Laplace transform nonlinear equation

Question

How can I apply the Laplace transform on a the following nonlinear PDE $$ \frac{\partial y}{\partial t}=\frac{\partial y^n}{\partial x}$$ where $n$ is a natural number?

When I say apply the Laplace transform, I mean apply if to both $t$ and $x$ independently. Is that possible?

Thank you in advance.

Answer 1

It is not suggested to solve by Laplace transform.

$\dfrac{\partial y}{\partial t}=\dfrac{\partial y^n}{\partial x}$

$\dfrac{\partial y}{\partial t}=ny^{n-1}\dfrac{\partial y}{\partial x}$

Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:

$\dfrac{dt}{ds}=1$ , letting $t(0)=0$ , we have $t=s$

$\dfrac{dy}{ds}=0$ , letting $y(0)=y_0$ , we have $y=y_0$

$\dfrac{dx}{ds}=-ny^{n-1}=-ny_0^{n-1}$ , letting $x(0)=f(y_0)$ , we have $x=f(y_0)-ny_0^{n-1}s=f(y)-ny^{n-1}t$ , i.e. $y=F(x+ny^{n-1}t)$

Answer 2

Take a look at the The Laplace Transform, Joel L. Schiff 1991 Springer Verlag, ISBN 0-387-98698-7, Examples on p182 onwards.