Boundary Value Problems of Linear Partial Differential Equations for Engineers and Scientists

The following sections are included:

• PREFACE

• TABLE OF CONTENTS

A certain number of partial differential equations which occur frequently in classical mathematical physics are derived in this chapter. Emphasis is here given to the physical motivation of the boundary value problems…

For a good many cases of the boundary value problem, it is not only convenient but also imperative to use an appropriate system of coordinates (cf. Ex. 1, Section 1.1). The elementary solutions, obtained by separation of variables, which satisfy all the homogeneous boundary conditions, form a set of orthogonal functions. A linear combination of these functions can, in most cases, represent an arbitrary function in the same sense as the Fourier series…

The methods developed in the preceding chapters fail when we dea1 with the boundary problems such as the type (ii) in Section 2 of the Chapter I and the heat conduction in a slab of infinite width. These problems natura11y introduce’ the general method of operational calculus, including the integral transformations and Cauchy’s residue theory. Laplace and Fourier transforms are treated in details in this chapter and their applications to the boundary value problems are illustrated by various concrete examples. Further applications of other integral transforms, such as Mellin transforms, Hankel transforms, etc., are found in the Chapter IV. In many cases, in order to find the inverse transforms, Cauchy’s residue theory is indispensable. For that purpose, a brief account of the complex functions is given in the appendix. It is meant for those who do not have any previous knowledge on that subject. The theory, started from the system of complex numbers, is carried on to the Cauchy’s expansion of a meromorphic function in terms of rational functions so that the students are able to apply the elementary theory of complex functions freely to the problems arisen in this course…

The present treatment of the subject is purely formal. Particular interests will be centered at general integral transforms with Fourier kernels. Up to now, we have employed Fourier and Laplace transforms as operational methods to solve the boundary value problems of partial differential equations. In the several concrete examples given later, we shall employ other integral transforms to reduce some boundary value problems into a pair of integral equations, the formal solution of which is obtained by Titchmarsh…

The first part of the present chapter deals with the boundary-value problems of the Laplace equation. The simplest case of the circle problem is solved by means of Fourier series. The formulas of Schwarz, Poisson, Boggio and Dini for the circle problem are derived. The method of conformal mapping is the introduced in attacking the general two-dimensional problems. In attacking higher dimensional problems, the notion of Green’s function and the method of images are used. Extension of these methods to other types of partial differential equations is illustrated by examples, such as the problems of vibrating airfoils in a supersonic flow and of the bending of an elastic plate under a concentrated load. The methods of a simple 1ayer and a double 1ayer and some important theorems concerning harmonic functions are included in exercises.

The following sections are included:

• APPENDIX THE OPERATIONAL METHOD (II) Cauchy’s Residue Theory

• REFERENCES