Budak / Samarskii / Tikhonov A Collection of Problems on Mathematical Physics

International Series of Monographs in Pure and Applied Mathematics
1. Auflage 2013
ISBN: 978-1-4831-8486-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 782 Seiten, Web PDF

A Collection of Problems on Mathematical Physics is a translation from the Russian and deals with problems and equations of mathematical physics. The book contains problems and solutions. The book discusses problems on the derivation of equations and boundary condition. These Problems are arranged on the type and reduction to canonical form of equations in two or more independent variables. The equations of hyperbolic type concerns derive from problems on vibrations of continuous media and on electromagnetic oscillations. The book considers the statement and solutions of boundary value problems pertaining to equations of parabolic types when the physical processes are described by functions of two, three or four independent variables such as spatial coordinates or time. The book then discusses dynamic problems pertaining to the mechanics of continuous media and problems on electrodynamics. The text also discusses hyperbolic and elliptic types of equations. The book is intended for students in advanced mathematics and physics, as well as, for engineers and workers in research institutions.

Autoren/Hrsg.

Budak, B. M.

Samarskii, A. A.

Tikhonov, A. N.

Sneddon, I. N.

Stark, M.

Ulam, S.

Inhaltsverzeichnis

1;Front Cover;1
2;A Collection of Problems on Mathematical Physics;4
3;Copyright Page;5
4;Table of Contents;6
5;TRANSLATION EDITOR'S NOTE;11
6;PREFACE;12
7;CHAPTER I. CLASSIFICATION AND REDUCTION TO CANONICAL FORM OF SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS;14
7.1;1. The equation for a function of two independent variables;14
7.2;2. The equation with constant coefficients for a function of n independent variables;16
8;CHAPTER II. EQUATIONS OF HYPERBOLIC TYPE;17
8.1;1. Physical problems reducible to equations of hyperbolic type; statement of boundary-value problems;17
8.2;2. Method of travelling waves (D'Alembert's method);30
8.3;3. Method of separation of variables;41
8.4;4. Method of integral representations;52
9;CHAPTER III. EQUATIONS OF PARABOLIC TYPE;60
9.1;1. Physical problems leading to equations of parabolic type; statement of boundary-value problems;60
9.2;2. Method of separation of variables;65
9.3;3. Method of integral representations and source functions;72
10;CHAPTER IV. EQUATIONS OF ELLIPTIC TYPE;84
10.1;1. Physical problems leading to equations of elliptic type, and the statement of boundary-value problems;84
10.2;2. Simplest problems for Laplace's and Poisson's equations;87
10.3;3. The source function;90
10.4;4. The method of separation of variables;94
10.5;5. Potentials and their application;105
11;CHAPTER V. EQUATIONS OF PARABOLIC TYPE;109
11.1;1. Physical problems leading to equations of parabolic type; statement of boundary-value problems;109
11.2;2. The method of separation of variables;111
11.3;3. The method of integral representations;119
12;CHAPTER VI. EQUATIONS OF HYPERBOLIC TYPE;128
12.1;1. Physical problems leading to equations of hyperbolic type; statement of boundary-value problems;128
12.2;2. The simplest problems; different methods of solution;133
12.3;3. The method of separation of variables;139
12.4;4. The method of integral representations;147
13;CHAPTER VII. EQUATIONS OF ELLIPTIC TYPE .u+cu= — f;153
13.1;1. Problems for the equation .u—X2u=—f;153
13.2;2. Some problems on natural vibrations;155
13.3;3. Propagation and radiation of sound;159
13.4;4. Steady-state electromagnetic vibrations;164
14;PART 1:
HINTS, ANSWERS AND SOLUTIONS;174
14.1;CHAPTER I. CLASSIFICATION AND REDUCTION TO CANONICAL FORM OF SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS;176
14.1.1;1. The Equation for a Function of Two Independent Variables;176
14.1.2;2. The Equation with Constant Coefficients for a Function of n
Independent Variables;181
14.1.3;CHAPTER II. .QUATIONS OF HYPERBOLIC TYPE;184
14.1.3.1;1. Physical Problems Reducible to Equations of Hyperbolic Type; Statement of Boundary-value Problems;184
14.1.3.2;2. Method of Travelling Waves (D'Alembert's Method);221
14.1.3.3;3. Method of Separation of Variables;261
14.1.3.4;4. Method of Integral Representations;309
14.2;CHAPTER III. EQUATIONS OF PARABOLIC TYPE;330
14.2.1;1. Physical Problems Leading to Equations of Parabolic Type; Statement of Boundary-value Problems;330
14.2.2;2. Method of Separation of Variables;344
14.2.3;3. Method of Integral Representations and Source Functions;364
14.3;CHAPTER IV. EQUATIONS OF ELLIPTIC TYPE;394
14.3.1;1. Physical Problems Leading to Equations of Elliptic Type and the Statement of Boundary-value Problems;394
14.3.2;2. Simplest Problems for Laplace's and Poisson's Equations;405
14.3.3;3. The Source Function;413
14.3.4;4. The Method of Separation of Variables;438
14.4;CHAPTER V. EQUATIONS OF PARABOLIC TYPE;519
14.4.1;1. Physical Problems Leading to Equations of Parabolic Type; Statement of Boundary-value Problems;519
14.4.2;2. The Method of Separation of Variables;524
14.4.3;3 . The Method of Integral Representations;559
14.5;CHAPTER VI. EQUATIONS OF HYPERBOLIC TYPE;578
14.5.1;1. Physical Problems Leading to Equations of Hyperbolic Type; Statement of Boundary-value Problems;578
14.5.2;2. The Simplest Problems; Different Methods of Solution;587
14.5.3;3. The Method of Separation of Variables;599
14.5.4;4. The Method of Integral Representations;634
14.6;CHAPTER VII. EQUATIONS OF ELLIPTIC TYPE .u+cu= — f;657
14.6.1;1. Problems for the Equation .u—.2u = — f;657
14.6.2;2. Some Problems on Natural Vibrations;662
14.6.3;3. Propagation and Radiation of Sound;689
14.6.4;4. Steady-State Electromagnetic Vibrations;716
15;PART 2:
SUPPLEMENT;754
15.1;I. Different orthogonal systems of coordinates;754
15.2;II. Some formulae of vector analysis;761
15.3;III. Special functions;762
15.4;IV. Tables of the error integral and roots of some characteristic equations;768
16;REFERENCES;772
17;INDEX;776