Given the following IBVP. $$\frac{\partial T}{\partial t} = \frac{\partial^2 T}{\partial x^2} + \frac{1}{x} \frac{\partial T}{\partial x}$$
Initial condition $$T(x,t=0)=0$$
BCs.
$$T(x=0,t)=1$$ $$\lim_{x \to \infty} T(x,t)=0$$
How to prove using the method of Variation of Parameters that the following solution
$$y(x,t) = g(t)*\frac{\partial }{\partial t}T(t) = \int_{0}^{t} g(t - \tau)\frac{\partial }{\partial \tau} T(\tau) d\tau$$
Solves the following IBVP?
$$\frac{\partial y}{\partial t} = \frac{\partial^2 y}{\partial x^2} + \frac{1}{x} \frac{\partial y}{\partial x}$$
Initial condition $$y(x,t=0)=0$$
BCs.
$$y(x=0,t)=g (t)$$ $$\lim_{x \to \infty} y(x,t)=0$$