I want to get nondimensional form of heat equation $u_t=a(x,t)u_{xx}$.

For the case of $a(x,t)=a(t)$, by setting $A(t)=\int_0^ta(\eta)d\eta$ and $t=\phi(\tau)$, where $\phi$ is the inverse mapping $\tau=A(t)$, one can obtain that $U_{\tau}=U_{xx}$.

I wonder that for the case of $a(x,t)=a(x)$, is there any transformation to reduce the heat equation into nondimensional form.

My work: I try coordinate transformation $x=\phi(\xi)$ but it give me nothing. I consider setting $B(x)=\int_0^xa(\xi)d\xi$ similar to above treatment again I couldn't obtain anything.

Question: Could anyone please help me to obtain nondimensional form or give me some hints? Please, I have no idea as to how exactly I can find a transformation.

Thanks in advance...


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