The Laplace Transform

The following sections are included:

• PREFACE

• CONTENTS

The following sections are included:

• Introduction

• Functions of a Bounded Variation

• The Stieltjes Integral

• Improper Stieltjes Integral

• The Laplace Transform

• Existence and Convergence

• Properties of the Laplace Transform

• The Inversion of the Laplace Transform

• The Convolution Theorem

• Instability of the Inverse of the Laplace Transform

• The Laplace Transform and Differential Equations

• Transient Solutions

• Generating Functions

• Problems

The following sections are included:

• Introduction

• Linear Differential Equations with Constant Coefficients

• The Laplace Transform Solution

• Systems of Linear Differential Equations

• Mismatched Solutions

• Behavior of the Higher Derivatives

• An Example

• Problems

The following sections are included:

• Introduction

• Examples

• Types of Differential-Difference Equations

• Existence and Uniqueness

• Exponential Solutions

• Laplace Transform Solutions

• Order of Growth of Solutions

• The Characteristic Roots

• Asymptotic Behavior of the Solution

• The Shift Theorem

• Equations of the Neutral and Advanced Types

• Linear Systems of Differential-Difference Equations

• A Limiting Case for Differential-Difference Equations

• Problems

The following sections are included:

• Introduction

• The Hyperbolic Partial Differential Equation

• The Parabolic Partial Differential Equation

• The Elliptic Partial Differential Equation

The following sections are included:

• Introduction

• The Formal Laplace Transform Solution

• Exponential Bounds on u(t)

• A Convolution Theorem

• Use of the Contour Integral Representation

• Some Important Results

• Systems of Renewal Equations

• Branching Processes

• Problems

The following sections are included:

• Introduction

• The Complex Laplace Transform

• Numerical Quadrature

• The Legendre Polynomials

• Numerical Inversion of the Laplace Transform

• Explicit Inversion Formula

• Instability of the Inverse of the Laplace Transform

• Tychonov Regularization

• Obtaining the Initial Approximation

• Change of the Time Scale

• Problems

The following sections are included:

• SUBJECT INDEX