A Course in Ordinary and Partial Differential Equations - 1st Edition - ISBN: 9781483230986, 9781483262628

A Course in Ordinary and Partial Differential Equations

1st Edition

Authors: Zalman Rubinstein
eBook ISBN: 9781483262628
Imprint: Academic Press
Published Date: 1st January 1969
Page Count: 488
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Description

A Course in Ordinary and Partial Differential Equations discusses ordinary differential equations and partial differential equations. The book reviews the solution of elementary first-order differential equations, existence theorems, singular solutions, and linear equations of arbitrary order. It explains the solutions of linear equations with constant coefficients, operational calculus, and the solutions of linear differential equations. It also explores the techniques of computing for the solution of systems of linear differential equations, which is similar to the solutions of linear equations of arbitrary order. The text proves that if the coefficients of some differential equations possess certain restricted types of singularities, the solution will have Taylor series expansions about the singular points. The investigator can calculate a divergent series whose partial sums numerically approximate the solution for large x if the point in question is infinity, of which the series will be a Taylor series of negative powers of x. The book also explains the Fourier transform, its applications to partial differential equations, as well as the Hilbert space approach to partial differential equations. The book is a stimulating material for mathematicians, for professors, or for students of pure and applied mathematics, physics, or engineering.

Table of Contents


Preface

Part I. Ordinary Differential Equations

Section 1 Classification and Solutions of First-Order Differential Equations

1.1 General Remarks

1.2 Solution of Elementary First-Order Differential Equations

1.3 Integration Factors

Section 2 Elementary Higher-Order Differential Equations

Exercises

Section 3 Existence Theorems

Exercises

Section 4 Singular Solutions

Exercises

Section 5 Linear Equations of Arbitrary Order

Section 6 Solutions of Linear Equations

6.1 Solution of Linear Equations with Constant Coefficients

6.2 Operational Calculus and Solutions of Linear Differential Equations

Exercises

Section 7 Linear Systems with Constant Coefficients

Exercises

Section 8 Infinite Series Solutions

Exercises

Section 9 Asymptotic Expansion of Solutions of Linear Differential Equations

Exercises

Section 10 Solutions of Differential Equations by Definite Integrals

Exercises

Section 11 Boundary Value Problems

11.1 Introduction

Exercises

11.2 Sturm-Liouville Systems

Exercises

Section 12 Green's Function

Exercises

Section 13 Expansion Theorems

Exercises

Section 14 Nonlinear Differential Equations

14.1 General Remarks

14.2 Autonomous Systems

Exercises

14.3 Stability of Solutions of Differential Equations

Exercises

Part II. Partial Differential Equations

Section 1 Introduction

Exercises

Section 2 Elementary Second-Order Partial Differential Equations

2.1 Classification of Second-Order Equations

Exercises

2.2 The Cauchy Problem and the Characteristic Surfaces

Exercises

Section 3 Second-Order Hyperbolic Differential Equations

Exercises

Section 4 Second-Order Elliptic Differential Equations

Exercises

Section 5 Second-Order Parabolic Differential Equations

Exercises

Section 6 The Fourier Transform and Its Applications to Partial Differential Equations

6.1 Definition and Elementary Properties of the Fourier Transform

6.2 Applications of the Fourier Transform to Elementary Differential Equations

Exercises

Section 7 Hilbert Space Approach to Partial Differential Equations

Exercises

Section 8 Distributions and Their Applications to Partial Differential Equations

Exercises

Index

Details

No. of pages:
488
Language:
English
Copyright:
© Academic Press 1969
Published:
Imprint:
Academic Press
eBook ISBN:
9781483262628

About the Author

Zalman Rubinstein