# Introduction to Partial Differential Equations for Scientists and Engineers Using Mathematica

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## Book Description

With special emphasis on engineering and science applications, this textbook provides a mathematical introduction to the field of partial differential equations (PDEs). The text represents a new approach to PDEs at the undergraduate level by presenting computation as an integral part of the study of differential equations. The authors use the computer software Mathematica® along with graphics to improve understanding and interpretation of concepts. The book also presents solutions to selected examples as well as exercises in each chapter. Topics include Laplace and Fourier transforms as well as Sturm-Liuville Boundary Value Problems.

## Table of Contents

**Fourier Series **The Fourier Series of a Periodic Function

Convergence of Fourier Series

Integration and Differentiation of Fourier Series

Fourier Sine and Fourier Cosine Series

*Mathematica*Projects

**Integral Transforms**

The Fourier Transform and Elementary Properties

Inversion Formula of the Fourier Transform

Convolution Property of the Fourier Transform

The Laplace Transform and Elementary Properties

Differentiation and Integration of the Laplace Transform

Heaviside and Dirac Delta Functions

Convolution Property of the Laplace Transform

Solution of Differential Equations by the Integral Transforms

**The Sturm-Liouville Problems**

Regular Sturm-Liouville Problem

Eigenvalues and Eigenfunctions

Eigenfunction Expansion

Singular Sturm-Liouville Problem: Legendre’s Equation

Singular Sturm-Liouville Problem: Bessel’s Equation

Partial Differential Equations

Partial Differential Equations

Basic Concepts and Definitions

Formulation of Initial and Boundary Problems

Classification of Partial Differential Equations

Some Important Classical Linear Partial Differential Equations

The Principle of Superposition

First Order Partial Differential Equations

First Order Partial Differential Equations

Linear Equations with Constant Coefficients

Linear Equations with Variable Coefficients

First Order Non-Linear Equations

Cauchy’s Method of Characteristics

Mathematica Projects

Hyperbolic Partial Differential Equations

Hyperbolic Partial Differential Equations

The Vibrating String and Derivation of the Wave Equation

Separation of Variables for the Homogeneous Wave Equation

D’Alambert’s Solution of the Wave Equation

Inhomogeneous Wave Equations

Solution of the Wave Equation by Integral Transforms

Two Dimensional Wave Equation: Vibrating Membrane

The Wave Equation in Polar and Spherical Coordinates

Numerical Solutions of the Wave Equation

Mathematica Projects

Parabolic Partial Differential Equations

Parabolic Partial Differential Equations

Heat Flow and Derivation of the Heat Equation

Separation of Variables for the One Dimensional Heat Equation

Inhomogeneous Heat Equations

Solution of the Heat Equation by Integral Transforms

Two Dimensional Heat Equation

The Heat Equation in Polar and Spherical Coordinates

Numerical Solutions of the Heat Equation

Mathematica Projects

Elliptic Partial Differential Equations

Elliptic Partial Differential Equations

The Laplace and Poisson Equations

Separation of Variables for the Laplace Equation

The Laplace Equation in Polar and Spherical Coordinates

Poisson Integral Formula

Numerical Solutions of the Laplace Equation

Mathematica Projects

**Appendix A. Special Functions**

**Appendix B. Table of the Fourier Transform of Some Functions**

**Appendix C. Table of the Laplace Transform of Some Functions**

## Author(s)

### Biography

Adzievski, Kuzman; Siddiqi, Abul Hasan

## Reviews

"The presentation is simple and clear, with no sacrifice of rigor. Throughout the text, the illustrations, numerous solved examples and the use of

Mathematicato visualize computations have been chosen to make the exposition as clear as possible. The book represents a good tool for facilitating the proper understanding of basic concepts and applications of PDEs."

—Zentralblatt MATH1282