Approximate the solution to the following partial differential equation using the Backward-Difference method.
∂u / ∂t − ∂2u / ∂x2 = 0, 0 < x < 2, 0 < t;
u(0, t) = u(2, t) = 0, 0 < t, u(x, 0) = sin π / 2 x, 0≤ x ≤ 2.
Use m = 4, T = 0.1, and N = 2, and compare your results to the actual solution u(x, t) = e−(π2/4)t sin π / 2 x.
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