4. Consider the initial boundary values problem U = Uzz u(0, t) = g(t), u(1, t)...

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4. Consider the initial boundary values problem U₁ = Uzz¹ u(0, t) = g(t), u(1, t) = 92(t), te [0, T] u(x,0) = f(x), τε [0,1] (x, t) € (0, 1) x (0,T] = D Let (₁, tn) be a grid point in a uniform rectangular grid, s.t. ; = iAr, tn = nAt for i = 0, 1,..., I and n = 0, 1,..., N where IAr = 1 and NAt = T, and let U be a numerical approximation of u(xi, tn). Assuming the exact solution is sufficiently smooth, show that the scheme Un+¹ - Un Un+¹ - 2U+¹ + Unt i+1 i-1 At AT² is unconditionally stable and U- u(xi, tn) = O(At + Ar²) as At, AT →0. 4. Consider the initial boundary values problem U₁ = Uzz¹ u(0, t) = g(t), u(1, t) = 92(t), te [0, T] u(x,0) = f(x), τε [0,1] (x, t) € (0, 1) x (0,T] = D Let (₁, tn) be a grid point in a uniform rectangular grid, s.t. ; = iAr, tn = nAt for i = 0, 1,..., I and n = 0, 1,..., N where IAr = 1 and NAt = T, and let U be a numerical approximation of u(xi, tn). Assuming the exact solution is sufficiently smooth, show that the scheme Un+¹ - Un Un+¹ - 2U+¹ + Unt i+1 i-1 At AT² is unconditionally stable and U- u(xi, tn) = O(At + Ar²) as At, AT →0.