2 = 1 where 2 is a laplacian operator with boundary conditions (0,y)=(1,y)=(x,0)

Question

2   = 1  where  2  is a laplacian operator

with boundary conditions (0,y)=(1,y)=(x,0)=(x,1)=0

this is 2-D poisson equation with source term equal to -1 since the general form is

 2   + f=0

the exact solution to this equation with the boundary conditions above has to be obtained.

as what i know this can be solved using fourier series with substitution u(x,y)=(x,y) - (1/2)*x^2

i tried to solve it but the fourier coefficients get very complicated and the solution doesnt satisfy the transformed boundary conditions in terms of u(x,y).

thank you for your time , Sir. I appreciate your help.

THE EXAMPLE FOR THIS PROBLEM IS IN THE BOOK TITLED 'ELEMENTARY PARTIAL DIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS BY LARRY C ANDREWS ISBN: 0-12-059510-9

LOOK FOR PAGE No: 378 PROBLEM 10

ALSO PAGE No: 477 PROBLEM 8

THEORY FOR THESE PROBLMES IS ALSO GIVEN FEW PAGES BEFORE THESE PROBLEMS ARE ASKED IN THE EXERCISE

Explanation / Answer

https://docs.google.com/viewer?a=v&q=cache:fMMPxFKrJRMJ:eprints.ma.man.ac.uk/894/02/0-19-852868-X.pdf+to+find+exact+solution+to+2+-+D+poisson+equation&hl=en&gl=in&pid=bl&srcid=ADGEESief1imrf9uKijPUY7NC2BFOYUp5hBWM_nhfsOvYLgsb9Cu3zThCF0yZqpnaBzo-8V3IwY6KvZzuuFucz9Fia1TRrSKs9h_AjG95f6gbweo9f0PoUuRTUBQYYe-m6s42RdO2AFw&sig=AHIEtbT8UD1QM1UNuborBy5dbRdl_VAnSg&pli=1

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