Solving Laplacian PDE with Separation of Variables

[sachi]

we are given the laplacian:
(d^2)u/(dx^2) + (d^2)u/(dy^2) = 0 where the derivatives are partial. we have the B.C's
u=0 for (-1<y<1) on x=0
u=0 on the lines y=plus or minus 1 for x>0
u tends to zero as x tends to infinity.

Using separation of variable I get the general solution

u = (ax+b)(cy+d) + sum over k of (Ak*sinky + Bk*cosky)*(Ck*exp(kx) + Dk*exp(-kx))

where a,b,c,d,Ak,Bk,Ck,Dk are constants. We can then say that Ck = 0 from B.C's, and I also think that we can say that a=b=c=d=0 as well (but I am not sure). I'm having trouble imposing the rest of the B.C's. the final solution is
u=sum over m of [Am*cos(m*Pi*y/2)*exp(-(m*Pi*x/2)

thanks very much

 

[HallsofIvy]

Since [itex]cos(\frac{n\pi}{2})= cos(-\frac{n\pi}{2})= 0[/itex] for n any positive integer, you can satisfy the boundary condition u(x,y)= 0 for |y|= 1 by choosing [itex]k= -\frac{n\pi}{2}[/itex]. Of course, the the coefficient of sin(kx) must be 0 for all k for the same reason.

 

[sachi]

Hi, thanks for the clarification. I'm now stuck on the next bit: determining the coefficients. I'm not sure what limits to integrate between, and keep getting all of my coefficients equal to zero.