You're dealing with a parabolic or hyperbolic problem and have developed the finite difference equations for the problem, which includes both discretized spatial and time variables. Now you have to solve these equations.

Use Excel and Solver to solve the finite difference equations in a manner similar to that discussed in Recipe 12.2.

In the previous recipe, I showed you how to leverage Solver to solve the finite difference equations, arriving at a steady state solution to an elliptic-type boundary value problem. You can apply the same techniques to solve time-dependent parabolic or hyperbolic types of problems.

Consider the time-dependent one -dimensional heat equation:

This is a parabolic equation, which can be used to model the time-dependent heat conduction in a metal rod, for example, subject to some prescribed initial temperature distribution and boundary conditions at the ends of the rod.

Let's assume we have a 1m rod that's insulated all around except at the ends. The ends of the rod are maintained at a constant 10°C and the initial temperature distribution along the length of the rod (at time *t* = 0) is as shown in Figure 12-7, row 13. Further assume that *c*
^{2} equals 1. Now we can use the finite difference method to solve for the temperature distribution along the length of the rod over time.

Erwin Kreyszig solves a ...

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