5 In the previous example, we used an integral representation of the modified Bessel function K0(.) to reexpress the transform so that we could take its inverse even though we had to write it as an integral. See Equation 2.1.45. Then we carried out the integration and obtained Equation 2.1.46. Yang, Latychev and Edwards 4 used a similar trick to invert the transform (2.1.47) Figure 2.1.6: The inverse of the Laplace transform given by Equation 2.1.31. 4 Yang, J.-W., K.Latychev, and R.N.Edwards, 1998: Numerical computation of hydrothermal fluid circulation in fractured Earth structures. Geophys. J. Int., 135, 627-- 649. Published by Blackwell Publishing.
Transform Methods for Solving Partial Differential Equations 70 where a, b and c are constants with a>0. In Chapter 4, we will show how to deal with Laplace transforms that contain multivalued functions. Even then, Equation 2.1.47 is tricky because it contains a square root of a square root. Let's see how Yang et al. solved this problem. They began by noting that (2.1.48) We may view Equation 2.1.48 as an integral representation of e-2r. Consequently, we can reexpress Equation 2.1.47 as (2.1.49) (2.1.50) Using the linearity property of Laplace transforms, (2.1.51) Because (2.1.52) and (2.1.53) we obtain the final result that (2.1.54) Figure 2.1.7 illustrates this inverse at various times t. • Example 2.1.6 Let us find the inverse5 of the Laplace transform (2.1.55) 5 Reprinted from Int. J. Solids Struct., 16, T.C. T. Ting, The effects of dispersion and dissipation on wave propagation in viscoelastic layered composities, pp. 903--911, ©1980, with kind permission from Pergamon Press Ltd., Headington Hill Hall, Oxford OX3 0BW, Appl. Mech., 19, 209--213. UK. See also Cole, J.D., and T.Y. Wu, 1952: Heat conduction in a compressible fluid. J.
Methods Involving Single-Valued Laplace Transforms 71 or (2.1.56) We begin by noting that (2.1.57) so that (2.1.58) To evaluate the first integral in Equation 2.1.58, we now deform the original Bromwich integral to the contour shown in Figure 2.1.8. The contribution from the integrals along the arcs AB and EF vanish as R→ . Along BC, z=re- i/3 and dz=dr e- i/3, while z=re i/3 and dz=dr e i/3 along DE. Then, (2.1.59) (2.1.60) Figure 2.1.7: The inverse of the Laplace transform given by Equation 2.1.47.
Transform Methods for Solving Partial Differential Equations 72 and (2.1.61) Combining Equation 2.1.59 through Equation 2.1.61, (2.1.62) (2.1.63) Consider now the integral (2.1.64) where z= - . If we set b= 2- , then (2.1.65) Figure 2.1.8: Contour used in the inversion of the Laplace transform given by Equation 2.1.55.
Methods Involving Single-Valued Laplace Transforms 73 (2.1.66) (2.1.67) where Ai( ) is the Airy function of the first kind. Therefore, the inverse is (2.1.68) In this form the inverse is now more amenable to physical interpretation or further asymptotic analysis. Figure 2.1.9 illustrates this inverse for 0≤ ≤1. Problems For the following transforms, use the inversion integral to find the inverse Laplace transform for constant a, M, R and . Figure 2.1.9: The inverse of the Laplace transform given by Equation 2.1.55.
Transform Methods for Solving Partial Differential Equations 74 11. Show that the inverse6 of the Laplace transform is where n is the nth root of tan( )=-b , b=m/ , and and m are real. This inverse is illustrated in the figure labeled Problem 11. 12. Show that the inverse of the Laplace transform is Problem 11 6 Taken from Arutunyan, N.H., 1949: On the research of statically indeterminate systems with vibrating support columns (in Russian). Prikl. Mat. Mek., 13, 399--500.
Methods Involving Single-Valued Laplace Transforms 75 where n is the nth positive root of J0( a)=0. This inverse is illustrated in the figure labeled Problem 12. 13. Show that the inverse7 of the Laplace transform is where n is the nth positive root of J0( a)=0. This inverse is illustrated in the figure labeled Problem 13. Problem 12 Problem 13 7 Taken from Wadhawan, M.C., 1974: Dynamic thermoelastic response of a cylinder. Pure Appl. Geophys., 112, 73--82. Published by Birkhäuser Verlag, Basel, Switzerland.
Transform Methods for Solving Partial Differential Equations 76 Problem 14 14. Show that the inverse of the Laplace transform is where n is the nth positive root of J1( )=0. This inverse is illustrated in the figure labeled Problem 14 with a=2. 15. Show that the inverse of the Laplace transform is where m is the mth positive root of Jn+1( )=0. This inverse is illustrated in the figure labeled Problem 15 with a/b=0.8. 16. Find the inverse 8 of the Laplace transform This transform has a simple pole at s=0 and an infinite number of essential singularities at sn=-(2n -1)2 2/4, where n=1, 2, 3,.... The most convenient 8 Taken from Roshal', A.A., 1969: Mass transfer in a two-layer porous medium. J. Appl. Mech. Tech. Phys., 10, 551--558.
Methods Involving Single-Valued Laplace Transforms 77 This inverse is illustrated in the figure labeled Problem 16. 17. Consider a function f(t) which has the Laplace transform F(z) which is analytic in the half-plane Re(z)>s0. Can we use this knowledge to find g(t) whose Laplace transform G(z) equals F[ (z)], where (z) is also analytic for Re(z)>s0? The answer to this question leads to the Schouten 9-Van der Pol10 theorem. Step 1: Show that the following relationships hold true: and Step 2: Using the results from Step 1, show that Problem 15 method for finding the inverse is to deform Bromwich's contour along the imaginary axis of the s-plane, except for an infinitesimally small semicircle around the simple pole. If we use this contour, show that 9 Schouten, J.P., 1935: A new theorem in operational calculus together with an application of it. Physica, 2, 75--80. 10 Van der Pol, B., 1934: A theorem on electrical networks with applications to filters. Physica, 1, 521--530.
Transform Methods for Solving Partial Differential Equations 80 Taking the Laplace transform of Equation 2.2.5 and Equation 2.2.7 and substituting the initial condition, we obtain (2.2.8) with the boundary conditions (2.2.9) The solution that satisfies this boundary-value problem is (2.2.10) Using tables, the inverse of the Laplace transform given by Equation 2.2.10 is (2.2.11) Figure 2.2.1 illustrates Equation 2.2.11 as a function of distance x and time t. • Example 2.2.2 Several slight modifications to Example 2.2.1 yield a more challenging problem, namely, (2.2.12) Figure 2.2.1: A plot of Equation 2.2.11 as a function of distance x and time t.
Methods Involving Single-Valued Laplace Transforms 81 with the initial condition (2.2.13) and boundary conditions (2.2.14) Taking the Laplace transform of Equation 2.2.12 and Equation 2.2.14 and substituting the initial condition, we obtain the boundary-value problem (2.2.15) with (2.2.16) The solution that satisfies this boundary-value problem is (2.2.17) Using tables, the first shifting theorem and convolution, the inverse of the Laplace transform given by Equation 2.2.17 is (2.2.18) (2.2.19) where =x2/(4 2). An alternative method for inverting Equation 2.2.17 is to apply the technique shown in Example 2.1.5. Using Equation 2.1.48 to replace the exponential in Equation 2.2.17, we obtain (2.2.20) Applying the second shifting theorem, (2.2.21) Eliminating the Heaviside function from Equation 2.2.21 leads directly to Equation 2.2.19.
Transform Methods for Solving Partial Differential Equations 82 • Example 2.2.3 To illustrate how the Laplace transform technique applies to problems over a finite domain, we solve a heat conduction problem11 in a plane slab of thickness 2L. Initially the slab has a constant temperature of unity. For 0<t, we allow both faces of the slab to radiatively cool in a medium which has a temperature of zero. If u(x, t) denotes the temperature, a2 is the thermal diffusivity, h is the relative emissivity, t is the time, and x is the distance perpendicular to the face of the slab and measured from the middle of the slab, then the governing equation is (2.2.22) with the initial condition (2.2.23) and boundary conditions (2.2.24) Taking the Laplace transform of Equation 2.2.22 and substituting the initial condition, (2.2.25) If we set s=a2q2, Equation 2.2.25 becomes (2.2.26) From the boundary conditions, U(x, s) is an even function in x and we may conveniently write the solution as (2.2.27) From Equation 2.2.24, (2.2.28) and (2.2.29) 11 Goldstein, S., 1932: The application of Heaviside's operational method to the solution of a problem in heat conduction. Z.Angew. Math. Mech., 12, 234--243.
Transform Methods for Solving Partial Differential Equations 84 or (2.2.40) We can further simplify Equation 2.2.40 by using h/ n=tan( nL) and hL= nL tan( nL). Substituting these relationships into Equation 2.2.40 and simplifying, (2.2.41) Figure 2.2.2 illustrates Equation 2.2.41. • Example 2.2.4 For this example, we use Laplace transforms to solve a partial differential equation in cylindrical coordinates. It differs from our previous problems in its use of Bessel functions. Let us solve (2.2.42) subject to the boundary conditions that u(0, t)=0 and u(1, t)=1, and the initial condition that u(r, 0)=0. Introducing the Laplace transform of u(r, t), (2.2.43) Figure 2.2.2: The temperature within a slab 0<x/L<1 at various times a2t/L2 if the faces of the slab radiate to free space at temperature zero and the slab initially has the temperature 1. The parameter hL=1.
Transform Methods for Solving Partial Differential Equations 86 because , I0(-i n)=J0( n) and I2(-i n)=-J2( n). Furthermore, we have J2( n)=-J0( n) because J2( n)+J0( n)=2J1( n)/ n=0. Consequently, upon summing the residues, the final solution is (2.2.53) where J1( n)=0 with n=1, 2, 3,.... Figure 2.2.3 illustrates Equation 2.2.53 as a function of distance r and time t. • Example 2.2.5 Let us solve the heat equation within an infinitely long cylindrical shell (2.2.54) where the interior surface is maintained at a constant temperature, u(a, t)=1, while the outside surface is kept at zero, u(b, t)=0. Initially, the shell has the temperature of zero, u(r, 0)=0. We begin by taking the Laplace transform of Equation 2.2.54 and find that (2.2.55) with U(a, s)=1/s and U(b, s)=0. The general solution to Equation 2.2.55 is (2.2.56) Figure 2.2.3: A plot of u(r, t) given by Equation 2.2.53 as a function of distance r and time t.
Transform Methods for Solving Partial Differential Equations 88 where we have used the properties that , , I0(z)K1(z)+I1(z)K0(z)=1/z, I0(xi)=J0(x) and K0(xi)=- i[J0(x)-iY0(x)]/2. Summing the residues leads to (2.2.64) This solution is illustrated in Figure 2.2.4 when b/a=4. • Example 2.2.6: Moving boundary In the previous examples, we showed that Laplace transforms are particularly useful when the boundary conditions are time dependent.12 Consider now the case when one of the boundaries is moving. We wish to solve the heat equation (2.2.65) subject to the boundary conditions (2.2.66) Figure 2.2.4: The temperature within a cylindrical shell a<r<b with b/a=4 at various times t if the inner surface is held at the temperature of 1 and outer surface has the temperature 0. Initially, the shell has the temperature of 0. 12 Taken from Redozubov, D.B., 1960: The solution of linear thermal problems with a uniformly moving boundary in a semiinfinite region. Sov. Phys. Tech. Phys., 5, 570--574.
Methods Involving Single-Valued Laplace Transforms 91 where q2=s/a2. Finally, the coefficients C1 and C2 are chosen so that the general solution satisfies the boundary conditions given by Equation 2.2.82, yielding (2.2.86) Using Bromwich's integral, Equation 2.2.86 may be inverted to yield (2.2.87) where kn denotes the nth root of J0(kb)=0. This solution is illustrated in Figure 2.2.5 when cb2=0.5. An alternative to the method of undetermined coefficients consists of expanding the nonhomogeneous term as a Fourier-Bessel expansion: 13 (2.2.88) where kn is again the nth root of J0(kb)=0, and (2.2.89) Figure 2.2.5: The temperature within a cylinder 0≤r<b at various times t if the surface is held at the temperature of 0 and the initial temperature distribution is 1-cr2. Here we have chosen cb2=0.5. 13 If you are not familiar with Fourier-Bessel series, see Duffy, D.G., 2003: Advanced Engineering Mathematics with MATLAB. Chapman & Hall/CRC, 818 pp. See Section 9.5.


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during the present war. The. of diabetes is perhaps of greater interest than the disappearance of obesity, of gout, and of chronic constipation noted during the same period and the increasing rarity of eclampsia in pregnant women. The balance is not equalised by the alleged increase in the frequency of peptic ulcer. Any observations of a similar character made in this country would be of interest and might throw useful light on the relation between habit and metabolic disorder. enemy countries cure
MUCH stress has been put, and rightly, on war diseases. That way progress has lain. But there have also been war cures, and of these we have heard less. The disappearance of functional nervous disease among the civil population during the last four years is, however, a matter of general comment. In Berlin we understand that migraine BRONCHO-SPIROCHÆTOSIS. disappeared completely from the month of August, THANKS to the work of G. A. Lurie, Galli- Valerio/ 1914, and, without doubt, the work of general H. and the subject of broncho-spirochsetosis Violle,3 in this has been country mitigated practitioners by the fact that they have seen fewer minor ail- has received considerable light. The condition - ments. Many of these, while real enough, are until recently was only of interest to the tropical such as the patient communicates only to his or practitioner, but lately it has become a serious her (generally her) family doctor, and the particular matter to medical men in Europe, cases of the family doctor has generally been away. Minor malady having been reported from Serbia and the The ailments cannot be assessed in actual figures, but Balkanic zone, Switzerland, and France. there is a certain amount of evidence accumulat- condition and its etiological agent were first ing in regard to definite morbid conditions described in Ceylon in 1905 by Castellani, who which have become numerically less frequent later named the causal agent SpoctsbfOMCHs. during the war. Dr. Herbert Elias and Dr. His researches were confirmed by S. Branch in Richard Singer laid before the Vienna Medical 1907 in the West Indies, by Jackson in the same 1 Society on Oct. 18th last a therapeutic study year in the Philippine Islands, and by several of the influence of war diet upon diabetes. Their other observers, whilst H. G. Waters in 1909 The conconclusion is based upon figures, and it is recorded numerous cases from India. later found in dition was All a remarkable one. practically every degrees of diabetes as it occurred in Vienna were favourably influenced part of the tropics, of special importance by the altered circumstances of war, in males being the work of Chalmers and O'Farrell, who almost without exception, in females frequently succeeded in reproducing the disease in monkeys, but by no means universally. The chart which we and the thorough investigation of Fantham on the morphology of the organism. Spirochaeta bronchi1&bgr;12,1,f I iOtA/I!:'.. I 1Ulh.1'-=. I 1UII.t'7 I 1017/1Q alis, according to the description given by all these authors, is so markedly polymorphic that the suggestion arises whether the term does not cover more than one species or variety of spirochsete. The classical investigation of Fantham4 seems to prove, however, that the various forms found belong, in reality, to only one species, differing from the spirochaates occurring commonly in the mouth. Fantham discovered the coccoid stage of the parasite, which he believes to be of fundamental significance in the dissemination of the malady. Clinically, three types of the condition are to be distinguished-the acute, the subacute, and the The acute chronic. develops abruptly ; reproduce here shows the decreasing number of the patient complains type of headache and rheudiabetics dying in the Vienna General Hospital matoid pains all over the body; he feels chilly during the last five years, the thick line indicating and coughs much, though there is but very scanty total deaths, the interrupted line the deaths occuror mucoid muco-purulent expectoration. The ring in coma. The authors express regret that they lasts from 3-6 days. This influenzagenerally cannot give the result in percentages, inasmuch as like of type broncho-spirochaetosis has given rise to the total diabetic admissions to hospital could not a suggestion that true influenza may be a form be ascertained, but of the general lesson to be of spirochoetosis. The subacute and c7wonic types drawn from the chart there appears to be no doubt. of broncho-spirochaetosis are of practical relevance, The better war prognosis in males as distinguished as they may closely simulate pulmonary tuberfrom that in females is a fact claimed by these culosis. In these types the patient often spits up observers as a new observation, coinciding with and this, together with wasting, an evening blood, their pre-war experience, for which they venture to rise of temperature, and, on examination, the preadvance no explanation. In the case of patients with sence of of dullness and crepitations, may slight glycosuria the difference between war and pre- suggest patches a diagnosis of phthisis. In a certain war experience was very striking. Whereas not one number of cases, however, the general condition of 29 slight diabetics before the war could be regarded remains good. The first patient seen by Castellani as cured, 33 out of 39 became sugar-free under war in Ceylon in 1905 was still alive in 1915, and Violle conditions. The experience -of Bouchardat during a number of cases in which the general the siege of Paris in 1870-71 is quoted as having quotes 1 Journal of been similar. G. Klemperer, P. F. Richter, and Tropical Medicine, Dec. 1st, 1915. 2 fur Schweizer-Aerzte, February, 1917. Correspondenzblatt others have noted the increased carbohydrate 3 Presse Médicale, July 11th, 1918; THE LANCET, 1918, ii., 775. 4 Annals of Parasitology, July. 1915. 1 Wiener klin. Wochenschr., Nov. 21st, 1918. 5 De Verbizier: Bull. Académie de Médecine, Séance 8 Octobre, 1918. ____
Report "WAR CURES."

redpajama_set_name: RedPajamaCommonCrawl


text:
Compelling Calgarians: Hayley Wickenheiser
Rita Mingo, For the Calgary Herald
Updated: January 2, 2019 4:00 AM PST
Four-time Olympic gold medallist Hayley Wickenheiser likes to stay busy. Among the greatest women's hockey players ever is working with the Toronto Maple Leafs in player development, attending medical school and raising her 18-year-old son. Jeff McIntosh / The Canadian Press
There don't seem to be enough hours in a day for Hayley Wickenheiser. Yet, in true high-performance fashion, she manages it all.
Wickenheiser, one of the greatest women's hockey players ever, has put her playing days behind her but remains entrenched in the sport.
In November at Canada Olympic Park, the ninth-annual Wickenheiser World Female Hockey Festival — known as WickFest — gave girls the opportunity to not only play but also take part in quality presentations.
"I'm really proud of it," said Wickenheiser, winner of four Olympic gold medals. "We had 110 teams this year in Calgary and we'll have another 15 in Surrey (B.C.) in January. We're talking about expanding it across North America.
"It's not a money-maker for anyone — we inject about $4 million a year into the Calgary economy — but it's great for young girls. They get access to Olympic athletes, to NHL players, the Stanley Cup was here, the prime minister was here, the best of the best in terms of nutrition and education seminars. I think it's becoming the go-to tournament."
In August, Wickenheiser was hired by the Toronto Maple Leafs as assistant director of player development. The flexibility of the job is perfect for her schedule.
"I can be working in Calgary with a Hitmen prospect drafted by the Leafs, to on the ice with the Marlies, on the ice with Leafs players rehabbing," she said. "I might be watching video, mainly the development of up-and-coming players. It really spans a lot of areas. . . . It was a position I couldn't say no to."
And she's currently attending medical school, which she will finish in a little over two years.
"I'm used to being really busy," said Wickenheiser, mother to 18-year-old Noah. "I think as an athlete, one thing you understand early is time management. There will always be adventures and new projects and fun things to create and do.
"That's the way I like it."
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Kinerase is a fairly new anti aging skin care product featuring an unusual active ingredient called N6-furfuryladenine. The manufacturers of Kinerase have invested a great deal of money in advertising and celebrity sponsorship. But how does Kinerase compare? Read on to find out.
A 1.4 ounce tube of Kinerase cream costs about $65.00.
Kinerase's main active ingredient is N6-furfuryladenine. As far as we have been able to determine, there's no research currently available on this particular compound (update: there has indeed been some research on N6-furfuryladenine, also known as kinetin - learn more). The manufacturers of Kinerase tell us that N6-furfuryladenine is a naturally occurring compound found in plant leaves that keep the leaves from drying out and from aging.
N6-furfuryladenine is said to be an effective moisturizer. Compare to hyaluronic acid, which is an incredibly powerful moisturizer and has a proven track record as an anti wrinkle ingredient. Recent research tentatively indicates that N6-furfuryladenine (kinetin) is a powerful natural antioxidant like CoQ10. Will the makers of other anti wrinkle creams follow Kinerase's lead and begin to use kinetin in their own formulations? Perhaps so, after there is more of an established body of research.
Kinerase also contains botanical compounds to help soothe the skin.
Clean and dry the skin. Apply a dime-sized drop of Kinerase cream to the fingertips. Rub the cream thoroughly into the skin. Apply morning and night for best results. Kinerase may be used on the skin of the face, neck, hands or chest.
Kinerase is an interesting product. Most anti wrinkle products include compounds that have been researched, at least tentatively, and therefore their effectiveness is more predictable.
Considering the lack of information available on N6-furfuryladenine, we recommend products that contain proven anti wrinkle and anti aging ingredients over Kinerase.
Update: some research has been performed on the chemistry and functions of N6-furfuryladenine, also known as kinetin.
Kinerase® is a registered trademark Valeant Pharmaceuticals International.


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text:
Flertalsdiktatur er en nedsættende betegnelse, som bruges om en beslutningsprocedure, der kun fuldt opfylder det ene krav til demokratisk procedure, at der er foregået en afstemning og en flertalsafgørelse. Udtrykket anvendtes vist nok første gang den 28. december 1936 af Jyllandsposten.

Theodor Geiger

Udtrykket toges op i 1937 af Theodor Geiger i artiklen "Om Forholdet mellem Liberalisme og Demokrati" i Nationaløkonomisk Tidsskrift, hvor han blandt andet skriver:
"det væsentlige er, at troen på saglig diskussion som kilde til en viljes syntese danner grundlag. Denne franske oplysningstanke synes mig at være utopisk. Der findes ikke nogen praktisk-politisk sandhed, som hele demos kan blive enige om; der findes kun interesseprægede offentlige meninger, repræsenteret af partierne. Indtager man denne pessimistiske holdning, så er parlamentarismen — fra et liberalt synspunkt — sikkert ikke ideal, men den eneste praktiske mulighed, sålænge der ikke er opfundet noget bedre. Det gør ikke noget, at der i virkeligheden ikke foregår nogen saglig diskussion; det vilde være nok, hvis man opgav det hykleriske krav om, at hvert parlamentsmedlem skal føle sig som folkets, ikke som sit partis repræsentant. Det er parlamentarismens ærliggørelse, ikke dens afskaffelse, vi trænger til. Partierne, som talerør for konkurrerende magtbestræbelser skal slås eller forliges — helst det sidste. — I nyere tid har der tit været tale om flertalsdiktatur; et fjollet og uærligt ord, som kun dårligt skjuler mindretallets afmægtige over — at være mindretal. Den politiske kurs efter flertallets vilje er ikke ideel, men i hvert fald bedre end en politik efter et fåtals ønsker. I øvrigt kan saglig diskussion ikke hidføre andet end et kompromis; det samme gør magtkonkurrencen på sin måde. Et flertal udnytter aldrig sin magt til bristepunktet, men tager af taktiske grunde på forhånd et hensyn til oppositionen. Og her får parlamentet — bortset fra det mere saglige kommissionsarbejde — en demokratisk værdi som scene for en "skindiskussion"; Regering og partier er nødt til at forsvare deres magtpolitiske og interesseprægede standpunkter med fornuftgrunde".
Af formuleringen ses, at Geiger anså udtrykket ikke som en afvigelse fra demokratiets principper men som et udtryk opfundet af det frustrerede mindretal, der ikke vil godkende flertallets afgørelse. Senere, i 1947, udvikler han denne betragtning derhen, at den gruppe, der den ene dag udgør flertallet, den næste kan komme i mindretal. Denne svingning mellem ulige flertal sikrer, at flertallet derfor vil tage hensyn til mindretallet "netop fordi flertallet ikke kan påregne at holde stillingen i al evighed. Flertallets hensyntagen til mindretallet er her ikke blot et moralsk spørgsmål, men garanteret gennem selve den demokratisk-parlamentariske mekanisme".

Hal Koch

Ganske anderledes så Hal Koch på forholdet. I sin bog "Hvad er demokrati?" fra 1945 gennemdrøfter han – uden ligefrem at anvende udtrykket – forskellen mellem demokrati og flertallets diktatur. Om demokratiet skriver han blandt andet:
"Dets væsen beror på dette, at man mødes og tales ved, at man gennem samtalen når til en bedre og rimeligere forståelse og derudfra formår at træffe en afgørelse, som ikke alene tjener en enkelt person eller klasse, men som tager rimeligt hensyn til helheden",
og han tilføjer:
"Ofte har man – til skade for demokratiets anseelse betragtet selve afstemningen som det væsentlige. Var en ting blot vedtaget ved afstemning, var der ingen tvivl om dens berettigelse. Så var demokratisk set alt i orden. Så let går det imidlertid ikke. Ellers ville der ikke – demokratisk set – være noget at bebrejde nazismen i Tyskland, og der findes adskillige stater, hvor enhver afstemning, hvis den da overhovedet er mulig, ville medføre et rent autoritært styre. Værre er det imidlertid, at man ud fra denne overvurdering af "afstemningen" næsten med indre nødvendighed tvinges over i den konsekvens, som mere end noget andet har bidraget til at bringe demokratiet i vanry, nemlig den påstand, at det altid er flertallet, som har ret. For enhver, der tænker sig lidt om, er dette jo en uhyrlighed".

Og sluttelig giver han denne rammende skildring af flertalsdiktaturet:
"Der findes et sted her i landet et sogneråd, i hvilket der er to grupper, en flertalgruppe på 7 medlemmer og en mindretalsgruppe på 4 medlemmer. Flertallet er således solidt, og der er ingen grund til at tro, at det skulle kunne rokkes i de første mange år. I møderne går det meget ofte således til, at flertallet kort og knapt fremsætter det forslag, som man i gruppen er blevet enig om. Derefter får oppositionen ordet. Det er jo en demokratisk forsamling! Den afgiver et langt og velbegrundet indlæg, som viser vanskelighederne ved sagen. Under dette indlæg kan ingen se, om flertalsgruppens medlemmer hører efter eller sover, men vi må vel gå ud fra, at de gør det første. Når oppositionen har fremsat sine indvendinger, kan det ske, at formanden eller en anden siger et par ord – gerne temmelig hvasse ord. Men man er sjælden nået ret langt, før flertalsgruppen foreslår, at man går over til afstemning. En videre drøftelse finder man som regel ikke nødvendig. Det er det heller ikke. For flertallet har afgjort sagen i forvejen og bryder sig ikke en døjt om modparten eller dens synspunkter. Men mødet slutter altid med en god demokratisk afstemning, hvor stemmerne mærkelig nok falder 7 mod 4, hvorefter man går hver til sit, overbevist om at have gjort et godt demokratisk arbejde. Har ordet ikke været frit, er sagen ikke blevet afgjort ved afstemning, og har flertallet ikke fået lov at bestemme? Hvad vil man mere?"

Erling Olsen

Senere blev udtrykket anvendt af Erling Olsen i artiklen "Interessegrupperne og den økonomiske politik" i Nationaløkonomisk Tidsskrift 1959, hvor han i en drøftelse af ulige interessegrupper og deres mulige indflydelse på den førte økonomiske politik blandt andet skriver:
"den økonomiske politik udformes til gavn for de mange og skade for de få, fordi de manges organisationer er mere slagkraftige end de fås. Muligheden for flertalsdiktatur er imidlertid en systematisk fejl ved demokratiet. Den kan rettes ved mindretalsgarantier".
Hos Erling Olsen betegner udtrykket således, som hos Hal Koch, en tilstand, hvor flertallet ikke tager hensyn til mindretallet. Og som Hal Koch er han tilhænger af tiltag for at modvirke dette.

1990-erne

Endnu senere er udtrykket anvendt af blandt andre Tim Knudsen (1993) og Hans Aage (1994) til beskrivelse af den tilstand, hvor et flertal "majoriserer" (det vil sige gennemtvinger sin vilje uden hensyn til mindretallet). For dem, som for Hal Koch, betegnes med udtrykket en tilstand, hvor flertallet dikterer sin vilje, og hvor ingen lytter til endsige tager hensyn til mindretal.

Litteratur

 Hal Koch: Hvad er demokrati? (Gyldendalske Boghandel 1945; 4. udgave, Haslev 1981);

Eksterne henvisninger

 Theodor Geiger: "Om Forholdet mellem Liberalisme og Demokrati" (Nationaløkonomisk Tidsskrift, 3. række, Bind 45; 1937)
 Theodor Geiger: "Demokratiet under Debat" (Nationaløkonomisk Tidsskrift, Bind 85; 1947)
 Søren Hansen: "Mellem demokrati og diktatur. Jyllands-Posten i 1930'erne" (Historie/Jyske Samlinger, Bind 1997; 1997)
 Tim Knudsen: "Det går nok – et essay om dansk statskultur og eurokratiet" (Politica, Bind 25; 1993)
 Erling Olsen: "Interessegrupperne og den økonomiske politik" (Nationaløkonomisk Tidsskrift, Bind 97; 1959)
 Hans Aage: "Demokrati i økonomisk belysning" (Nationaløkonomisk Tidsskrift, Bind 132; 1994)

Noter

Se også
 Demokrati
 Diktatur
 Demokratisk diktatur
 Populisme

Diktatur

redpajama_set_name: RedPajamaWikipedia


text:
Bipartisan legislation seeks to end Virginia medical malpractice damages limit
Published Sunday, Jan. 10, 2021, 10:07 am
(© BillionPhotos.com – stock.adobe.com)
Attorney Joseph Cammarata, State Sen. Bill Stanley and State Sen. Scott Surovell are leading efforts to Cancel the Cap on the recovery of damages in medical malpractice cases in Virginia and help victims receive complete justice.
Stanley introduced Senate Bill 1107, which "eliminates the cap on the recovery in actions against health care providers for medical malpractice where the act or acts of malpractice occurred on or after July 1, 2021."
Virginia's General Assembly will consider the bipartisan bill when the legislative session convenes on Jan. 13.
Cammarata, a nationally known civil trial lawyer and partner with Chaikin, Sherman, Cammarata & Siegel, was driven to pursue the initiative for personal reasons. In 2015 his longtime friend and neighbor, Nancy, underwent lower back surgery to fix her chronic back pain. After the surgeon negligently severed an artery in her stomach, she coded for 23 minutes, depriving her brain of oxygen.
Nancy suffered from severe and permanent cognitive, emotional and behavioral disabilities and now requires 24-hour care.
At trial, attorneys detailed Nancy's lifetime care needs and economic losses totaling $9.3 million. The jury returned a verdict for $35.6 million; however, current Virginia law places a limit on the amount of money medical malpractice victims can recover, limiting Nancy's recovery to $2.2 million.
Because Nancy's medical bills far exceed the cap, she effectively was allowed zero compensation.
"Medical malpractice insurance companies are happy with a system that imposes artificial limits on responsibility, but it's unfair to medical malpractice victims," Cammarata said. "If someone is seriously injured due to negligence, the victim cannot get complete financial relief for their harms and losses. This ultimately places a burden on taxpayers who pay for the victims' care through taxpayer-funded public health insurance."
Nancy's tragedy inspired Cammarata to help future medical malpractice victims in Virginia by changing a law he views as unjust. After hearing Nancy's story, Stanley agreed to be the chief patron of SB 1107, with Surovell supporting the bill as a co-patron.
In a special edition of Medical Malpractice Diaries, Cammarata discussed Cancel the Cap with host Deirdre Dickson-Gilbert of the National Medical Malpractice Advocacy Association. He said the proposed bill would allow seriously injured Virginians to be fairly and fully compensated for harm caused by the wrongdoer, ensure Virginia taxpayers don't have to foot the bill for malpractice, reduce the burden on the courts and public health insurance system, end the protection of medical providers from full accountability, and eliminate placing arbitrary values on human life.
"This is not a political issue; it is a human issue," he said. "Nancy will not benefit from this legislation, but its passing will allow future medical malpractice victims in Virginia to receive full justice."
To learn more about Cancel the Cap (SB 1107), visit chaikinandsherman.com/cancel-the-cap/.


text:
Home/News/'He Was the C.S. Lewis of Our Day': Remarks by Vice President Pence at the Memorial for Ravi Zacharias
'He Was the C.S. Lewis of Our Day': Remarks by Vice President Pence at the Memorial for Ravi Zacharias
Full Remarks by Vice President Mike Pence at the Memorial Service for Ravi Zacharias
THE VICE PRESIDENT: Pastor Giglio, Governor Kemp, Secretary Perdue, Secretary Carson, Senator Scott, honored guests, members of the RZIM team, to all those looking on across America and across the world, and especially to Margie, Sarah, Naomi, Nathan, and the entire Zacharias family: It is deeply humbling for my wife, Karen, and me to be here, among so many friends and distinguished leaders, to give thanks for the life and legacy of Ravi Zacharias.
The Bible says "the Lord is close to the brokenhearted," and that is our prayer for you on this day and in all the long days that lie ahead.
Yesterday, President Trump said, by "making the intellectual case for Christianity, Ravi Zacharias was instrumental in helping millions of people around the world come to know the love and mercy of Jesus Christ." And so on behalf of the First Family, our family, and a grateful nation, Karen and I offer our deepest sympathies to your family and to all, like us, who had the privilege to know and cherish this good man, this godly man, this man for all seasons.
On the death of Abner, King David said, "Do you not know that a prince and a great man has fallen in Israel this day?" Ravi Zacharias was such a man. And like so many gathered here, and so many looking on from afar, he was my friend. And as I had the privilege to share with Margie just a few short days ago, God put him at my side at just the right time, with a word of wisdom and kindness and encouragement that I'll never forget.
And so while it is my great privilege to be here today to speak on behalf of so many Americans, Karen and I mark his passing with a sense of personal loss as well. And we thank you for the privilege to be here.
And, of course, his passing comes at a time when so many families here and around the world are grieving the loss of loved ones who have succumbed amidst the coronavirus epidemic. As our nation begins the process of healing and recovering, we will miss Ravi's comforting voice and his insights as we pick up the pieces of our national life and move forward.
And during our last conversation from his hospital bed, not surprisingly, Ravi was more concerned for those who were struggling in the midst of this pandemic than he was for himself.
So I know Ravi would approve if, on this day, we also made a point to remember and pray for all who have endured heartbreak during these unprecedented times.
So for his family and families around America and around the world grieving the loss of a loved one, let's all pray that those who mourn will be comforted with the absolute assurance that he had: that the day will come when God will wipe away every tear from your eyes and theirs, and in every sense, in these challenging times, that He will heal our land.
Ravi Zacharias's story has been an inspiration to millions of people. On a bed of suicide at 17, void of needing, emptied of hope, he opened a Bible and he read the words that are now displayed on the wall outside this church: "Because I live, you shall live also." And lying on that hospital bed, he opened his heart and his life to Jesus Christ, now to leave no stone unturned in his pursuit of the truth. And, that, he surely did.
In the book of Isaiah, the word says, "I took you from the ends of the Earth. From its farthest corners, I called you." And so it was for Ravi Zacharias. Over the past five decades, called from that hospital bed, Ravi's voice and ministry impacted countless lives, transformed and strengthened people all over America and all over the world with the gospel of Jesus Christ.
Ravi Zacharias was a man of faith who could rightly handle the word of truth like few others in our time. And he will be missed. His travels would take him to stand in venues that were oftentimes unfriendly and unwelcoming. But he'd always answer the arguments of the skeptic or the cynic, giving the reason for the hope that he had, but doing so with gentleness and respect, recognizing at the other end of the question was always a person made in the image of God.
Ravi said, and I quote, "I have reminded myself over the years to never forget that behind every question is a questioner. And behind every questioner is a network of assumptions, hurts, struggles, and often prejudices." He spoke truth, he spoke with kindness and a deep and abiding care for every person who would listen.
On the death of Billy Graham, a little more than two years ago, Ravi remarked, and I quote, "A great voice has been lost, but the message goes on." And those two things are just as true today.
In the late Reverend Billy Graham, it's been observed that God gave us the greatest Evangelist of the 20th century. In Ravi Zacharias, God gave us the greatest Christian Apologist of this century. (Applause.)
He was the C.S. Lewis of our day. Crisscrossed the globe to every Mars Hill he could find, to answer skeptics, move obstacles of unbelief, armed with intellect, girded with truth and love.
And what he said on that day two years ago is just as true today: The work of RZIM will go on. (Applause.)
When giants fall, the impact is felt near and far. So it was in Atlanta, around America and around the world on May the 19th, when this gentle giant of the faith, on whose shoulders so many of us have leaned, was with us no more.
So we mourn with those who mourn, and we grieve with those who grieve, but we do not grieve like those who have no hope. Because by faith, we believe that Jesus lives. And by faith, we know that Ravi lives also. (Applause.)
Ravi Zacharias fought the good fight. He finished the race. He kept the faith and clung to the power of the cross in the empty grave. And now we know there is laid up for him a crown, which the Lord will reward to him on that day.
May God bless the memory of Ravi Zacharias. May God comfort his precious family as they mourn the passing of this good and great man. And may God grant us all the grace to walk forward with hearts full of gratitude for the life that we honor today: a life that, just a few short days ago — and a man who undoubtedly heard, "Well done, good and faithful servant."
Thank you, Ravi. And until we meet again, may God hold you in the palm of his hand. God bless you. (Applause.)
Day Lewis memorial pence president Ravi Remarks vice Zacharias
No pause for coronavirus, 'Soul Patrol' has stayed on track
Some camps going digital to reach students this summer


Guess what my FAVORITE sound in the whole world is? It's the sound of my kids laughing. You're probably the same way. So in addition to the regular awesomeness of Christmas morning with kids, throw in some good laughs and everyone will be so happy they won't know what to do with themselves.
These gifts will ensure that you get to hear that glorious sound of your children's laughter not only on Christmas morning, but for months beyond. Some of these choices are just plain silly. Some are cheap, and you could throw them in a stocking. Some are-- ahem-- a bit crude. You get the idea. Take a look below, then add some laughter to your Christmas morning.
1. Chattering Teeth - I know my girls would love these. They'd chomp chomp all over our coffee table with plenty of giggles in the background.
2. Snake in a Nut Can- I can't believe that they actually still make these! Any middle-school aged boy would be thrilled to own this classic gag.
3. Rubber Chicken - Can you imagine the look on your child's face if they were to pull a rubber chicken out of a gift bag? Or maybe have him peeking out of a stocking?
4. Captain Underpants - I'll admit that I've never read these. BUT I did teach middle school and I can tell you that early adolescent reluctant readers could not put these books down, and they always chuckled when they read them. C'mon-- his name is Captain Underpants!
5. The Book With No Pictures - This book is by B.J. Novak, who is most well-known for his role on The Office. There aren't many books that both kids and adults can genuinely laugh at together, but this book is one of them! It is downright funny, and I look forward to reading it to my girls. They laugh. I laugh. We laugh together.
6. Saturday Morning Cartoons: 1960s Vol. 1 - Children's cartoons are not like they used to be! These classics are hilarious. You may want to preview them first, to make sure that they're okay for your kids.
7. The Big Book of Laugh Out Loud Jokes for Kids - Kids of ALL ages love knowing and telling jokes. This book is a fantastic collection of all types of jokes, and you can choose some to teach your kids, or just read them to one another and get some chuckles.
8. Looney Toons: Golden Collection - This is the ultimate Looney Toons collection. This is the kind of stuff that I grew up on-- if you need a pick-me-up, there's nothing like watching someone get whacked with a frying pan! Again, these cartoons are not nearly as sanitized as today's cartoons are, so you may want to check them out yourself before showing them to your kids.
9. Blah! Button - Push the yellow button and it says what all of our kids want to say to us anyway, "Blah, Blah, Blah!" It rotates through several different voices so you never know what you're going to get.
10. Whoopee Cushions - Um. Yes, please.
11. Multi Voice Changer - This megaphone changes your child's voice into several different alternatives. Any child, from age 2 - 22 would love it!
12. Food Face Dish - This is just hilarious. I think that this plate is marketed towards young children, but I think any boy or girl would love having one. Let their creativity soar!
13. Pop Rocks - Do you remember what these feel like in your mouth?! Just imagine your son or daughter experiencing them for the first time. Or the second or third time. They're great!
14. Sippin' Specs - My girls love crazy straws, so they're getting some of these specs in their stockings this year. I can just hear them squeal as they feel the coldness of a drink going all around their face. Plus they'll just look silly wearing such a strange contraption.
15. Ball Popper - For babies, the funniest thing is typically something that takes them by surprise. This ball popper will have your baby laughing, crawling to retrieve balls, and waiting anxiously for the next one to pop up. My girls have both loved this toy!
16. Jack-In-The-Box - There is nothing funnier.
17. Pop n' Push Car - Your baby will giggle as the little balls inside pop-pop-pop as he drives the little car around!
18. Horse Hopper - Lena's best friend has this toy, and it is always the first thing she goes to when we go over there to play. She bounces along and laughs like crazy when she topples over.
19. Indoor Snowball Fight - Take a break in the middle of your Christmas morning festivities for a snowball fight! These are SO much fun-- they're very lightweight and squishy, but have enough weight to them that you can throw them across the room. No one will get hurt, but you can all attack one another!
20. Foam Pogo Jumper - Lena got one of these pogo jumpers for her 4th birthday back in the summer and it is still one of her favorite things to play with. It squeaks as she bounces, and she likes to see how many bounces she can get in a row without falling off.
These choices don't need descriptions of their own. They'd all make great stocking stuffers and your kids would think you are awesome. The teeth, clown noses, and jumbo glasses come in multi-packs, so they'd be perfect for a large family or you could consider splitting them with a friend!
This post is part of an AMAZING collaboration of over 75 bloggers. Click on through to see TONS of other gift guides, so that you can be sure to find the perfect gift your the children in your life. Many of these bloggers have made other gift guides in the past, and may make one or two more this year. So in order to not to miss any of those great resources, you'll want to follow the Pinterest board where you can see them in all their glory.
Great selection of gifts to get your kids laughing. Add some laughter to your Christmas or holiday morning! Choices for all ages and all price points.


This backstory begins when Kate realizes her husband of seventeen years has kept some big secrets, secrets that span continents and decades. As she unravels one secret darker than the other, she learns and grows and finds herself walking away from her corporate self and to a new life with a new-age order of nuns. This book has been self-published, and is only available through the Sister's webshop. If you want a signed copy, request one when you order. If you want something special in the inscription, please say so in the notes.


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text:
Cassiopeia (afkorting Cas) is een sterrenbeeld dat op de breedte van de Benelux het gehele jaar is te zien, het is circumpolair. Het sterrenbeeld heeft de vorm van de letter W. Het ligt tussen rechte klimming  en  en declinatie +55° en +75°.

Sterren

(in volgorde van afnemende helderheid)
 Schedar (α, alpha Cassiopeiae)
 Caph (β, beta Cassiopeiae)
 Cih (γ, gamma Cassiopeiae)
 Ruchbah (δ, delta Cassiopeiae)
 Segin (ε, epsilon Cassiopeiae)
 Achird (η, eta Cassiopeiae)
 Marfak (θ, theta Cassiopeiae)

Telescopisch waarneembare objecten in Cassiopeia, opgenomen in de New General Catalogue en Index Catalogue

New General Catalogue (NGC)

NGC 103, NGC 110, NGC 129, NGC 133, NGC 136, NGC 146, NGC 147, NGC 185, NGC 189, NGC 225, NGC 278, NGC 281, NGC 358, NGC 366, NGC 381, NGC 433, NGC 436, NGC 457, NGC 559, NGC 581, NGC 609, NGC 629, NGC 637, NGC 654, NGC 657, NGC 659, NGC 663, NGC 743, NGC 771, NGC 886, NGC 896, NGC 1027, NGC 1343, NGC 7438, NGC 7635, NGC 7654, NGC 7788, NGC 7789, NGC 7790, NGC 7795, NGC 7801

Index Catalogue (IC)

IC 10, IC 11, IC 59, IC 63, IC 155, IC 166, IC 289, IC 1590, IC 1747, IC 1795, IC 1805, IC 1824, IC 1831, IC 1848, IC 1851, IC 1871, IC 5366

Bezienswaardigheden
Het lijstje van deep-sky objecten in Cassiopeia is relatief lang, mede doordat het sterrenbeeld in de Melkweg ligt.
 Hoewel de eigenlijke intergalactische ruimte in het sterrenbeeld Cassiopeia geblokkeerd wordt door de storende invloed van het melkwegstelsel, zijn hier toch twee sterrenstelsels ontdekt, nl. Maffei 1 en Maffei 2. Beide sterrenstelsels zijn ontdekt door de Italiaanse astrofysicus Paolo Maffei (1926-2009).
 Cassiopeia bevat de resten van de door Tycho Brahe bestudeerde supernova SN 1572 en de sterke radiobron Cassiopeia A.
 Het sterrenbeeld bevat ook de Zielnevel en Hartnevel.
 NGC 457 is een open sterrenhoop in de vorm van een uil met lichtgevende ogen, vandaar de bijnaam Owl Cluster.
 Messier 52 is een open sterrenhoop met een oranjerode centrale ster.
 Messier 103 is een open sterrenhoop met, evenals Messier 52, een roodkleurige ster.
 WZ Cassiopeia is een typische koele koolstofster (Cool Carbon Star), en daarmee een van de meest roodkleurige sterren in de noordelijke sterrenhemel. WZ Cassiopeia (of WZ Cas) kan beschouwd worden als een dubbelster (ΟΣΣ 254), en bijgevolg kan de extreem rode kleur van de ene component vergeleken worden met het wit van de andere component (een opvallend kleurcontrast).
 Melotte 15 is een open sterrenhoop in de Running Dog Nebula (Lopende Hond Nevel of Hartnevel).
 Eta Cassiopeia (η Cassiopeia, Achird) is een dubbelster met een opmerkelijk kleurcontrast (lila-blauw en saffraan-geel). Deze dubbelster is bekend als het Paasei (Easter Egg Double).
 NGC 133 is een door de IAU officieel erkende open sterrenhoop die eerder als een telescopisch asterism gecatalogiseerd dient te worden. Het bestaat nl. uit een rijtje van slechts 4 sterren. Toch is dit sterrenrijtje observaties m.b.v. een telescoop waard.
 Stock 23 is een telescopisch asterism op het grensgebied tussen Cassiopeia en Camelopardalis (Giraffe). Dit object is bekend als Pazmino's Cluster.
 NGC 657 is een open sterrenhoop met de dubbelster ΟΣ 35 in het midden ervan.
 Σ 3053 is een dubbelster met een opmerkelijk kleurcontrast. De kleuren ervan worden beschreven als Orange/Blue, of Very Yellow/Blue (Burnham).
 NGC 7789 is een open sterrenhoop met de bijnaam Caroline's Rose.
 NGC 278 is een extragalactisch sterrenstelsel met een opmerkelijke helderheid (High Surface Brightness galaxy).

Aangrenzende sterrenbeelden
(met de wijzers van de klok mee)
 Cepheus
 Hagedis (Lacerta)
 Andromeda
 Perseus
 Giraffe (Camelopardalis)

Mythologie
In de Griekse mythologie is Cassiopeia de koningin van de Ethiopiërs. Ze was de vrouw van Cepheus en moeder van Andromeda.

Zie ook
Cassiopeia-test
 The clickable Cassiopeia


Traditional style and elegance will fill your home office with the addition of this stunning wall unit. Six doors, three drawers, and fourteen shelves make it easy to find storage space for all of your books and nicknacks neatly away. A handy desk nestled in between the storage space creates the ideal place for studying or researching. The handsome birch veneers are covered in a Light Vintage Sherry finish that adds warmth and depth to the piece that will liven up your home with traditional elegance. Drawer and door framing complete the vintage look and are accented beautifully by metal drawer and door pulls for the perfect look.
The Stanford Complete Wall Unit with Built In Desk by Parker House is available in the area from FMG - Local Home Furnishing.
STA#425 Glass Door Cabinet 2 22.125" 14.625" 84.25"
STA#463H Library Glass Door Hutch 1 40" 14.62" 54.125"
STA#461D Library Desk 1 40" 21" 30.125"
The Stanford collection is a great option if you are looking for Traditional furniture in the area.


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