# A 2nd Order PDE Using Similarity Method

1. May 14, 2017

### keropi452

Hi All,

Does anybody know how to solve the following PDE? I tried a similarity solution method where eta = y/f(x) (which I can do successfully without the C * U term) but was unsuccessful.

Thank you very much in advance!

2. May 14, 2017

### Ssnow

If it is a PDE, it will be $A\frac{\partial U}{\partial x}=B\frac{\partial^2 U}{\partial y^2}+C\cdot U$ ...

3. May 14, 2017

### keropi452

True - Sorry about that. Please take the d's to mean partial differential. Thank you for that catch.

4. May 14, 2017

### Ssnow

If you assume that $A\not=0$ you can write your equation as $\frac{\partial U}{\partial x}=\frac{B}{A}\frac{\partial^2 U}{\partial y^2}+\frac{C}{A}\cdot U$ that is an example of diffusion reaction equation with $R(U)=\frac{C}{A}U$, see

https://en.wikipedia.org/wiki/Reaction–diffusion_system

where you call $x=t$ and $y=x$, here the reaction term is simply $\frac{C}{A}U$...

Ssnow

5. May 14, 2017

### keropi452

Thank you very much for your response and observation. Are you possibly aware of any closed form solutions to the diffusion reaction eq with R(U) = CU/A?

6. May 14, 2017

### pasmith

The substitution $u = e^{Cx/A}v$ results in a standard diffusion equation for $v$.