Question

Find the solution by Fourier series of the heat equation with nonhomogeneous boundary conditions. Assume that the initial condition is given byf(x) uo u(0,t) = uo, u(L,t) = u1, t > 0 u(x,0) = f(x), 0 < x < L

Answer 1

Given that, u_t = c^2 u_xx + q(x): 0 < x < L u(0, t) = u0: u(1, t) = u_L: t > 0 u(x, 0) = f(x) , 0 < x < L This is heat equation with non homogeneous boundary conditions so, we will have to choose w(*) show that w(0, t) = u_0: w(L, t) = u_L w(u, t) = u_0 + (u_L - u_0)/L x and choose vacpts that w(x, t) = u(x, t) - w(x, t) v_t = c^t v_xx + q(x) v(0, t) = 0: v(L, t) = 0 v(x, 0) = f(x) can be written as v_t (x, t) = c^2 v_xx (x, t) , 0 < x < L v_1 (0, t) = 0: v_1 (L, t) = 0 rightarrow (A) v_1 (x, 0) = f(x) and, v_2t (x, t) = c^2 v_2xx + q(x): 0 < x < L: t > 0 v_2 (0, t) = 0: v_2 (L, t) = 0