somehow, I got this partial differential equation but I don't know how should I start.

$$ a\frac{\partial f(x,t)}{\partial x}\left[ \frac{\partial g(x,t)}{\partial t}+bg(x,t)\left[g(x,t)-f(x,t)+C\right] +bd\frac{\partial g(x,t)}{\partial x} -h\frac{\partial^2 g(x,t)}{\partial x^2}+\frac{g(x,t)}{j}-k\right] + l\frac{\partial g(x,t)}{\partial x}\left[ \frac{\partial f(x,t)}{\partial t}+mf(x,t)\left[g(x,t)-f(x,t)+C\right] -md\frac{\partial f(x,t)}{\partial x} -n\frac{\partial^2 f(x,t)}{\partial x^2}+\frac{f(x,t)}{p}-k\right] =0 $$

where a, b, C, d, h, j, k, l, m, n, and p are constants.

Does the laplace tranform work in this PDE?

How do I laplace transform

$$ \frac{\partial g(x,t)}{\partial x} \frac{\partial f(x,t)}{\partial t} $$


Or is there any way I can solve other way?


This is a nonlinear PDE, so Laplace transform is unlikely to help.

Also, it's one equation but with two dependent variables $f$ and $g$. You might make a more-or-less arbitrary choice for one of them, and hope to solve (at least locally) for the other.

Some easy solutions are obtained by assuming $f$ or $g$ depends only on $t$.


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