Hint: Interpret the physical meaning of the boundary conditions in terms of wave

Question

Hint: Interpret the physical meaning of the boundary conditions in terms of waves, and then reason what the answer to this one should be in terms of your solutions from parts (a) and (b).

(a) Solve the following Dirichlet problem for Laplace's equation in a square region: Findu(x,y),0leq xleqpi,yleq 0leqpisuch that frac{partial^2 u}{partial x^2} + frac{partial^2u}{partial y^2} = 0,qquad u(x,0)=0, u(0,y)=u(pi,y)=0,qquad u(x,pi)=sin x-4sin 2x+8sin 11 x. u(x,y)= (b) Solve the following Dirichlet problem for Laplace's equation in the same square region: Findu(x,y),0leq xleqpi,yleq0leqpisuch that frac{partial^2u}{partial x^2}+frac{partial^2u}{partial y^2}=0,qquad u(0,y)=0, u(pi,y)=sin 2 y+6sin 11 y,qquad u(x,0)=u(x,pi)=0. u(x,y)= (c) Find the solution to the Dirichlet problem: Findu(x,y),0leq xleqpi,yleq0leqpisuch that frac{partial^2u}{partial x^2}+frac{partial^2u}{partial y^2}=0,qquad u(0,y)=0,qquad u(x,0)=0, u(pi,y)=sin 2 y+6sin 11 y,qquad u(x,pi)=sin x-4sin 2x+8sin 11 x. Hint: Interpret the physical meaning of the boundary conditions in terms of waves, and then reason what the answer to this one should be in terms of your solutions from parts (a) and (b). u(x,y)=

Explanation / Answer

try solving it through convolution method nd u will have ur answer

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