Equations in Mathematical Physics : A Practical Course

by Pikulin, Victor P.; Pohozaev, Stanislav I.

• ISBN13: 9783764365011
• ISBN10: 3764365013
• Format: Hardcover
• Copyright: 2001-10-01
• Publisher: Springer Verlag

Summary

Many physical processes in fields such as mechanics, thermodynamics, electricity, magnetism or optics are described by means of partial differential equations. The aim of the present book is to demontstrate the basic methods for solving the classical linear problems in mathematical physics of elliptic, parabolic and hyperbolic type. In particular, the methods of conformal mappings, Fourier analysis and Green's functions are considered, as well as the perturbation method and integral transformation method, among others. Every chapter contains concrete examples with a detailed analysis of their solution.The book is intended as a textbook for students in mathematical physics, but will also serve as a handbook for scientists and engineers.

Table of Contents

Preface	vii
Introduction	1	(6)
Elliptic problems	7	(74)
The Dirichlet problem for the Laplace equation in an annulus	7	(3)
Examples of Dirichlet problems in an annulus	10	(1)
The interior and exterior Dirichlet problems	11	(3)
The Poisson integral for the disc. Complex form. Solution of the Dirichlet problem when the boundary condition is a rational function R (sin ϕ, cos ϕ)	14	(4)
The interior and exterior Dirichlet problems	18	(2)
Boundary value problems for the Poisson equation in a disc and in an annulus	20
Boundary value problems for the Laplace and Poisson equations in a rectangle	19	(8)
Boundary value problems for the Laplace and Poisson equations in a bounded cylinder	27	(6)
Boundary value problems for the Laplace and Poisson equations in a ball	33	(10)
Boundary value problems for the Helmholtz equations	43	(1)
Boundary value problem for the Helmoltz equation in a cylinder	44	(2)
Boundary value problems for the Helmoltz equation in a disc	46	(3)
Boundary value problems for the Helmoltz equation in a ball	49	(5)
Guided electromagnetic waves	54	(1)
The method of conformal mappings (for the solution of boundary value problems in the plane)	55	(5)
The Green function method	60	(8)
Other methods	68	(5)
Problems for independent study	73	(3)
Answers	76	(5)
Hyperbolic problems	81	(80)
The travelling-wave method	81	(12)
The method of selection of particular solutions	93	(3)
The Fourier integral transform method	96	(15)
The Laplace integral transform method	111	(4)
The Hankel integral transform method	115	(6)
The method of standing waves. Oscillations of a bounded string	121	(3)
Some examples of mixed problems for the equation of oscillations of a string	124	(6)
The Fourier method. Oscillations of a rectangular membrane	130	(6)
The Fourier method. Oscillations of a circular membrane	136	(5)
The Fourier method. Oscillations of a beam	141	(4)
The perturbation method	145	(4)
Problems for independent study	149	(6)
Answers	155	(6)
Parabolic problems	161	(44)
The Fourier integral transform method	161	(9)
The Lapalce integral transform method	170	(4)
The Fourier method (method of separation of variables)	174	(16)
A modification of the method of separation of variables for solving the Cauchy problem	190	(7)
Problems for independent study	197	(5)
Answers	202	(3)
References	205	(2)
Index	207