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Chapter 10 Introduction to Partial Differential Equations

# Chapter 10 Introduction to Partial Differential Equations -...

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1 Chapter 10 Introduction to Partial Differential Equations Contents Introduction ..................................................................................................................................... 2 10.1 Classification of Second-Order Partial Differential Equations ............................................... 2 10.2 Using Dimensionless Variables in the Solution of PDE ......................................................... 5 10.3 Solving Parabolic Partial Differential Equations .................................................................... 7 10.3.1 Explicit Method ........................................................................................................ 10 Example 10.1 Solving the Heat Equation Using an Explicit Method .................................. 12 10.3.2 The Implicit Method ................................................................................................. 18 Example 10.2 Solving the Heat Equation Using the Implicit Method ................................ 20 10.3.3 The Crank-Nicolson Method .................................................................................... 23 10.3.4 Method of Lines ........................................................................................................ 25 Example 10.3 Solution to the Heat Equation Using the Method of Lines ........................... 28 10.4 Solving Elliptic Partial Differential Equations ..................................................................... 30 Example 10.4 Solution to Laplace’s Equation with Constant Boundary Conditions .......... 34 10.5 Summary .............................................................................................................................. 39 Exercises ....................................................................................................................................... 40

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2 Introduction No application of numerical methods in science and engineering is more exciting than the solution of partial differential equations (PDE) and systems of partial differential equations. Development of the computational power to solve the PDE that constitute the mathematical models of global climate change, blood flow near artificial valves, drug metabolism in the liver, evolution of severe storms, production of oil reservoirs, performance of financial markets, or atmospheric reentry of space vehicles, has given scientists and engineers an indispensable new tool that has transformed modern technology. While we have found Excel and Visual Basic to be a powerful combination for many numerical tasks, it is not an adequate tool for solving partial differential equations of interest to scientists and engineering. Comparing the numerical solution of an ordinary differential to the numerical solution of a partial differential equation is like comparing a straight line to a square, a cube, or even a hypercube. The addition of new dimensions to the equations greatly increases the demand for CPU speed, storage capacity, and the ability to represent the solution in an accessible manner. Consequently, this chapter will focus on providing only an introduction to the fundamental techniques used in the solution of PDE. The techniques we will study are quite simple, but they will serve to present the concepts and terminology essential for the study of more advanced techniques or the use of more powerful software packages. Excel with VB is more than adequate to introduce us to this central topic of modern science and engineering. For simplicity we will mostly limit our study to second order PDE. Extension of these techniques to first order or to higher order PDE is surprisingly straightforward, so the student should not be concerned about the adequacy of this introduction to prepare them for using even the most sophisticated simulations. The examples we will use are examples of linear PDE, but these examples will prepare the student for extension of the approach to nonlinear equations. The method of lines in particular is well suited for application to nonlinear PDE.
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Chapter 10 Introduction to Partial Differential Equations -...

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