# Methods of Contour Integration, Volume 3

## 1st Edition

**Editors:**H. A. Lauwerier W. T. Koiter

**Authors:**M. L. Rasulov

**eBook ISBN:**9781483275000

**Imprint:**North Holland

**Published Date:**1st January 1967

**Page Count:**440

**View all volumes in this series:**North-Holland Series in Applied Mathematics and Mechanics

## Table of Contents

Preface

Introduction

Part One The Residue Method

Chapter 1 Dini's Theorem Generalised

1. Lagrange's Formula and Systems of Integral Equations

2. Generalised Existence Theorem

3. Existence Theorems

Chapter 2 Asymptotic Representations of Solutions of Linear Differential Equations with a Complex Parameter

1. Formal Solutions of First-Order Systems

2. Asymptotic Representations of Solutions of a System of First-Order Equations

3. Asymptotic Representations of Solutions of a Single Equation of Higher Order

Chapter 3 Expansion of Vector-Valued Functions

1. Boundary-Value Problems for a System of First-Order Equations with Piece-Wise Coefficients

2. Theorem on the Expansion in Series of Residues of Solutions of Boundary-Value Problems with a Parameter for Systems of Ordinary Differential Equations with Discontinuous Coefficients

3. Derivation of the Solution of the Spectral Problem for a Single Equation of Higher Order with Discontinuous Coefficients

Chapter 4 Solution of One-Dimensional Mixed Problems for Systems of Equations with Discontinuous Coefficients

1. Mixed Problems with Boundary Conditions Containing Time Derivatives

2. Mixed Problems with Boundary Conditions Containing No Time Derivatives

3. The Mixed Problem with Separate Variables

Chapter 5 Residue Method for Solving Multi-Dimensional Mixed Problems

1. Procedure for Solving Multi-Dimensional Mixed Problems

2. Residue Method of Separating Variables

3. Formula for Expanding an Arbitrary Function in a Series of Residues of a Solution of a Certain Class of Multi-Dimensional Spectral Problems

4. Problems in Subterranean Hydromechanics

Part Two The Contour-Integral Method

Chapter 6 Contour-Integral Method of Solving One-Dimensional Mixed Problems for Second-Order Equations with Discontinuous Coefficients

1. Equations Containing Only First-Order Time Derivatives

2. Asymptotic Representation of the Solution of a Spectral Problem Outside a δ-Neighbourhood of the Spectrum

3. Solution of the Mixed Problem (6.1.1) - (6.1.3) with Parabolicity in the Sence of Petrovskiy

4. Expansion of an Arbitrary Function in a Series of Residues of the Spectral Problem: Necessary and Sufficient Conditions for the Correct Formulation of Problem (6.1.1) - (6.1.3)

5. Solution of Mixed Problems for Equations Containing First-Order Time Derivatives: Necessary and Sufficient Conditions

Chapter 7 Solution of One-Dimensional Mixed Problems for Linear Differential Equations with Discontinuous Coefficients and Time-Dependent Boundary Conditions

1. Asymptotic Representation of the Solution of a Spectral Problem Outside a δ-Neighbourhood

2. Solution of Mixed Problems for Equations Containing Only First-Order Time Derivatives

Chapter 8 Solution of a Multi-Dimensional Spectral Problem for a Single Elliptic Equation with a Large Complex Parameter

1. Fundamental Solution and its Bounds

2. Formulae for the Saltus in the Potentials of a Single and a Double Layer

3. Solution of the Spectral Problem for a Homogeneous Equation and Bounds for it

4. Bound for the Regular Part of the Green's Function of the Spectral Problem

Chapter 9 Multi-Dimensional Mixed Problem for a Parabolic Equation with Time-Dependent Boundary Conditions

1. Mixed Problem for a Homogeneous Equation with Homogeneous Initial Condition

2. Mixed Problem for a Non-Homogeneous Equation with Homogeneous Boundary but Non-Homogeneous Initial Conditions

3. Actual Solution of Mixed Problems

Chapter 10 Multi-Dimensional Mixed Problem for Parabolic Equations with Discontinuous Coefficients and Time-Dependent Boundary Conditions

1. A Mixed Problem and Corresponding Spectral Problem

2. Fundamental Solution of the Spectral Problem and a Bound for its Absolute Value

3. Solution of the Spectral Problem and a Bound for its Absolute Value

4. Solution of the Mixed Problem for Equations with Discontinuous Coefficients

References

Index

## Description

Methods of Contour Integration contains two parts: (1) a systematic exposition of the computational method for solving boundary and mixed problems, and (2) the contour-integral method for investigating general linear mixed problems. The first part includes formulae for expanding arbitrary vector-valued functions in series from integral residues of solutions of boundary-value problems for systems of ordinary differential equations with discontinuous coefficients. These formulae give residue representations of solutions of the corresponding one-dimensional mixed problems for equations with discontinuous coefficients. The book also explains a computational method of separating the variables which is a generalization of the ordinary method of separating variables to the case of nonself-adjoint operators. In part two, the text discusses one-dimensional mixed problems for equations with discontinuous coefficients. Under regular boundary conditions, it proves the existence of solutions for these problems and the representability of the solutions in the form of contour integrals with a complex parameter. The text points out that the contour-integral method is also applicable to parabolic equations and to equations in which the coefficients are functions of time. The book is ideal for mathematicians, students, and professor of calculus and advanced mathematics.

## Details

- No. of pages:
- 440

- Language:
- English

- Copyright:
- © North Holland 1967

- Published:
- 1st January 1967

- Imprint:
- North Holland

- eBook ISBN:
- 9781483275000