How Is the Laplace Equation Applied to Semi-Infinite Plates in Physics?

[sigh1342]

I want to find some application of the laplace equation on semi-infinite plate on physics
where the PDE is looke like
$$u_{xx}+u_{yy}=f , for a<x<\infty , c<y<d$$
$$u(a,y)=g(y), u(x,c)=f_{1}(x), u(x,d)=f_{2}(x)$$
$$\lim_{x->\infty} f(x)=\lim_{x->\infty} f_{1}(x)=\lim_{x->\infty} f_{2}(x)=0 $$
Thank you :)

 

[Chestermiller]

I want to find some application of the laplace equation on semi-infinite plate on physics
where the PDE is looke like
$$u_{xx}+u_{yy}=f , for a<x<\infty , c<y<d$$
$$u(a,y)=g(y), u(x,c)=f_{1}(x), u(x,d)=f_{2}(x)$$
$$\lim_{x->\infty} f(x)=\lim_{x->\infty} f_{1}(x)=\lim_{x->\infty} f_{2}(x)=0 $$
Thank you :)

One possible physical situation that equations of this form could be consistent with is a steady state conductive heat transfer problem in which the temperature distributions are fixed along the edges, and heat is being generated within the plate (as characterized by the function f(x,y)). In this case u is temperature, or temperature relative to some reference state (e.g., u = 0 at x->∞).