Introduction to Numerical Methods in Differential Equations

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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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.

• Boundary value problem
• MATLAB
• arithmetic
• differential equation
• partial differential equation
• partial differential equations
• ordinary differential equations

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

  Mark H. Holmes