**Partial Differential Equations for Mathematical Physicists**

**Contents **

Preface ix

Acknowledgments xi

Author xiii

1 Preliminary concepts and background material 1

1.1 Notations and de_nitions . . . . . . . . . . . . . . . . . . . . 2

1.2 Generating a PDE . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 First order PDE and the concept of characteristics . . . . . . 8

1.4 Quasi-linear _rst order equation: Method of characteristics . . . . .. . . . . . 9

1.5 Second order PDEs . . . . . . . . . . . . . . . . . . . . . . . 18

1.6 Higher order PDEs . . . . . . . . . . . . . . . . . . . . . . . 19

1.7 Cauchy problem for second order linear PDEs . . . . . . . . 21

1.8 Hamilton-Jacobi equation . . . . . . . . . . . . . . . . . . . . 27

1.9 Canonical transformation . . . . . . . . . . . . . . . . . . . . 28

1.10 Concept of generating function . . . . . . . . . . . . . . . . . 31

1.11 Types of time-dependent canonical transformations . . . . . 33

1.11.1 Type I Canonical transformation . . . . . . . . . . . . 33

1.11.2 Type II Canonical transformation . . . . . . . . . . . 34

1.11.3 Type III Canonical transformation . . . . . . . . . . . 35

1.11.4 Type IV Canonical transformation . . . . . . . . . . . 35

1.12 Derivation of Hamilton-Jacobi equation . . . . . . . . . . . . 38

1.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2 Basic properties of second order linear PDEs 45

2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.2 Reduction to normal or canonical form . . . . . . . . . . . . 47

2.3 Boundary and initial value problems . . . . . . . . . . . . . . 60

2.4 Insights from classical mechanics . . . . . . . . . . . . . . . . 70

2.5 Adjoint and self-adjoint operators . . . . . . . . . . . . . . . 73

2.6 Classi_cation of PDE in terms of eigenvalues . . . . . . . . . 75

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3 PDE: Elliptic form 81

3.1 Solving through separation of variables . . . . . . . . . . . . 83

3.2 Harmonic functions . . . . . . . . . . . . . . . . . . . . . . . 90

3.3 Maximum-minimum principle for Poisson’s and Laplace’s equations . . . . . 92

3.4 Existence and uniqueness of solutions . . . . . . . . . . . . . 93

3.5 Normally directed distribution of doublets . . . . . . . . . . 94

3.6 Generating Green’s function for Laplacian operator . . . . . 97

3.7 Dirichlet problem for circle, sphere and half-space . . . . . . 100

3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4 PDE: Hyperbolic form 109

4.1 D’Alembert’s solution . . . . . . . . . . . . . . . . . . . . . . 110

4.2 Solving by Riemann method . . . . . . . . . . . . . . . . . . 113

4.3 Method of separation of variables . . . . . . . . . . . . . . . 117

4.4 Initial value problems . . . . . . . . . . . . . . . . . . . . . . 121

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5 PDE: Parabolic form 137

5.1 Reaction-di_usion and heat equations . . . . . . . . . . . . . 137

5.2 Cauchy problem: Uniqueness of solution . . . . . . . . . . . . 140

5.3 Maximum-minimum principle . . . . . . . . . . . . . . . . . 141

5.4 Method of separation of variables . . . . . . . . . . . . . . . 143

5.5 Fundamental solution . . . . . . . . . . . . . . . . . . . . . . 154

5.6 Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6 Solving PDEs by integral transform method 165

6.1 Solving by Fourier transform method . . . . . . . . . . . . . 165

6.2 Solving by Laplace transform method . . . . . . . . . . . . . 172

6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

A Dirac delta function 185

B Fourier transform 203

C Laplace transform 213

Bibliography 221

Index 223

**Preface **

This **book** aims at providing an introduction to partial di_erential equations (PDEs) and expects to serve as a textbook for the young graduate students of theoretical physics and applied mathematics who have had an initial training in introductory calculus and ordinary di_erential equations (ODEs) and an elementary grasp of classical mechanics. The presentation of the book does not always follow the conventional practice of classifying the PDE and then taking up the treatment of the representative Laplace equation, wave equation and heat conduction equation one by one. On the contrary, certain basic features of these equations are presented in the introductory chapter itself with an aim to provide an appropriate formulation of mathematical methods associated with them as and when they emerge from the class of PDE they belong to.

Numerous examples have been worked out in the book which aim to illustrate the key mathematical concepts involved and serve as a means to improve the problemsolving skills of the students. We have purposefully kept the complexities of the mathematical structure to a minimum, often at the expense of general and abstract formulation, but put greater stress on the applicationside of the subject. We believe that the book will provide a systematic and comprehensive coverage of the basic theory of PDEs.

The book is organized into six chapters and three appendices. Chapter 1 contains a general background of notations and preliminaries required that would enable one to follow the rest of the book. In particular, it contains a discussion of the _rst order PDEs and the di_erent types of equations that one often encounters in practice. The idea of characteristics is presented and also second and higher order PDEs are introduced. We also comment briey on the Cauchy problem and touch upon the classi_cation of a second order partial di_erential equation in two variables.

Chapter 2 shows how a PDE can be reduced to a normal or canonical form and the utility in doing it. We give some insights from the classical mechanics as well. We also discuss the construction of the adjoint and selfadjoint operators.

The elliptic form of a partial di_erential equation is discussed in Chapter 3. Topics like method of separation of variables for the two-dimensional plane polar coordinates, three-dimensional spherical polar coordinates and cylindrical polar coordinates, harmonic functions and their properties, maximumminimum principle, existence, uniqueness and stability of solutions, normally directed distribution of doublets, Green’s equivalent layer theorem, generation of Green’s function, Dirichlet’s problem of a circle, sphere and half-space are considered.

PDEs of hyperbolic type are taken up in Chapter 4 which begins with the D’Alembert’s solution of the linear second order wave equation. The more general Riemann’s method is introduced next. The role of Riemann function is pointed out to help us solve the PDE in the form of a quadrature.

Solutions by the method of separation of variables is illustrated for the threedimensional wave equation both for the spherical polar and cylindrical coordinates. This chapter also consists of detailed discussions of the initial value problems related to the three-dimensional and two-dimensional wave equations. We give here the basic derivation of the Poisson/Kircho_ solution for the three-dimensional homogeneous equation supplemented by a set of inhomogeneous conditions and then focus on the solution of the inhomogeneous wave equation by means of the superposition principle, which gives us the Poisson formula. Hadamard’s method of descent is employed to solve completely the two-dimensional counterpart.

Parabolic equations in Chapter 5 is our next topic of inquiry. This chapter covers the Cauchy problem for the heat equation wherein we discuss the uniqueness criterion of the solution, the method of separation of variables for the Cartesian coordinates, spherical polar and cylindrical polar coordinates, and derive the fundamental solution and give a formulation of the Green’s function.

Chapter 6 is concerned with solving di_erent types of PDEs by the integral transform method focusing on the use of Fourier transform and Laplace transforms only. Some problems are worked out for their asymptotic nature.

Finally in the three appendices we address respectively some important issues of the delta function, the Fourier transform and Laplace transform. I wish to remark that we added a summary at the end of each chapter for the lay reader to have a quick look at the materials that have been covered and also appended a reasonable collection of homework problems relevant to the chapter. In addition, we solved in each chapter a wide range of problems to clarify the basic ideas involved. It is believed that these will help the readers in furthering the understanding of the subject.