Introductory Laplace Transform with Applications

By Dalpatadu and Singh

One of the first applications of the modern Laplace transform was by Bateman in 1910 who used it to transform Rutherfords equations in his work on radioactive decay. The modeling of complex engineering and physical problems by linear differential equations has made the Laplace transform an indispensable mathematical tool for engineers and scientists. The method of Laplace transform for solving linear differential equations is very popular in the disciplines of electrical engineering, environmental engineering, hydrology, and petroleum engineering. This book presents some applications of Laplace transforms in these disciplines. Algorithms for the numerical inversion of Laplace transform are given, and a computer program in R for the Stehfest algorithm is included.

• Language: English
• Publisher: Trafford Publishing
• Release date: Jul 17, 2015
• ISBN: 9781490760698

Book preview

CONTENTS

Preface

1. Introduction

Applications of the Laplace Transform in Electrical Engineering

Applications of the Laplace Transform in Environmental Engineering

Application of the Laplace Transform in Hydrology

Applications of the Laplace Transforms in Petroleum Engineering

2. The Laplace Transform and its Properties

3. Applications of the Laplace Transform

Example 3.1

Example 3.2

Example 3.3

Example 3.4

Example 3.5

Example 3.6

Example 3.7

Example 3.8

Example 3.9

Example 3.10

Example 3.11

Example 3.12

Example 3.13

Example 3.14

Example 3.15

Example 3.16

Example 3.17

Example 3.18

Example 3.19

Example 3.20

4. Numerical Inversion of the Laplace Transform

4.1: Gaver-Stehfest Method

4.2: An R code for the Method of Gaver-Stehfest

4.3: Introduction to R

4.0.1 DOWNLOAD R

4.0.2 COMPUTATIONS IN R

4.4: Introduction to RStudio

About the Authors

PREFACE

Numerous books on theory and applications of the Laplace transform are available; most of these books are written primarily for mathematicians. By contrast, this book attempts to present the material in a simplified manner, and examples from various disciplines from existing literature are included to demonstrate the method of Laplace transform. A computer program in the R programming language for numerical inversion of the Laplace transform is included.

The target audience for this book is a researcher in one of the following disciplines: petroleum engineering, environmental engineering, contaminant hydrology, electrical engineering, finance and economics. This book can be used as supplemental material in undergraduate and graduate classes in engineering mathematics, and also as a handbook of Laplace transforms by practicing engineers and scientists.

1. INTRODUCTION

Integral transforms were introduced by Euler in 1763 as a tool for solving second-order linear differential equations, and Spitzer in 1878 gave the name of Laplace transform to the integral¹

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One of the first applications of the modern Laplace transform was by Bateman in 1910 who used it to transform Rutherford’s equations in his work on radioactive decay². The modeling of complex engineering and physical problems by linear differential equations has made the Laplace transform an indispensable mathematical tool for engineers and scientists. The method of Laplace transform for solving linear differential equations is very popular in the disciplines of electrical engineering, environmental engineering, hydrology, and petroleum engineering. The Laplace transform method is also used in statistics for deriving the distribution of sums of independent random variables. Applications of the Laplace transformcan be found in finance and economics. We briefly describe some applications of Laplace transforms in these disciplines.

Applications of the Laplace Transform in Electrical Engineering

The Laplace transform plays an important role in the discipline of control systems engineering. A control system is typically analyzed by computing the Laplace transforms of different functions of time, and then using inversion to find the solution to the problem. The method of Laplace transform is widely used in digital signal processing³as well.

Applications of the Laplace Transform in Environmental Engineering

Due to the complexity of the channels through which ground water flows, it is difficult to consider a model of the movement of ground water at a microscopic level. The French engineer Darcy, however, developed an equation that effectively averages the microscopic complexities, and provides a macroscopic model of ground water movement. Darcy’s law is central to the derivation of equations used to model the flow of groundwater. The mathematical models, based on the physics of ground water flow and boundary conditions imposed by the ground water basin in question, usually take the form of a boundary value problem⁴, which are solved either analytically or computationally⁵. Numerical techniques are the basis for digital computer simulation of transient ground water flow in aquifers. The two most widely used numerical methods for solving ground water equations are the finite difference and the finite element techniques.

Application of the Laplace Transform in Hydrology

The Laplace transform is commonly used in subsurface contaminant hydrology⁶. The problems of contaminant transport in fractured porous formation have attracted considerable attention especially when dealing with the disposal of radioactive waste in underground repositories⁷. Using Laplace transform techniques, numerousproblems arising in the study of migration of radionuclides in fractures as well as in the surrounding rock have been investigated. In the studies of radionuclear waste repositories, the problem of nuclide migration has been well investigated⁸. The Laplace transform has been used to solve the problem of migration of radionuclides in fissured rock under the influence of micropore diffusion and longitudinal dispersion⁹, the case of constant source strength¹⁰, and for contaminant transport with radioactive decay in fractured porous rock under radial flow condition¹¹.

Analytical solutions for steady state tracer transport with precipitation-dissolution reactions in multiple-fracture systems have been obtained using the method of Laplace