# Any idea for this nonlinear equation?

1. Feb 21, 2010

### 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 couldnt find any analytical or semianalytical solution for this equation, do you have any idea?

2. Feb 24, 2010

### kosovtsov

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

3. Feb 24, 2010

### 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.

4. Feb 24, 2010

### 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

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

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

Last edited: Feb 24, 2010
5. Feb 24, 2010

### fery

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

6. Feb 24, 2010

### kosovtsov

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

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

7. May 20, 2010

### PhysicsHigh

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.

8. May 20, 2010

### fery

If you are asking me, I dont 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.