Mixed Boundary Value Problems© 2008 ppt

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Mixed Boundary Value Problems

Dean G Duffy

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Library of Congress Cataloging-in-Publication Data

Duffy, Dean G.

Mixed boundary value problems / Dean G Duffy.

p cm (Chapman & Hall/CRC applied mathematics & nonlinear science series ; 15)

Includes bibliographical references and index.

ISBN 978-1-58488-579-5 (alk paper)

1 Boundary value problems Numerical solutions 2 Boundary element methods I Title II Series.

1 Overview 1

1.1 Examples of Mixed Boundary Value Problems 1

1.2 Integral Equations 13

1.3 Legendre Polynomials 22

1.4 Bessel Functions 27

2 Historical Background 41

2.1 Nobili’s Rings 41

2.2 Disc Capacitor 45

2.3 Another Electrostatic Problem 50

2.4 Griffith Cracks 53

2.5 The Boundary Value Problem of Reissner and Sagoci 58

2.6 Steady Rotation of a Circular Disc 72

3.1 Dual Fourier Cosine Series 84

3.2 Dual Fourier Sine Series 102

3.3 Dual Fourier-Bessel Series 109

3.4 Dual Fourier-Legendre Series 126

3.5 Triple Fourier Sine Series 157

4 Transform Methods 163

4.1 Dual Fourier Integrals 165

4.2 Triple Fourier Integrals 202

4.3 Dual Fourier-Bessel Integrals 210

4.4 Triple and Higher Fourier-Bessel Integrals 284

4.5 Joint Transform Methods 318

5 The Wiener-Hopf Technique 347

5.1 The Wiener-Hopf Technique When the Factorization Contains No Branch Points 353

5.2 The Wiener-Hopf Technique When the Factorization Contains Branch Points 397

6 Green’s Function 413

6.1 Green’s Function with Mixed Boundary-Value Conditions 413

6.2 Integral Representations Involving Green’s Functions 417

6.3 Potential Theory 436

7 Conformal Mapping 443

7.1 The Mapping z = w + a log(w) 443

7.2 The Mapping tanh[πz/(2b)] = sn(w, k) 445

7.3 The Mapping z = w + λ w2− 1 447

7.4 The Mapping w = ai(z − a)/(z + a) 449

7.5 The Mapping z = 2[w − arctan(w)]/π 452

7.6 The Mapping k w sn(w, k w ) = k z sn(K z z/a, k z ) 457

I am indebted to R S Daniels and M A Truesdale of the Defense College

of Management and Technology for their aid in obtaining the portrait of Prof.Tranter My appreciation goes to all the authors and publishers who allowed

me the use of their material from the scientific and engineering literature.Finally, many of the plots and calculations were done using MATLAB.R

The MathWorks Inc

24 Prime Park WayNatick, MA 01760-1500Phone: (508) 647-7000Email: info@mathworks.comwww.mathworks.com

Dean G Duffy received his bachelor of science in geophysics from CaseInstitute of Technology (Cleveland, Ohio) and his doctorate of science in me-teorology from the Massachusetts Institute of Technology (Cambridge, Mas-sachusetts) He served in the United States Air Force from September 1975 toDecember 1979 as a numerical weather prediction officer After his militaryservice, he began a twenty-five year (1980 to 2005) association with NASA

at the Goddard Space Flight Center (Greenbelt, Maryland) where he focused

on numerical weather prediction, oceanic wave modeling and dynamical teorology He also wrote papers in the areas of Laplace transforms, antennatheory and mechanical engineering In addition to his NASA duties he taughtengineering mathematics, differential equations and calculus at the UnitedStates Naval Academy (Annapolis, Maryland) and the United States MilitaryAcademy (West Point, New York) Drawing from his teaching experience,

me-he has written several books on transform methods, engineering matme-hematicsand Green’s functions This present volume is his fourth book for Chapman

& Hall/CRC Press

Purpose This book was conceived while I was revising my

engineer-ing mathematics textbook I noticed that in many engineerengineer-ing and scientificproblems the nature of the boundary condition changes, say from a Dirichlet

to a Neumann condition, along a particular boundary Although these mixedboundary value problems appear in such diverse fields as elasticity and biome-chanics, there are only two books (by Sneddon1and Fabrikant2) that address

this problem and they are restricted to the potential equation The purpose

of this book is to give an updated treatment of this subject

The solution of mixed boundary value problems requires considerablemathematical skill Although the analytic solution begins using a conven-tional technique such as separation of variables or transform methods, themixed boundary condition eventually leads to a system of equations, involv-ing series or integrals, that must be solved The solution of these equationsoften yields a Fredholm integral equation of the second kind Because theseintegral equations usually have no closed form solution, numerical methodsmust be employed Indeed, this book is just as much about solving integralequations as it involves mixed boundary value problems

Prerequisites The book assumes that the reader is familiar with the

conventional methods of mathematical physics: generalized Fourier series,transform methods, Green’s functions and conformal mapping

Holland, 283 pp.

Their Applications in Engineering Kluwer Academic, 451 pp.

book for anyone in the physical sciences, engineering, or applied mathematics.

Chapter Overview The purpose of Chapter 1 is twofold The firstsection provides examples of what constitutes a mixed boundary value prob-lem and how their solution differs from commonly encountered boundary valueproblems The second part provides the mathematical background on integralequations and special functions that the reader might not know

Chapter 2 presents mixed boundary value problems in their historicalcontext Classic problems from mathematical physics are used to illustratehow mixed boundary value problems arose and some of the mathematicaltechniques that were developed to handle them

Chapters 3and4are the heart of the book Most mixed boundary valueproblems are solved using separation of variables if the domain is of limitedextent or transform methods if the domain is of infinite or semi-infinite extent.For example, transform methods lead to the problem of solving dual or tripleFourier or Bessel integral equations We then have a separate section for each

of these integral equations

Chapters 5through7are devoted to additional techniques that are times used to solve mixed boundary value problems Here each technique ispresented according to the nature of the partial differential or the domain forwhich it is most commonly employed or some other special technique.Numerical methods play an important role in this book Most integralequations here require numerical solution All of this is done using MATLAB

some-and the appropriate code is included MATLABis also used to illustrate thesolutions

We have essentially ignored brute force numerical integration of mixedboundary value problems In most instances conventional numerical methodsare simply applied to these problems Because the solution is usually dis-continuous along the boundary that contains the mixed boundary condition,analytic techniques are particularly attractive

An important question in writing any book is what material to include

or exclude This is especially true here because many examples become verycumbersome because of the nature of governing equations Consequently weinclude only those problems that highlight the mathematical techniques in

a straightforward manner The literature includes many more problems thatinvolve mixed boundary value problems but are too complicated to be includedhere

Features Although this book should be viewed primarily as a source

book on solving mixed boundary value problems, I have included problemsfor those who truly wish to master the material As in my earlier books, Ihave included intermediate results so that the reader has confidence that he

or she is on the right track

n (x) Hankel functions of first and second kind and of order n

I n (x) modified Bessel function of the first kind and order n

J n (x) Bessel function of the first kind and order n

K n (x) modified Bessel function of the second kind and order n

P n (x) Legendre polynomial of order n

Chapter 1 Overview

In the solution of differential equations, an important class of problemsinvolves satisfying boundary conditions either at end points or along a bound-ary As undergraduates, we learn that there are three types of boundary con-ditions: 1) the solution has some particular value at the end point or along

a boundary (Dirichlet condition), 2) the derivative of the solution equals aparticular value at the end point or in the normal direction along a boundary(Neumann condition), or 3) a linear combination of Dirichlet and Neumannconditions, commonly called a “Robin condition.” In the case of partial dif-ferential equations, the nature of the boundary condition can change along aparticular boundary, say from a Dirichlet condition to a Neumann condition

The purpose of this book is to show how to solve these mixed boundary value

problems

1.1 EXAMPLES OF MIXED BOUNDARY VALUE PROBLEMS

Before we plunge into the details of how to solve a mixed boundary valueproblem, let us examine the origins of these problems and the challenges totheir solution

• Example 1.1.1: Separation of variables

Mixed boundary value problems arise during the solution of Laplace’sequation within a specified region A simple example1 is

The interesting aspect of this problem is the boundary condition given by

Equation 1.1.4 For x between 0 and c, it satisfies a Dirichlet condition which becomes a Neumann condition as x runs between c and π.

The problem posed by Equation 1.1.1 to Equation 1.1.4 is very similar

to those solved in an elementary course on partial differential equations Forthat reason, let us try and apply the method of separation variables to solve

it Assuming that u(x, y) = X(x)Y (y), we obtain



ycos

n −1 2



x

, (1.1.7)

with n = 1, 2, 3, Because the most general solution to our problem consists

of a superposition of these particular solutions, we have that

exp

−n −1 2



ycos

n −1 2



x

(1.1.8)

for problems with discontinuous boundary conditions in heat conduction and diffusion with

reaction Indust Eng Chem Fund., 24, 64–77.

Substituting this general solution into the boundary condition given by tion 1.1.4, we obtain

cos

n −1 2

along y = 0 is the solution of this dual Fourier cosine series given by Equation

1.1.9 and Equation 1.1.10 This solution of these dual Fourier series will beaddressed inChapter 3

• Example 1.1.2: Transform methods

In the previous problem, we saw that we could apply the classic method

of separation of variables to solve mixed boundary value problems where thenature of the boundary condition changes along a boundary of finite length.How do we solve problems when the boundary becomes infinite or semi-infinite

in length? The answer is transform methods

Let us solve Laplace’s equation2

and

The interesting aspect of this problem is the boundary condition given byEquation 1.1.13 It changes from a Neumann condition to a Dirichlet condition

along the boundary x = 1.

To solve this boundary value problem, let us introduce the Fourier cosinetransform

in-process dressing I: Two-dimensional modeling J Appl Phys., 87, 3151–3158.

which automatically fulfills the boundary condition given by Equation 1.1.12.Then, the differential equation given by Equation 1.1.11 and boundary con-dition given by Equation 1.1.14 become

Equation 1.1.18 and Equation 1.1.19 are a set of dual integral equations where

A(k) is the unknown In Chapter 4we will show how to solve this kind ofintegral equation

• Example 1.1.3: Wiener-Hopf technique

In the previous example we showed how mixed boundary value problemscan be solved using transform methods Although we have not addressed thequestion of how to solve the resulting integral equations, the analysis leading

up to that point is quite straightforward To show that this is not alwaystrue, consider the following problem:3

Weaver, 1979: H-polarization induction in two thin half-sheets Geophys J R Astr Soc.,

56, 419–438.

Before tackling the general problem, let us find the solution at large

|x| In these regions, the solution becomes essentially independent of x and

Equation 1.1.20 becomes an ordinary differential equation in y The solution

Why are these limiting cases useful? If we wish to use Fourier transforms

to solve the general problem, then u(x, y) must tend to zero as |x| → ∞ so

that the Fourier transform exists Does that occur here? No, because u(x, y) tends to constant, nonzero values as x → −∞ and x → ∞ Therefore, the

use of the conventional Fourier transform is not justified

Let us now introduce the intermediate dependent variable v(x, y) so that

This substitution therefore yields a v(x, y) that tends to zero as x → ∞.

Unfortunately, v(x, y) does not tend to zero as x → −∞ Consequently, once

again we cannot use the conventional Fourier transform to solve this mixedboundary value problem; we appear no better off than before In Chapter

5, we show that is not true and how the Wiener-Hopf technique allows us tosolve these cases analytically

• Example 1.1.4: Green’s function

In Example 1.1.2 we used transform methods to solve a mixed boundaryvalue problem that eventually lead to integral equations that we must solve

An alternative method of solving this problem involves Green’s functions asthe following example shows

Consider the problem4

and f (ξ) is an unknown function such that u(ξ, 0) = f (ξ) if |ξ| ≤ 1.

To find g(x, y |ξ, η), we first take the Fourier transform of Equation 1.1.32

through Equation 1.1.34 with respect to x This yields

nπy

L

(1.1.36)

constriction resistance of a strip contact spot on a thin film J Phys D Appl Phys., 32,

930–936 Published by IOP Publishing Ltd.

Once again, we have reduced the mixed boundary value problem to findingthe solution of an integral equation Chapter 6is devoted to solving Equation1.1.46 as well as other mixed boundary value problems via Green’s function.

• Example 1.1.5: Conformal mapping

Conformal mapping is a mathematical technique involving two complex

variables: z = x + iy and t = r + is Given an analytic function t = g(z), the domain over which Laplace’s equation holds in the z-plane is mapped into some portion of the t-plane, such as an upper half-plane, rectangle or circle It

is readily shown that Laplace’s equation and the Dirichlet and/or Neumann

conditions in the z-plane also apply in the t-plane For this method to be useful, the solution of Laplace’s equation in the t-plane must be easier than

in the z-plane.

For us the interest in conformal mapping lies in the fact that a solution

to Laplace’s equation in the xy-plane is also a solution to Laplace’s equation

in the rs-plane Of equal importance, if the solution along a boundary in the xy-plane is constant, it is also constant along the corresponding boundary

in the rs-plane For these reasons, conformal mapping has been a powerful

method for solving Laplace’s equation since the nineteenth century Let ussee how we can use this technique to solve a mixed boundary value problem.Let us solve Laplace’s equation5

Consider the transformation t = − cos(πz/a) As Figure 1.1.1 shows,

this transformation maps the strip 0 < x < a, 0 < y < ∞ into the half-plane

0 < (t): The boundary x = a, y > 0 is mapped into (t) > cosh(πc/a),

(t) = 0 while the x-axis lies along −1 < (t) < 1, (t) = 0.

Consider next, the fractional linear transformation

s = αt + β

disk electrodes Phys Rev., 148, 170–175.

Figure 1.1.1: Lines of constant values of of x/a (solid line) and y/a (dashed lines) in the

t-plane given by the conformal mapping t = − cos(πz/a).

denotes one of the Jacobian elliptic functions.6 This maps the half-plane

(s) > 0 into a rectangular box with vertices at (K, 0), (K, K ), (−K, K )

and (−K, 0), where K and K  are the real and imaginary quarter-periods,

respectively We show this conformal mapping inFigure 1.1.3when c/a = 1 and D = 1.

Why have we introduced these three conformal mappings? After applyingthese three mappings, our original problem, Equation 1.1.47 through Equation

Handbook of Mathematical Functions, M Abromowitz and I A Stegun, Eds., Dover, 567–

586.

Figure 1.1.2: Same asFigure 1.1.1 except that we have the additional mapping given

by Equation 1.1.51 with c/a = 1 or k = 0.430 by Equation 1.1.55 If D = 1, α = 2.325,

β = −9.344, γ = 1, and δ = 6.018.

Figure 1.1.3: Same as Figure 1.1.2 except that we have the additional mapping s =

u(ξ, η) Next, using the MATLAB procedure ellipj, we find the values ofR

the Jacobian elliptic functions to compute s Because ζ is complex, we use

where k  = 1− k Next, we use Equation 1.1.51 to compute t given s

Fi-nally, z = arccos( −t)/π Thus, for a particular value of x and y, we have u(x, y). Figure 1.1.4illustrates this solution In Chapter 7, we will explorethis technique further

• Example 1.1.6: Numerical methods

Numerical methods are necessary in certain instances because the etry may be simply too complicated for analytic techniques These techniquesare similar to those applied to solve most partial differential equations How-ever, because most of the solutions are discontinuous along the boundary, afew papers have examined the application of finite differences to mixed bound-ary value problems.7

conditions J Franklin Inst., 277, 11–30; Bramble, J H., and B E Hubbard, 1965:

Ap-proximation of solutions of mixed boundary value problems for Poisson’s equation by finite

differences J Assoc Comput Mach., 12, 114–123; Thuraisamy, V., 1969: Approximate

solutions for mixed boundary value problems by finite-difference methods Math Comput.,

23, 373–386.

0 0.2 0.4 0.6 0.8 1

0 0.5 1

1.5 2

Although there is an exact solution8to this problem, we will act as if there

is none and solve it purely by numerical methods Introducing a grid with

nodal points located at r n = n∆r and z m = m∆z, where n = 0, 1, 2, , N and m = 0, 1, 2, , M , and applying simple second-order, finite differences

to represent the partial derivatives, Equation 1.1.61 can be approximated by

nu-merical solution for the potential J Phys A, 18, 1337–1342 See also Schwarzbek, S M.,

and S T Ruggiero, 1986: The effect of fringing fields on the resistance of a conducting

film IEEE Microwave Theory Tech., MTT-34, 977–981.

where n = 1, 2, , N − 1 and m = 1, 2, , M − 1 Axial symmetry yields for

where a = I∆r and h = H∆z In Equation 1.1.64 through Equation 1.1.68,

we have denoted u(r n , z m ) simply by u n,m

Although this system of equations could be solved using techniques fromlinear algebra, that would be rather inefficient; in general, these equationsform a sparse matrix For this reason, an iterative method is best A simple

one is to solve for u n,m in Equation 1.1.64 Assuming ∆r = ∆z, we obtain

where m = 1, 2, , M − 1 Here, we denote the value of u n,m during the

ith iteration with the subscript i This iterative scheme is an example of the

Gauss-Seidel method It is particularly efficient because u n −1,m and u n,m −1

have already been updated

Figure 1.1.5illustrates this numerical solution by showing u n,mat variouspoints during the iterative process Initially, there is dramatic change in the

solution, followed by slower change as i becomes large.

1.2 INTEGRAL EQUATIONS

An integral equation is any equation in which the unknown appears in the integrand Let ϕ(t) denote the unknown function, f (x) is a known function, and K(x, t) is a known integral kernel, then a wide class of integral equations

can be written as

f (x) =

 b a

or

ϕ(x) = f (x) +

 b a

0.001 0.01

0.01 0.05

0.2 0.2

0.5 0.5 0.7 0.7 0.9 0.9

r

10,000 iterations

Figure 1.1.5: A portion of the numerical solution of Equation 1.1.61 through Equation

1.1.63 using the Gauss-Seidel scheme to solve the finite differenced equations The

param-eters used in this example are N = M = 200, ∆r = ∆z = 0.01, a = 0.4, and h = 0.2.

A common property of these integral equations is the fixed limits in the gral These integral equations are collectively called “Fredholm integral equa-tions,” named after the Swedish mathematician Erik Ivar Fredholm (1866–1927) who first studied them Equation 1.2.1 is referred to as a Fredholmintegral equation of the first kind while Equation 1.2.2 is a Fredholm integralequation of the second kind; integral equations of the second kind differ fromintegral equations of the first kind in the appearance of the unknown outside

inte-of the integral In an analogous manner, integral equations inte-of the form

f (x) =

 x a

and

ϕ(x) = f (x) +

 x a

 t a

 b t

 t a

 b t

Edinburgh Math Soc., Ser 2 , 13, 271–272.

A second application occurs if we set h(τ ) = τ2 In this case we have the

integral equation (Abel-type integral equation)

 t a

 b t

 t a

 t a

1

√

t − η

 η a

F  (ξ)

√

η − ξ dξ dη = π[F (a) − F (x)]. (1.2.21)

Hafen10 derived Equation 1.2.20 and Equation 1.2.21 in 1910.

t which has the

Laplace transformL[K(t)] = √ πe −k2/ (4s) / √

tanh(βx) + tanh(βt) tanh(βx) − tanh(βt) dt = x h , 0≤ x < 1, (1.2.34)

where 2hβ = π How does Equation 1.2.31 help us here? If we introduce the variables tanh(βt) = tanh(β)T and tanh(βx) = tanh(β)X, then Equa-

tion 1.2.34 transforms into an integral equation of the form Equation 1.2.31.Substituting back into the original variables, we find that

h(t) = 1

π2

d dt

dual integral equations and series Glasgow Math J., 11, 9–20.

d dy

If f (x) is a constant, the equation has no solution We will use Equation

1.2.39 and Equation 1.2.40 inChapter 6

if(µ) > (ν) > −1 Akhiezer13used Equation 1.2.41 along with the result14

that the integral equation

Academic Press, Formula 6.596.6.

Nauk USSR, 98, 333–336.

Press, Formula 1.8.71.

because 0≤ t ≤ 1 < x < ∞ Therefore, the introduction of the integral

defi-nition for C(k) results in Equation 1.2.45 being satisfied identically Turning

with 0 < x < 1. Therefore, the solution to the dual integral equations,Equation 1.2.44 and Equation 1.2.45, consists of Equation 1.2.46 and

It is straightforward to show that this choice for S(k) satisfies Equation 1.2.58

identically When we perform an analysis similar to Equation 1.2.50 throughEquation 1.2.53, we find that

if−1 < p < 0 To obtain Equation 1.2.61, we integrated Equation 1.2.57 with

respect to x which yields

Applying Equation 1.2.42 and Equation 1.2.43, we have that

for−1 < p < 0 Therefore, the solution to the dual integral equations

Equa-tion 1.2.57 and EquaEqua-tion 1.2.58 has the soluEqua-tion EquaEqua-tion 1.2.59 along with

Equation 1.2.63 or Equation 1.2.64 depending on the value of p.

1.3 LEGENDRE POLYNOMIALS

In this book we will encounter special functions whose properties will berepeatedly used to derive important results This section focuses on Legendrepolynomials

Legendre polynomials15 are defined by the power series:

(1− x2)dy

dx + n(n + 1)y = 0, (1.3.3)

that arose in the separation-of-variables solution of partial differential tions in spherical coordinates Several of their properties are given in Table1.3.1

pr´ esent´ es ` a l’Acad Sci pars divers savants, 10, 411–434 The best reference on Legendre

polynomials is Hobson, E W., 1965: The Theory of Spherical and Ellipsoidal Harmonics.

Chelsea Publishing Co., 500 pp.

Table 1.3.1: Some Useful Relationships Involving Legendre Polynomials

18, 161–194.

-1.0 -0.5 0.0 0.5 1.0

x

-1.0 -0.5 0.0 0.5 1.0

Figure 1.3.1: The first four Legendre functions of the first kind.

We begin by noting that

sin(η)J0[z sin(η/2)]



cos(x) − cos(η) dη. (1.3.10)

to the reduction of triple cosine series Glasgow Math J., 14, 198–201.

They began by using the formula18

Here the hypergeometric function in Watson’s formula is replaced with

Leg-endre polynomials Multiplying Equation 1.3.12 by sin(η)/



x

, 0 < x < π, and r(x) = 1.

2 The series given in Problem 1 are also expansions in Legendre polynomials

In that light, show that



t

n +1 2

,

University Press, 804 pp See Equation (3) in Section 5.21.

t

n +1 2

P n [cos(t)] {sin[(n + 1)x] − sin(nx)}

and the results from Problem 1, show that

Setting ξ = h, then ξ = −h, and finally adding and subtracting the

resulting equations, show20that

1 + 2h cos(θ) + h2−1/2

for m = 0 and 1.

capac-itor Sov Tech Phys., 7, 1041–1043.

Equation 1.2.63 or Equation 1.2.64 depending on the value of p.

1.3 LEGENDRE POLYNOMIALS

In this book we will encounter special