Re: laplace equation

In article <75MDf.2199$MJ.463@fed1read07>, quat <spam@xxxxxxxx> wrote:
>Solve:
>
>u_xx + u_yy = 0 for 0 < x < B, 0 < y < A
>u(x,y) = u(B,y) = u(x,A)=0
>u(x,0) = f(x)
>
>My professor told us we can start with separation of variables to get the
>correct sine/cosine series that satisfies the boundaries, but he also showed
>us some tricks. For instance, if you are given the boundary function is
>zero, then it will be a sine series since sin(0) = sin(pi) = 0. On the
>other hand, if you are given the boundary derivatives as zero, then guess a
>cosine series.
>
>Anyway, I am trying to apply that here. However, the first problem I have
>is the domain of x. In the book, they derive the general formula on [0,
>pi]. So I think to do a change of scale and let:
>
>x' = pi/B * x
>
>Then guess u(x,t) = infiniteSum( c_n*sin(n*pi/B *x) * sinh(n(A-y)) ),
>
>I have the answer to this problem and the sin(n*pi/B *x) looks good.
>However, the sinh does not: they have
>
>sinh(n*pi(A-y)/B)
>
>So it seems that the change of scale needs to affect the y somehow. But I
>am not sure of the relationship. y doesn't need to be scaled down because
>there is not the restriction of the domain on y like there is for x.

Hint: for what constant c would sin(n pi x/B) sinh(c (A - y)) satisfy
the equation u_xx + u_yy = 0 ?

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada


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