Introduction to Partial Differential Equations for Scientists and Engineers Using Mathematica

Features

 

 

Covers basic theory, concepts, and applications of PDEs in engineering and science Includes solutions to selected examples as well as exercises in each chapter Uses Mathematica along with graphics to visualize computations, improving understanding and interpretation Provides adequate training for those who plan to continue studies in this area

 

Summary

 

With a special emphasis on engineering and science applications, this textbook provides a mathematical introduction to PDEs at the undergraduate level. It takes a new approach to PDEs by presenting computation as an integral part of the study of differential equations. The authors use Mathematica® along with graphics to improve understanding and interpretation of concepts. They also present exercises in each chapter and solutions to selected examples. Topics discussed include Laplace and Fourier transforms as well as Sturm-Liouville boundary value problems.

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 
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 
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 
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
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
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