Solving Ordinary and Partial Boundary Value

This book provides an elementary, accessible introduction for engineers and scientists to the concepts of ordinary and partial boundary value problems, acquainting readers with fundamental properties and with efficient methods of constructing solutions or satisfactory approximations.Discussions include:· ordinary differential equations· classical theory of partial differential equations· Laplace and Poisson equations· heat equation· variational methods of solution of corresponding boundary value problems· methods of solution for evolution partial differential equationsThe author presents special remarks for the mathematical reader, demonstrating the possibility of generalizations of obtained results and showing connections between them. For the non-mathematician, the author provides profound functional-analytical results without proofs and refers the reader to the literature when necessary.Solving Ordinary and Partial Boundary Value Problems in Science and Engineering contains essential functional analytical concepts, explaining its subject without excessive abstraction.

Introduction

1

(4)

1 Ordinary Differential Equations with Boundary Conditions; Eigen-value Problems

5

(50)

1.1 Technical Motivation

5

(5)

1.2 Relation Between the Problem (1.1.3) and (1.1.4) and the Problem Corresponding to the Zero Vertical Loading of the Bar

10

(1)

1.3 Brief Summary of the Space L(2)(a,b)

11

(7)

1.4 Eigenvalue Problems

18

(2)

1.5 Basic Properties of Eigenvalues and Eigenfunctions

20

(7)

1.6 Nonhomogeneous Equations with Boundary Conditions

27

(10)

1.7 The Finite-Difference Method for Ordinary Differential Equations

37

(3)

1.8 Convergence of the Finite-Difference Method

40

(1)

1.9 Application of the Finite-Difference Method in Eigenvalue Problems

41

(2)

1.10 Problems 1.10.1 to 1.10.15

43

(12)

2 Partial Differential Equations: Classical Approach

55

(30)

2.1 Basic Concepts; Examples of Equations Frequently Encountered in Applications; The Heat-Conduction Equation Derived

55

(7)

2.2 Boundary Value Problems (Equations with Boundary Conditions); The Dirichlet Problem for Laplace and Poisson Equations; The Maximum Principle for Harmonic Functions and its Consequences

62

(9)

2.3 The Heat-Conduction Equation

71

(6)

2.4 Problems 2.4.1 to 2.4.12

77

(8)

3 Variational Methods of Solution of Elliptical Boundary Value Problems; Generalized Solutions and Their Approximations; Weak Solutions

85

(70)

3.1 The Equation Au = f

85

(5)

3.2 Comparison Functions, Domain of Definition of Operator A; Symmetrical, Positive, and Positive-Definite Operators

90

(14)

3.3 Theorem on Minimum of Functional of Energy

104

(7)

3.4 Generalized Derivatives; The Energetic Space, the Sobolev Space; Generalized Solutions, Weak Solutions

111

(1)

3.4.1 Functions of One Variable

111

(11)

3.4.2 Functions of Several Variables

122

(1)

3.5 The Ritz and Galerkin Methods; The Finite-Element Method

122

(15)

3.6 Eigenvalue Problems for Equations of Order 2k

137

(4)

3.7 Problems 3.7.1 to 3.7.16

141

(14)

4 The Finite-Difference Method for Partial Differential Equations; The Method of Discretization in Time (the Rothe Method)

155

(28)

4.1 The Finite-Difference Method (the Method of Finite Differences, the Net Method) for Partial Differential Equations

155

(5)

4.2 The Finite-Difference Method for the Heat Equation

160

(1)

4.2.1 The Explicit Scheme

160

(3)

4.2.2 The Implicit Scheme

163

(3)

4.3 The Method of Discretization in Time (the Rothe Method, the Method of Lines)

166

(5)

4.4 Problems 4.4.1 to 4.4.9

171

(12)

5 The Fourier Method

183

(14)

5.1 The Fourier Method for One-Dimensional Vibration Problems

183

(5)

5.2 Problems 5.2.1 to 5.2.8

188

(9)

References

197

(2)

Index

199

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Solving Ordinary and Partial Boundary Value Problems in Science and Engineering

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Table of Contents

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Solving Ordinary and Partial Boundary Value Problems in Science and Engineering

Summary

Table of Contents

Supplemental Materials

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