Any idea for this nonlinear equation?

[fery]

Hi

I have a nonlinear equation for diffusion of multiphase fluids in porous media, and it is like
1/2(Laplacian(P^2)+d(p)/dy=d(p)/dt
I couldn't find any analytical or semianalytical solution for this equation, do you have any idea?

 

[kosovtsov]

Can you write out your DE more clearly. Independent and dependent (P and/or p) variables first of all.

 

[fery]

Sure, P = P(x,y,t), Laplacian = d/dx2 + d/dy2, As I said this nonlinear equation represents diffusion of capillary pressure in porous media.

 

[kosovtsov]

If I understand rightly, your PDE is

$$\frac{1}{2}(\frac{\partial^2 P^2}{\partial x^2}+\frac{\partial^2 P^2}{\partial y^2})+\frac{\partial P}{\partial y}=\frac{\partial P}{\partial t}$$

I do not think that it is easy to find the general solution to the PDE, but you can find some particular solutions of the type

$$P = \frac{1}{C_5 +C_6 \tanh(C_1+C_2 x+iC_2 (y+t))}$$

$$P = \frac{\sqrt{2}}{k}e^{-\frac{kx}{2}}\sqrt{e^{2kx}C_1 -C_2 }\sqrt{C_3 \sin[k(y+t)]-C_4 \cos[k(y+t)]}$$

 

[fery]

Thanks you, I am not quite sure about the second type, can you fix the Parenthesis.

 

[kosovtsov]

$$P = \frac{1}{C_5 +C_6 \tanh[C_1+C_2 x+iC_2 (y+t)]}$$

$$P = \frac{\sqrt{2}}{k}e^{-\frac{kx}{2}}\sqrt{e^{2kx}C_1 -C_2 }\sqrt{C_3 \sin[k(y+t)]-C_4 \cos[k(y+t)]}$$

 

[PhysicsHigh]

$$P = \frac{1}{C_5 +C_6 \tanh[C_1+C_2 x+iC_2 (y+t)]}$$

$$P = \frac{\sqrt{2}}{k}e^{-\frac{kx}{2}}\sqrt{e^{2kx}C_1 -C_2 }\sqrt{C_3 \sin[k(y+t)]-C_4 \cos[k(y+t)]}$$

Since your in such an advanced math and know a lot about it, does that make sense when you look at it? I mean is it hard for you or scary? if you understand what i mean i know its a sill question and maybe for this thread.

 

[fery]

If you are asking me, I don't think these functions represent the general solution of the equation. And they are not orthogonal so if they would be the general solution, we would never be able to find particular solution for them.