Question

3. Solve the initial-boundary value problem. ur(0,t) 0, u(4,t) 0, A, 0 2 u(z,0) =

show all the work

Answer 1

"Given partial differential u/ partial differential t = 3 partial differential^2 u/ partial differential x^2, 0 < x < 4 and u_x (0, t) = 0, u_x (4, t) = 0 u(x, 0) = {A: 0 lessthanorequalto x lessthanorequalto 2 0: 2 < x lessthanorequalto 4 let u(x, + 1) = X (x) T (+) be the solution of given problem now partial differential u/partial differential t = X T' and partial differential^2 2 u/partial differential x^2 = "" T Therefore X T' = 3 X"" T Rightarrow T'/T = 3X""/X = -lambda^2 (say) where lambda is separation constant we have T'/T = -lambda^2 and 3 X""/X = -X^2 Rightarrow T' + lambda^2 T = 0 and 3X"" + lambda^2 X = 0 Rightarrow T(t) = c_1 e^- lambda^2 t and X(x) = c_2 cos lambda/squareroot 3 x + c_3 sin lambda/squareroot 3 x where c_1, c_2 and c_3 are arbitrary Therefore our solution becomes"