Solution to multiplied heat equation

Question

I know how to solve the standard heat equation but if I were introduce a function g of safe and time into the mix as: $$ \partial_t f(t,x)= g(t,x)\partial^2_x f(t,x), $$ How can I solve this problem? Is it possible to use a clever change of variables in x?

Answer 1

Case $1$: $g(t,x)=a(t)b(x)$

The PDE is obviously separable.

Let $f(t,x)=T(t)X(x)$ ,

Then $T'(t)X(x)=a(t)b(x)T(t)X''(x)$

$\dfrac{T'(t)}{a(t)T(t)}=\dfrac{b(x)X''(x)}{X(x)}$

Case $2$: $g(t,x)=h(ax+bt)$ , $a,b\in\mathbb{C}$

Let $\begin{cases}x_1=x\\t_1=ax+bt\end{cases}$ ,

Then $f_x=f_{x_1}(x_1)_x+f_{t_1}(t_1)_x=f_{x_1}+af_{t_1}$

$f_{xx}=(f_{x_1}+af_{t_1})_x=(f_{x_1}+af_{t_1})_{x_1}(x_1)_x+(f_{x_1}+af_{t_1})_{t_1}(t_1)_x=f_{x_1x_1}+af_{x_1t_1}+af_{x_1t_1}+a^2f_{t_1t_1}=f_{x_1x_1}+2af_{x_1t_1}+a^2f_{t_1t_1}$

$f_t=f_{x_1}(x_1)_t+f_{t_1}(t_1)_t=bf_{t_1}$

$\therefore bf_{t_1}=h(t_1)(f_{x_1x_1}+2af_{x_1t_1}+a^2f_{t_1t_1})$