(a) (b) Let k, L> 0 and consider the non-homogeneous heat equation ut(x, t) - kuxx...

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(a) (b) Let k, L> 0 and consider the non-homogeneous heat equation ut(x, t) - kuxx (x, t) = H(x, t) over the interval [0, L] subject to general initial and Dirichlet conditions u(x,0) = f(x), u(0, t) = a(t), u(L, t) = b(t). Show that at most one solution exists for any given choice of a, b, ƒ, H. Derive a formula for the solution of the problem in part (a) when H(x, t) = f(x) = 0, a(t) = b(t) = t. (a) (b) Let k, L> 0 and consider the non-homogeneous heat equation ut(x, t) - kuxx (x, t) = H(x, t) over the interval [0, L] subject to general initial and Dirichlet conditions u(x,0) = f(x), u(0, t) = a(t), u(L, t) = b(t). Show that at most one solution exists for any given choice of a, b, ƒ, H. Derive a formula for the solution of the problem in part (a) when H(x, t) = f(x) = 0, a(t) = b(t) = t.

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a Proof of Uniqueness To show that at most one solution exists well employ a proof by contradiction ... View the full answer