PDE with non-constant coefficient

[FrankST]

Dear All,

I have a PDE like:

A * d2w/dy2 + B * 1/x * d2w/dx2 + C * w = 0

where , w = w(x,y), A & B & C are constants.

Is there any analytical solution for this PDE?

If not, is finite difference is the right numerical tools to solve it?

Thanks,

Frank

 

[kosovtsov]

Your PDE

$$A\frac{\partial^2 w(x,y)}{\partial y^2}+\frac{B}{x}\frac{\partial^2 w(x,y)}{\partial y^2}+Cw(x,y) = 0$$

can be solved by the Laplace transform. The general solution is as follows

$$w(x,y)=\int_{-\infty}^{-\infty} F_1 (\omega) AiryAi [-((A\omega^2+C)/B)^{1/3}x]+F_2 (\omega) AiryBi [-((A\omega^2+C)/B)^{1/3}x]\exp(y\omega)d\omega ,$$

where $$F_1 (\omega) , F_2 (\omega)$$ are arbitrary functions.

 

[kosovtsov]

I'm sorry for misprint. The right answer is

$$w(x,y)=\int_{-\infty}^{-\infty} \{F_1 (\omega) AiryAi [-((A\omega^2+C)/B)^{1/3}x]+F_2 (\omega) AiryBi [-((A\omega^2+C)/B)^{1/3}x]\}\exp(y\omega)d\omega ,$$