Question

2 ef 4P eorevm P2x2 ci

Answer 1

Application of Footer transform to solve Boundary value problem One dimension heat equation is partial differential y/partial differential t = k partial differential^2 y/partial differential x^2 = -infinity < x < infinity, t > 0 ---(I) leftarrow__infinity X rightarrow_infinity u(x, t) with u(x, o) = f(x) -infinity < x < infinity, u(0, t) = 0: t > 0 since rod is infinite (-infinity < x < infinity) we can apply here Fourier transform with respect to x if rod is semi infinite (0 lessthanorequalto x < infinity) we can apply Fourier sine (or Fourier cosine) transform depending upon given conditions on u(x, f) and u(x, t) we write F(u(x, t) = u(w, t) such that F(partial differential^2 u/ partial differential x^2) = -w^2 u(w, t) and F(partial differential u/partial differential t) = partial differential/partial differential t (u(w, t)) Taking Fourier transform of (I)_n and writing F(u(x, t)) = u(w, t) we have \r\n partial differential/partial differential t u(w, t) = k (-w^2 u(w, t))