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Solve Laplace’s equ… - QuestionCove
OpenStudy (errrwut):

Solve Laplace’s equation inside a rectangle 0 <= x <= L, 0 <= y <= H, with the following boundary conditions

1 year ago
OpenStudy (errrwut):

u(0,y) = g(y), u(L,y)=0, \[\frac{\partial u}{\partial x} (x,0)=0\] , u(x,H) = 0

1 year ago
OpenStudy (kainui):

I think it helps to draw a picture, although it's unnecessary it can really help to organize your thoughts. Make sure to use separation of variables to save yourself. You can also use the picture you draw to help you flop the boundary conditions over to the ODEs you get from the separation. \[u(x,y)=X(x)Y(y)\]

1 year ago
OpenStudy (unklerhaukus):

\[\nabla^2u(x,y)=0\\[1ex] 0<x<L,\qquad\ \ u(0,y) = g(y), \qquad u(L,y)=0\\ 0<y<H,\qquad u_x(x,0)=0,\qquad \ \ \ \ u(x,H) =0\] \begin{align} u_{xx}+u_{yy}&=0\\ && u(x,y) &= X(x)Y(y)\\ && u_{xx}&=X''Y\\ && u_{yy}&=XY''\\ X''Y+XY''&=0\\[1ex] X''/X=-Y''/Y&=\text{const.}=-\lambda \end{align}

1 year ago
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