(a) Solve by hand the nonlinear PDE ut =uxx+(u)3 for all x
Chapter 8, Problem 12(choose chapter or problem)
(a) Solve by hand the nonlinear PDE ut =uxx+(u)3 for all x using the standard forward difference scheme with (u)3 treated as (un j)3. Use s = 1 4, t =1, and initial data u0 j =1 for j = 0 and u0 j =0for j =0. Solve for u3 0. (b) Compare your answer to the same problem without the nonlinear term. (c) Exactly solve the ODE dv/dt =(v)3 with the condition v(0)=1. Use it to explain why u3 0 is so large in part (a). (d) Repeat part (a) with the same initial data but for the PDE ut = uxx(u)3. Compare with the answer in part (a) and explain.
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