CONTENTS -- PREFACE -- Chapter 1 THE LAPLACE TRANSFORM -- 1.1 Introduction -- 1.2 Functions of a Bounded Variation -- 1.3 The Stieltjes Integral -- 1.4 Improper Stieltjes Integral -- 1.5 The Laplace Transform -- 1.6 Existence and Convergence -- 1.7 Properties of the Laplace Transform -- 1.8 The Inversion of the Laplace Transform -- 1.9 The Convolution Theorem -- 1.10 Instability of the Inverse of the Laplace Transform -- 1.11 The Laplace Transform and Differential Equations -- 1.12 Transient Solutions -- 1.13 Generating Functions -- Problems -- Chapter 2 ORDINARY DIFFERENTIAL EQUATIONS -- 2.1 Introduction -- 2.2 Linear Differential Equations With Constant Coefficients -- 2.3 The Laplace Transform Solution -- 2.4 Systems of Linear Differential Equations -- 2.5 Mismatched Solutions -- 2.6 Behavior Of The Higher Derivatives -- 2.7 An Example -- Problems -- Chapter 3 DIFFERENTIAL-DIFFERENCE EQUATIONS -- 3.1 Introduction -- 3.2 Examples -- 3.3 Types of Differential-Difference Equations -- 3.4 Existence and Uniqueness -- 3.5 Exponential Solutions -- 3.6 Laplace-Transform Solutions -- 3.7 Order Of Growth Of Solutions -- 3.8 The Characteristic Roots -- 3.9 Asymptotic Behavior of the Solution -- 3.10 The Shift Theorem -- 3.11 Equations of the Neutral and Advanced Types -- 3.12 Linear Systems Of Differential-Difference Equations -- 3.13 A Limiting Case For Differential-Difference Equations -- Problems -- Chapter 4 PARTIAL DIFFERENTIAL EQUATIONS -- 4.1 Introduction -- 4.2 The Hyperbolic Partial Differential Equation -- 4.3 The Parabolic Partial Differential Equation -- 4.4 The Elliptic Partial Differential Equation -- Chapter 5 THE RENEWAL EQUATION -- 5.1 Introduction -- 5.2 The Formal Laplace Transform Solution -- 5.3 Exponential Bounds on u(t) -- 5.4 A Convolution Theorem -- 5.5 Use of the Contour Integral Representation -- 5.6 Some Important Results.

5.7 Systems of Renewal Equations -- 5.8 Branching Processes -- Problems -- Chapter 6 NUMERICAL INVERSION OF THE LAPLACE TRANSFORM -- 6.1 Introduction -- 6.2 The Complex Laplace Transform -- 6.3 Numerical Quadrature -- 6.4 The Legendre Polynomials -- 6.5 Numerical Inversion of the Laplace Transform -- 6.6 Explicit Inversion Formula -- 6.7 Instability of the Inverse of the Laplace Transform -- 6.8 Tychonov Regularization -- 6.9 Obtaining the Initial Approximation -- 6.10 Change of the Time Scale -- Problems -- SUBJECT INDEX.