Geared toward students of applied rather than pure mathematics, this volume introduces elements of partial differential equations. Its focus is primarily upon finding solutions to particular equations rather than general theory.Topics include ordinary differential equations in more than two variables, partial differential equations of the first and second orders, Laplace's equation, the wave equation, and the diffusion equation. A helpful Appendix offers information on systems of surfaces, and solutions to the odd-numbered problems appear at the end of the book. Readers pursuing independent study will particularly appreciate the worked examples that appear throughout the text.
Author(s): Ian N. Sneddon
Series: Dover Books on Mathematics
Publisher: Dover Publications
Year: 2006
Language: English
Pages: C, viii, 327, B
Tags: Математика;Дифференциальные уравнения;Дифференциальные уравнения в частных производных;
Cover
ELEMENTS OF PARTIAL DIFFERENTIAL EQUATIONS
Copyright
ISBN 0486452972
QA374.S6 2006 515'.353-dc22
LCCN 2006045445
PREFACE
CONTENTS
Chapter 1 ORDINARY DIFFERENTIAL EQUATIONS IN MORE THAN TWO VARIABLES
1. Surfaces and Curves in Three Dimensions
PROBLEMS
2. Simultaneous Differential Equations of the First Order and the First Degree in Three Variables
3. Methods of Solution of dx/P = dy/Q = dz /R
PROBLEMS
4. Orthogonal Trajectories of a System of Curves on a Surface
PROBLEMS
5. Pfaffian Differential Forms and Equations
PROBLEMS
6. Solution of Pfaffian Differential Equations in Three Variables
(a) By Inspection
(b) Variables Separable.
(c) One Variable Separable
(d) Homogeneous Equations
(e). Natani's Method
(f) Reduction to an Ordinary Differential Equation.
PROBLEMS
7. Carathéodory's Theorem
8. Application. to Thermodynamics
MISCELLANEOUS PROBLEMS
Chapter 2 PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER
1. Partial Differential Equations
2. Origins of First -order Partial Differential Equations
PROBLEMS
3. Cauchy's Problem for First -order Equations
4. Linear Equations of the First Order
PROBLEMS
5. Integral Surfaces Passing through a Given Curve
PROBLEMS
6. Surfaces Orthogonal to a Given System of Surfaces
PROBLEMS
7. Nonlinear Partial Differential Equations of the First Order
PROBLEMS
8. Cauchy's Method of Characteristics
PROBLEMS
9. Compatible Systems of First -order Equations
PROBLEMS
10. Charpit's Method
PROBLEMS
11. Special Types of First -order Equations
PROBLEMS
12. Solutions Satisfying Given Conditions
PROBLEMS
13. Jacobi's Method
PROBLEMS
14. Applications of First -order Equations
PROBLEMS
MISCELLANEOUS PROBLEMS
Chapter 3 PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER
1. The Origin of Second -order Equations
PROBLEMS
2. Second -order Equations in Physics
PROBLEMS
3. Higher -order Equations in Physics
PROBLEMS
4. Linear Partial Differential Equations with Constant Coefficients
PROBLEMS
5. Equations with Variable Coefficients
PROBLEMS
6. Characteristic Curves of Second -order Equations
PROBLEMS
7. Characteristics of Equations in Three Variables
PROBLEMS
8. The Solution of Linear Hyperbolic Equations
PROBLEMS
9. Separation of Variables
PROBLEMS
10. The Method of Integral Transforms
PROBLEMS
11. Nonlinear Equations of the Second Order
PROBLEMS
MISCELLANEOUS PROBLEMS
Chapter 4 LAPLACE'S EQUATION
1. The Occurrence of Laplace's Equation in Physics
PROBLEMS
2. Elementary Solutions of Laplace's Equation
PROBLEMS
3. Families of Equipotential Surfaces
PROBLEMS
4. Boundary Value Problems
PROBLEMS
5. Separation of Variables
PROBLEMS
6. Problems with Axial Symmetry
PROBLEMS
7. Kelvin's Inversion Theorem
PROBLEMS
8. The Theory of Green's Function for Laplace's Equation
PROBLEMS
9. The Relation of Dirichlet's Problem to the Calculus of Variations
10. "Mixed" Boundary Value Problems
PROBLEMS
11. The Two -dimensional Laplace Equation
PROBLEMS
12. Relation of the Logarithmic Potential to the Theory of Functions
PROBLEMS
13. Green's Function for the Two -dimensional Equation
PROBLEMS
MISCELLANEOUS PROBLEMS
Chapter 5 THE WAVE EQUATION
1. The Occurrence of the Wave Equation in Physics
PROBLEMS
2. Elementary Solutions of the One -dimensional Wave Equation
PROBLEMS
3. The Riemann- Volterra Solution of the One -dimensional Wave Equation
PROBLEMS
4. Vibrating Membranes: Application of the Calculus of Variations
PROBLEMS
5. Three -dimensional Problems
PROBLEMS
6. General Solutions of the Wave Equation
PROBLEMS
7. Green's Function for the Wave Equation
PROBLEMS
8. The Nonhomogeneous Wave Equation
PROBLEMS
9. Riesz's Integrals
PROBLEMS
10. The Propagation of Sound Waves of Finite Amplitude
PROBLEMS
MISCELLANEOUS PROBLEMS
Chapter 6 THE DIFFUSION EQUATION
1. The Occurrence of the Diffusion Equation in Physics
PROBLEMS
2. The Resolution of Boundary Value Problems for the Diffusion Equation
PROBLEMS'
3. Elementary Solutions of the Diffusion Equation
PROBLEMS
4. Separation of Variables
PROBLEMS
5. The Use of Integral Transforms
PROBLEMS
6. The Use of Green's Functions
PROBLEMS
7. The Diffusion Equation with Sources
PROBLEMS
MISCELLANEOUS PROBLEMS
APPENDIX SYSTEMS OF SURFACES
1. One- parameter Systems
2. Two -parameter Systems
3. The Edge of Regression
4. Ruled Surfaces
SOLUTIONS TO THE ODD-NUMBERED PROBLEMS
INDEX
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