Hi I have a specific problem, that is given a 1-d semi-infinite nonhomogeneous heat equation
$$\frac{\partial^2 T}{\partial x^2}+\frac{1}{k}g(x,t)=\frac{1}{\alpha}\frac{\partial T}{\partial t}, \quad 0<x<\infty, t>0$$
for any given $t$, is there a way to get the location of maximum i.e. $x_{max}$, for ${T_{max}(x,t)}$ without solving the equation fully?
For example, giving $g(x,t)$ and boundary condition is $T(0,t)=f(t)$
The solution can be explicitly derived using Laplace transform, Greens' function etc..
But without deriving the solution first, if i just want to find $x_{max}$ associated with $T_{max}(x,t)$ for any given $t$, is there a technique?