The problem is to solve this equation:
$$ (x,y)ϵ[0,L_1]x[0,∞] $$ $$ Δu = 0 $$ $$ u(x,0) = φ(x) $$ $$ u(0,y) = α(x) $$ $$ u(L_1,y) = β(x) $$
The strategy adopted is to limit the length of the strip to a fixed value $L_2$ and then obtain the solution on the infinite strip by sending $L_2$ to infinity. I had no problem calculating the solution of the finite strip $[0,L_1]x[0,L_2]$, however I encountered problems during the limit.
Lets start with this secondary problem
$$ (x,y)ϵ[0,L_1]x[0,L_2] $$ $$ Δu_{L_2} = 0 $$ $$ u_{L_2}(x,0) = φ(x) $$ $$ u_{L_2}(x,L_2) = ψ(x) = α(L_2) + x/L_1·[ β(L_2) - α(L_2) ] $$ $$ u_{L_2}(0,y) = α(x) $$ $$ u_{L_2}(L_1,y) = β(x) $$
First I solve it in the special case $ u_{L_2}(x,0) = φ(x) $ and $u(x,y)=0$ otherwise on the border.
The special case $ u_{L_2}(0,y) = α(x) $ and $u(x,y)=0$ otherwise on the border is calculated by swapping the role of $x$ and $y$.
Simmetry considerations allow to solve the special case $ u_{L_2}(x,L_2) = ψ(x) $ and $u(x,y)=0$ otherwise on the border; the special case $ u_{L_2}(L_1,y) = β(x) $and$ u(x,y)=0$ follows.
To sum up, if $φ_n$, $α_n$ and $β_n$ are the Fourier coefficients of $φ(x)$, $α(y)$ and $β(y)$, then the solution of the secondary problem is
$$ u_{L_2}(x,y) = ∑_{n=1}^∞ sin( \frac{nπx}{L_1})· exp(- \frac{nπy}{L_1}) ·φ_n $$ $$ + ∑_{n=1}^∞ sin(\frac{nπx}{L_1})· exp(- \frac{nπ(L_1-y)}{L_1}) · \frac{2}{nπ} [ \frac{α(L_2)-β(L_2)}{L_2} + α(L_2)·(1-(-1)^n) ] $$ $$ + ∑_{n=1}^∞ sin( \frac{nπy}{L_2})· exp(- \frac{nπx}{L_2}) ·α_n $$ $$ + ∑_{n=1}^∞ sin(\frac{nπy}{L_2})· exp(- \frac{nπ(L_2-x)}{L_2}) · β_n $$
What I would like to do now is to affirm that $$ u(x,y) = lim_{L_2→∞} u_{L_2}(x,y) $$
My problem is that $$ \lim_{L_2→∞} exp(- \frac{nπx}{L_2}) = 1 $$ $$ \lim_{L_2→∞} exp(- \frac{nπ(L_2-x)}{L_2})= exp(-nπ) $$ $$ ∄ \lim_{L_2→∞} sin( \frac{nπy}{L_2}) $$
The first infinite sum is unchanged by the limit. The second transforms into $ ∑_{n=1}^∞ sin(\frac{nπx}{L_1})· exp(- \frac{nπ(L_1-y)}{L_1}) · \frac{2}{nπ} lim_{L_2→∞}α(L_2)·(1-(-1)^n) $, which is zero supposing that $\lim_{y→∞}α(y) = 0$. However I have no idea how to solve the limits of the third and the fourth infinite sums.