Question

8.3 Consider the periodic wave equation for t E R and x e T. Suppose the initial conditions are or (0, x) h(x), for g є Cm +1 (T) and h є Cm(T), for m E N. 152 8 Fourier Series (a) Assuming that u(t, x) can be represented as a Fourier series u(t, x)- av(l)ekx, kEZ (8.48) find an expression for ak() in terms of the Fourier coefficients of g and h.

Answer 1

partial differential^2 u/ partial differential t^2 = partial differential^2 u/partial differential x^2 equation (1) u(x, 0) = g(x) partial differential u/partial differential t (x, 0) = h(x): u(v, f) = 0 u(l, t) Let u(x, t) = X(x) T(t) equation (2) be solution of (1) partial differential u/partial differential x = X'(x) T(t): partial differential^2 u/partial differential x^2 = X''(x) T(t) partial differential u/partial differential x = X(x) T''(t): partial differential^2 u/partial differential t^2 = X(x) T''(t)} equation (3) using (3) in (1) we get X(x) T''(t) = T(t) X''(x) T''(t)/T(t) = X''(x)/X(x) = lambda equation (4) T''(t) = lambda T(t) & X''(x) = lambda X(x) equation (5) which are linear ODE with constant coeff. Then solution of (3) is given by (i) X = C_1 e^kx + C_2 e^-kx, T = C_3 e^kt + C_4 e^-kt when lambda = k^2 > 0 (ii) X = C_5 cos kx + C_6 sin kx, T = C_7 cos kt + C_8 sin kt when lambda = k^2 < 0
we use variable separable method