# About solving heat equation in half plane

Hi guys,

I have trouble when solving the following heat transport equation in half plane in frequency domain.

$$(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2})\theta(x,y)=i\beta\theta(x,y),-\infty<x<+\infty,y\geq 0; \theta(|x|\rightarrow\infty,y)=\theta(x,y\rightarrow\infty)=0; \frac{\partial}{\partial y}\theta(x,y)|_{y=0}=f(x)$$,

where $$i=\sqrt{-1}$$ is the unit of imaginary number, $$\beta$$ is a positive real constant, f(x) is a real function.

I tried to solve it in polar coordinate. $$\theta$$ can be expanded as the sum series of Bessel function of the second type, K. But the problem is K is divergent around the origin.

I really appreciate any help from you guys. Thanks.

jasonRF
Gold Member
since your domain is semi-infinite in the y dimension and you have a boundary condition at y=0, I would try Laplace transform wrt y. Should give you an ODE you can deal with.

good luck.

jason

You are right. It can be converted to an ODE using Lapace transformation. But the difficulty is transported to using the boundary condition to determine the coefficients which depends on the Laplace variable s. Plus, even if the coefficients can be determined, I need to do the inverse Laplace transform to get the final result, which is almost equally changeable.

Thanks anyway.