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# 12.3. Solving Initial Value Problems

## Problem

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.

## Solution

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

## Discussion

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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