the two‐point boundary value problem problemusing a range of mesh sizes, starting with , and going as far as . Comment on your results.Solution: For larger values of , the approximate solution is erratic and wildly oscillatory. It isn't until about or so that the solution begins to settle down.7 Generalize the solution of the two‐point boundary value problem to the case where and . Apply this to the solution of the problemwhich has exact solutionSolve this for a range of values of the mesh. Do we get the expected accuracy?
8 Consider the problem of determining the deflection of a thin beam, supported at both ends, due to a uniform load being placed along the beam. In one simple model, the deflection as a function of position along the beam satisfies the boundary value problemHere is a constant that depends on the material properties of the beam, is the length of the beam, and depends on the material properties of the beam as well as the size of the load placed on the beam. For a six‐foot‐long beam, with and , what is the maximum deflection of the beam? Use a fine enough grid that you can be confident of the accuracy of your results. Note that this problem is slightly more general than our example (2.28)‐(2.29); you will have to adapt our method to this more general case.Solution: See Figure 2.6.
9 Repeat the above problem, except this time use a three‐foot‐long beam. How much does the maximum deflection change, and is it larger or smaller?
10 Repeat the beam problem again, but this time use a 12‐foot‐long beam.
11 Try to apply the ideas of this section to the solution of the nonlinear boundary value problem defined byFigure 2.6 Exact solution for Exercise 2.7.8.Write out the systems of equations for the specific case of . What goes wrong? Why can't we proceed with the approximate solution?Solution: The problem is that the system of equations that is produced by the discretization process is nonlinear, so our inherently linear algorithm won't work.
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