Question

7. Consider the boundary value problem for the Laplace equation on the strip u(0, y) u(n,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) = Σ Yn (y) sin nx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y) = Σ Y, (y) sin na. the Laplace equation and the boundary conditions. (i.e. find Yn (y).) that satisfies the extra conditions u (x,0) = f (x), linnu (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

Given U_xx + u_y y = 0 0 < x < pi u(0,y) = u(pi,y) = 0 u(0,y) = 0 = > x(0) y(y) = 0 u(pi,y) = 0 x(pi) y(y) = 0 i e y(0) = 0 a) By separation of variables let u(x,y) = x(x) y(y) be the solution partial differential^2 u/partial differential x^2 = y d^2 x/d x^2 partial differential^2 y /d y^2 = x^2 y /d y^2 Hence y d^2 x/d x^2 + x d^2 y/d y^2 = 0 1/x d^2 x/d x^2 + 1/y d^2 y/d y^2 = 0 \r\n x''/x - y''\/y = -lambda^2 x''/x = -lambda^2 x'' + lambda^2 x = 0 with x(0) = 0 x(pi) = 0 -y''/y = -lambda^2 =. y'' - lambda^2 y = 0 x'' + lambda^2 x = 0 (d^2 + lambda^2) x = 0 m^2 + lambda^2 = 0 m = plusminus i lambda Therefore X(x) = C F = C_1 cos lambda x + c_2 sin lambda x let put x = 0 X(0) = c_1 cos(0) + c_2 sin 0 c_1 = 0