[itex]\nabla[/itex]^2(Z)=0(adsbygoogle = window.adsbygoogle || []).push({});

Z= 0 for x=0, y=0

Z= x(1-x) for y=0

Z=0 for y=infinity

Range 0<x<1 and y>0 (suppose strictly speaking should be x=1 and x=0 too)

So all I want to do is solve this

Use separation of variables:

X''/X = a^2 = -Y''/Y

Gives X = Aexp(ax) + Bexp(-ax) and Y=Ccos(ay) + Dsin(ay)

Or completely free to swap these around to give

Y=Aexp(ay)+ Bexp(-ay) and X= Ccos(ax) +Dsin(ax)

which I shall do as get further with boundary conditions

Know that Z=XY

As at y=infinity, Z=0 ==> A=0

At x=0, Z=0 ==>C = 0

At x=1, Z=0 ==> a=n*pi where n is an integer

so have Z=Esin(n*pi*x)exp(-n*pi*y) where E is a new constant

but how on earth do I make this compatible with the remaining boundary condition for y=0

==> Z=x(1-x) = Esin(n*pi*x) ??????

Clearly must have gone wrong somewhere???

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# Solve laplace's equation on semi infinite strip

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