Question

7. Consider the boundary value problem for the Laplace equation on the strip u (0, y) u (т, y) = 0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x, y) -ZYn (v)sinnx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y)-Yn (y) sin n. the Laplace equation and the boundary conditions. (i.e. find Yn. (3).) that satisfies the extra conditions u(z,0) = f(x), n a(z, y) = 0 for all 0<x<π. You should derive a solution formula where certain coefficients are determined in terms of the Fourier coefficients of f(x)

Answer 1

Boundary conditions are u(0, y) = 0 = X(0) gamma (y) 1 alpha X(0) = 0 u(pi, y) = 0 = X(pi) gamma (y) 1 alpha X(pi) = 0 by separation of variables let u(x, y) = X(x) gamma(y) be the solution partial differential^2 4/partial differential x^2 = gamma d^2 X/dx^2, partial differential^2 u/partial differential y^2, partial differential^2 u/partial differential y^2 = X d^2 gamma/dy^2 hence gamma d^2X/dx^2 + d^2 gamma/dy^2 = 0 1/X d^2X/dx^2 + 1/gamma d^2 gamma/dy^2 = 0 X''/X = -y''/y - lambda^2 X'' + lambda^2 X = 0 with x(0) = 0, X(pi) = 0 and y'' - lambda y = 0(D^2 + lambda^2) X = 0 m = plusmines I lambda y(x) = CF = C_1 cos lambda x + C-2 sin lambda x when x = 0 0 = C_1 + 0 Rightarrow C_1 = 0