Use the Forward-Difference method to approximate the solution to the following parabolic partial differential equations. a. u t 2u x 2 = 0, 0 < x < 2, 0 < t; u(0, t) = u(2, t) = 0, 0 < t, u(x, 0) = sin 2x, 0 x 2. Use h = 0.4 and k = 0.1, and compare your results at t = 0.5 to the actual solution u(x, t) = e42t sin 2x. Then use h = 0.4 and k = 0.05, and compare the answers. b. u t 2u x 2 = 0, 0 < x < , 0 < t; u(0, t) = u(, t) = 0, 0 < t, u(x, 0) = sin x, 0 x . Use h = /10 and k = 0.05, and compare your results at t = 0.5 to the actual solution u(x, t) = et sin x.

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