- Ou 0 Y 0 2 Find The Solution To The Following Laplace Equation V 0 0 X 3 0 Y 1 Where The Boundary Conditions Are 1 (58.1 KiB) Viewed 39 times
ou (0,y) = 0, 2. Find the solution to the following Laplace equation v=0, 0<x<3, 0<y<1 where the boundary conditions are given by де du (x,0) = 0, (X, 1) - 1 - Cos (**). ax де ay (a) Use separation of variables u(x,y) = x (x)Y(y) to write ordinary differential equations for X () and Y(y) in terms of a separation constant p. (b) The homogeneous boundary conditions for the problem in the r-direction constrain the possible values of p and functional forms of X(r). What are these possible values and functions? (c) Calculate the form of Y(y), and write down the general form of u(x,y) as a superposition of the eigenfunctions. (d) Enforce the boundary condition at y = 1, and determine what the coefficients must be using orthogonality relations. Express your final answer in terms of u(x,y). (Hint: There will only be two terms in the final answer.]