1st Edition
Introduction to Partial Differential Equations for Scientists and Engineers Using Mathematica
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.
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
Biography
Adzievski, Kuzman; Siddiqi, Abul Hasan
"The presentation is simple and clear, with no sacrifice of rigor. Throughout the text, the illustrations, numerous solved examples and the use of Mathematica to 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 MATH 1282
