Question

Let u be the solution to the initial boundary value problem for the Heat Equation, фа(t, x)-5 &n(t, x), t E (0,00), x E (0, 1); with initial condition 2 r-, 1 and with boundary condition:s n(t, 0)=0, rn(t, 1-0. Find the solution u using the expansion with the normalization conditions vn (0)-1, wn a. (3/10) Find the functions wz, with indexn> 1 b. (3/10) Find the functions v, with index n> 1. c. (4/10) Find the coefficients cn, with indexn> 1.

Answer 1

"Given that r_t u(t, x) = 5 partial difference_x^2 u(t, x), t elementof (0, infinity), x elementof (0, 1)} Initial Condition u(0, x) = f(x) = {x x elementof [0, 1/2] 1/2, x elementof [1.2, 1] let x(0) = 1, x'(0) = 0 u(x, t) = X(x) T(t) u'(x, t) - X'(x) T(t) u_xx (x, t) = X''(x) T(t) u_t'(x, t) = X(x) T'(t) X(x) T'(t) = X''(x) T(t) X(x) T'(t)/T(t) = 2 X""(x) T'(t)/T(t) = 2 X''(x)/x(x) 1/2 T'(t)/T(t) = X""(x)/X(x) \r\n since x''(x)/x(x) = lambda (i) since lambda = 0 x''(x)/x(x) = 0 x''(x) = 0 x(x) = c_1 x + x_2 x(0) = 0 rightarrow c_2 = 0 x'(5) = 0 rightarrow c_1 = 0 X(x) = 0 (ii) since lambda = sigma^2 x''(x)/X(x) = sigma^2 x''(x) = sigma^2 x(x) x''(x) - sigma^x(x) = 0 X(X) = e6mx x'(x) = m e^mx \r\n x''(x) = m^2 e^mx (m^2 - sigma^2) e^mx = 0 m = t sigma x(x) = c_1 e^sigma x + c_2 -sigma x"