Partial Differential Equations for Mathematical Physicistsis intended for graduate students, researchers of theoretical physics and applied mathematics, and professionals who want to take a course in partial differential equations. This book offers the essentials of the subject with theprerequisite being only an elementary knowledge of introductory calculus, ordinary differential equations, and certain aspects of Read more... Abstract: Partial Differential Equations for Mathematical Physicistsis intended for graduate students, researchers of theoretical physics and applied mathematics, and professionals who want to take a course in partial differential equations. This book offers the essentials of the subject with theprerequisite being only an elementary knowledge of introductory calculus, ordinary differential equations, and certain aspects of classical mechanics. We have stressed more the methodologies of partial differential equations and how they can be implemented as tools for extracting their solutions rather thandwelling on the foundational aspects. After covering some basic material, the book proceeds to focus mostly on the three main types of second order linear equations, namely those belonging to the elliptic, hyperbolic, and parabolic classes. For such equations a detailed treatment is given of the derivation of Green's functions, and of the roles of characteristics and techniques required in handling the solutions with the expected amount of rigor. In this regard we have discussed at length the method of separation variables, application of Green's function technique, and employment of Fourier and Laplace's transforms. Also collected in the appendices are some useful results from the Dirac delta function, Fourier transform, and Laplace transform meant to be used as supplementary materials to the text. A good number of problems is worked out andan equallylarge number of exercises has been appended at the end of each chapter keeping in mind the needs of the students. It is expected that this book will provide a systematic and unitary coverage of the basics of partial differential equations. Key Features An adequate and substantive exposition of the subject. Covers a wide range of important topics. Maintainsmathematical rigor throughout. Organizes materials in a self-contained way with each chapter ending with a summary. Contains a large number of worked out problems

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Bijan Kumar Bagchi

Taylor & Francis Group

Taylor & Francis Ltd

Psychology Press Ltd

CRC Press

United Kingdom and Ireland, United Kingdom

CRC Press (Unlimited), Boca Raton, 2020

Boca Raton, FL, 2020

lg2711903

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Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Acknowledgments
Author
1. Preliminary concepts and background material
1.1 Notations and definitions
1.2 Generating a PDE
(a) Eliminating arbitrary constants from a given relation
(b) Elimination of an arbitrary function
1.3 First order PDE and the concept of characteristics
1.4 Quasi-linear first order equation: Method of characteristics
(a) Lagrange's method of seeking a general solution
(b) Integral lines and integral surfaces
1.5 Second order PDEs
1.6 Higher order PDEs
1.7 Cauchy problem for second order linear PDEs
1.8 Hamilton-Jacobi equation
1.9 Canonical transformation
1.10 Concept of generating function
1.11 Types of time-dependent canonical transformations
1.11.1 Type I Canonical transformation
1.11.2 Type II Canonical transformation
1.11.3 Type III Canonical transformation
1.11.4 Type IV Canonical transformation
1.12 Derivation of Hamilton-Jacobi equation
1.13 Summary
2. Basic properties of second order linear PDEs
2.1 Preliminaries
2.2 Reduction to normal or canonical form
2.3 Boundary and initial value problems
(a) Different types of boundary and initial value problems
(b) Applications
Dirichlet problem on a rectangular domain: 0 ≤ x ≤ a, 0 ≤ y ≤ b
Neumann problem on a rectangular domain: 0 ≤ x ≤ a, 0 ≤ y ≤ b
Cauchy initial value problem on an interval: 0 ≤ x ≤ l, t > 0
(c) Well-posedness
Hadamard example
2.4 Insights from classical mechanics
2.5 Adjoint and self-adjoint operators
Two-dimensional case
2.6 Classification of PDE in terms of eigenvalues
2.7 Summary
3. PDE: Elliptic form
3.1 Solving through separation of variables
(a) Two dimensions: plane polar coordinates (r, θ)
(b) Three dimensions: spherical polar coordinates (r, θ, φ)
(c) Cylindrical polar coordinates (r, θ, z)
3.2 Harmonic functions
Gauss' mean value theorem
3.3 Maximum-minimum principle for Poisson's and Laplace's equations
3.4 Existence and uniqueness of solutions
Theorem
3.5 Normally directed distribution of doublets
3.6 Generating Green's function for Laplacian operator
3.7 Dirichlet problem for circle, sphere and half-space
(a) Circle
(b) Sphere
(c) Half-space
3.8 Summary
4. PDE: Hyperbolic form
4.1 D'Alembert's solution
4.2 Solving by Riemann method
4.3 Method of separation of variables
(a) Three dimensions: spherical polar coordinates (r, θ, φ)
(b) Cylindrical polar coordinates (r, θ, z)
4.4 Initial value problems
(a) Three dimensional wave equation
(b) Two dimensional wave equation
4.5 Summary
5. PDE: Parabolic form
5.1 Reaction-diffusion and heat equations
5.2 Cauchy problem: Uniqueness of solution
5.3 Maximum-minimum principle
5.4 Method of separation of variables
(a) Cartesian coordinates (x; y; z)
(b) Three dimensions: spherical polar coordinates (r, θ, φ)
(c) Cylindrical polar coordinates (r, θ, z)
5.5 Fundamental solution
5.6 Green's function
5.7 Summary
6. Solving PDEs by integral transform method
6.1 Solving by Fourier transform method
6.2 Solving by Laplace transform method
6.3 Summary
Appendix A: Dirac delta function
Dirac delta function δ(x)
Other results using delta function
Test function
Green's function
Appendix B: Fourier transform
Fourier transform
Convolution theorem and Parseval relation
Appendix C: Laplace transform
Laplace transform
Inversion theorem for Laplace transform
Asymptotic form for Laplace's inversion integral
Bibliography
Index

Partial Differential Equations for Mathematical Physicists is intended for graduate students, researchers of theoretical physics and applied mathematics, and professionals who want to take a course in partial differential equations. This book offers the essentials of the subject with the prerequisite being only an elementary knowledge of introductory calculus, ordinary differential equations, and certain aspects of classical mechanics. We have stressed more the methodologies of partial differential equations and how they can be implemented as tools for extracting their solutions rather than dwelling on the foundational aspects. After covering some basic material, the book proceeds to focus mostly on the three main types of second order linear equations, namely those belonging to the elliptic, hyperbolic, and parabolic classes. For such equations a detailed treatment is given of the derivation of Green's functions, and of the roles of characteristics and techniques required in handling the solutions with the expected amount of rigor. In this regard we have discussed at length the method of separation variables, application of Green's function technique, and employment of Fourier and Laplace's transforms. Also collected in the appendices are some useful results from the Dirac delta function, Fourier transform, and Laplace transform meant to be used as supplementary materials to the text. A good number of problems is worked out and an equally large number of exercises has been appended at the end of each chapter keeping in mind the needs of the students. It is expected that this book will provide a systematic and unitary coverage of the basics of partial differential equations.
Key Features
An adequate and substantive exposition of the subject. Covers a wide range of important topics. Maintains mathematical rigor throughout. Organizes materials in a self-contained way with each chapter ending with a summary. Contains a large number of worked out problems.

2020-07-28