Numerical Solutions for Partial Differential Equations

Problem Solving Using Mathematica

Victor Grigor'e Ganzha, Evgenii Vasilev Vorozhtsov

• Publisher: CRC Press
• Year: 2017
• eBook ISBN: 9781351427500
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Partial differential equations (PDEs) play an important role in the natural sciences and technology, because they describe the way systems (natural and other) behave. The inherent suitability of PDEs to characterizing the nature, motion, and evolution of systems, has led to their wide-ranging use in numerical models that are developed in order to analyze systems that are not otherwise easily studied. Numerical Solutions for Partial Differential Equations contains all the details necessary for the reader to understand the principles and applications of advanced numerical methods for solving PDEs. In addition, it shows how the modern computer system algebra Mathematica® can be used for the analytic investigation of such numerical properties as stability, approximation, and dispersion.

Chapter 1

INTRODUCTION TO MATHEMATICA

1.1 GENERAL INFORMATION ABOUT MATHEMATICA

The development of the program package Mathematica was initiated by Stephen Wolfram in the 1980s. Now versions of Mathematica are available for a wide variety of computer systems, including: Apple Macintosh, IBM PC compatibles (MS-DOS and Microsoft Windows), Hewlett-Packard/Apollo, NeXT, Sony, Sun, and a number of other computer systems. Mathematica Version 2, as described in Wolfram’s book¹, was first released in January 1991.

The Mathematica software package unites a number of functions in the same interactive environment:

1. Numerical computations,

2. Symbolic computations,

3. Computer graphics.

The Mathematica program itself, or the kernel, now includes over 300,000 lines of the source text. It is written in the object oriented extension of the programming language C developed for Mathematica². By using a pre-compiler program a C-program is generated, which can be compiled into an executable computer code on different computer systems. In this way it became possible to extend the same huge program during a short time for various types of computers (from Macintosh and PC to supercomputers).

There are now a number of handbooks on Mathematica, see, in particular, Reference 1, Reference 2, Reference 3, Reference 4, Reference 5, Reference 6, Reference 7, Reference 8, Reference 9, Reference 10, Reference 11, Reference 12, Reference 13, Reference 14, Reference 15, Reference 16, Reference 17, Reference 18, Reference 19, Reference 20, Reference 21. The handbooks^2,19,20,21 show how the Mathematica graphics functions can be used in various situations. There have also appeared a number of books, in which Mathematica is applied for the solution of a number of applied problems in the natural sciences, see, in particular, Reference 22, Reference 23, Reference 24, Reference 25, Reference 26, Reference 27, Reference 28, Reference 29, Reference 30, Reference 31, Reference 32, Reference 33, Reference 34, Reference 35, Reference 36, Reference 37, Reference 38. It should be noted that none of the existing books addresses systematically of the application of the Mathematica for the development, investigation, and computer implementation of the numerical methods for solving elementary PDEs. Only Chapter 5 of the book²⁵ is devoted to the numerical solution of the Korteweg-de Vries equation.

We have described in our book the applications of Mathematica for the numerical solution of the partial differential equations. The elements of computer graphics with the Mathematica system are also included to help the reader to better understand the physics of the phenomena, which are modeled on a computer with the aid of finite difference schemes. The necessary explanations on the practical use of the Mathematica functions will be given whenever Mathematica is used. The outstanding feature of Mathematica is that it uses symbolic expressions to provide a very general representation of mathematical and other structures. The generality of symbolic expressions allows Mathematica to cover a wide variety of applications...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• 1 INTRODUCTION TO MATHEMATICA
• 2 FINITE DIFFERENCE METHODS FOR HYPERBOLIC PDEs
• 3 FINITE DIFFERENCE METHODS FOR PARABOLIC PDEs
• 4 NUMERICAL METHODS FOR ELLIPTIC PDEs
• APPENDIX GLOSSARY OF PROGRAMS