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Introduction to Numerical Methods in Differential Equations

  • Textbook
  • © 2007

Overview

Editors:
  1. Mark H. Holmes

    1. Academic Science of the Material Science and Engineering, Rensselaer Polytechnic Institute, Troy

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  • Over 100 illustrations
  • Numerous exercise sets, MATLAB computer codes (for both students and instructors), instructor overheads, and movies included
  • Includes supplementary material: sn.pub/extras

Part of the book series: Texts in Applied Mathematics (TAM, volume 52)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xi
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  2. Initial Value Problems

    Pages 1-43

  3. Two-Point Boundary Value Problems

    Pages 45-82

  4. Diffusion Problems

    Pages 83-126

  5. Advection Equation

    Pages 127-154

  6. Numerical Wave Propagation

    Pages 155-180

  7. Elliptic Problems

    Pages 181-222

  8. Back Matter

    Pages 223-239
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Keywords

About this book

The title gives a reasonable ?rst-order approximation to what this book is about. To explain why, let’s start with the expression “di?erential equations.” These are essential in science and engineering, because the laws of nature t- ically result in equations relating spatial and temporal changes in one or more variables.Todevelopanunderstandingofwhatisinvolvedin?ndingsolutions, the book begins with problems involving derivatives for only one independent variable, and these give rise to ordinary di?erential equations. Speci?cally, the ?rst chapter considers initial value problems (time derivatives), and the second concentrates on boundary value problems (space derivatives). In the succeeding four chapters problems involving both time and space derivatives, partial di?erential equations, are investigated. This brings us to the next expression in the title: “numerical methods.” This is a book about how to transform differential equations into problems that can be solved using a computer.The fact is that computers are only able to solve discrete problems and generally do this using ?nite-precision arithmetic. What this means is that in deriving and then using a numerical algorithmthecorrectnessofthediscreteapproximationmustbeconsidered,as must the consequences of round-o? error in using ?oating-point arithmetic to calculatetheanswer.Oneoftheinterestingaspectsofthesubjectisthatwhat appears to be an obviously correct numerical method can result in complete failure. Consequently, although the book concentrates on the derivation and use of numerical methods, the theoretical underpinnings are also presented andusedinthedevelopment.

Editors and Affiliations

  • Academic Science of the Material Science and Engineering, Rensselaer Polytechnic Institute, Troy

    Mark H. Holmes

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