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Excel Scientific and Engineering Cookbook by David M Bourg

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11.4. Shooting Boundary Value Problems

Problem

You need to numerically solve a boundary value problem where you're given an ordinary differential equation and boundary conditions in the problem domain.

Solution

You can use the shooting method to solve the boundary value problem in Excel.

Discussion

The shooting method is a well-known iterative method for solving boundary value problems . Consider this example:

This is a second-order equation subject to two boundary conditions, or a standard two-point boundary value problem .

Let y 2 = u and y 1 = du/dt = dy 2/dt to reduce this second-order equation to two first-order equations:

The shooting method attempts to solve this sort of problem as an initial value problem using a marching algorithm like Euler's method or the Runge-Kutta method, as discussed earlier in this chapter. The problem is that the initial conditions are not fully specified; du/dt at t = 0 is unknown. This is the same as saying y 1(0) in the reduced system is unknown. Therefore, the shooting method requires you to make an initial guess at the unknown initial condition and then implement a marching algorithm over the problem domain. Once that is complete, you have to check to see if the values obtained at the end of the marching process satisfy the given boundary conditions ...

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