Let smooth functions $f(x) , g(t)$ are given
solve the heat equation on the semi infinite domain $(a,\infty) \times (0,T)$.
for simplicity, we can let $a = 0$.
\begin{eqnarray}
&&u_t(x,t) = u_{xx}(x,t) \quad a<x<\infty , \quad 0<t<T \\
&&u(x,0) = f(x), \quad a<x\\
&&u(a,t) = g(t), \quad 0<t<T \\
&& lim_{x\rightarrow \infty} u(x,t) = 0 .
\end{eqnarray}
i need the closed form solution to the problem subject to $f(x) , g(t)$
Thanks for any help in advance
On
For the last condition to be true one has to make some assumptions on $f$. Such as $\lim_{x\to +\infty} f(x) = 0$. For example, if $f\equiv1$ then $\lim_{x\to +\infty} u(x,t) = 1$. The solution can be expressed as a sum of two potentials: $G*f+2Wg$, where $G$ is the Green function of the first boundary value problem and $Wg$ is double layer potential for the heat equation.
As a hint, I suspect you'll want to use Duhamel's principle where the solution to the problem with time-variant conditions is the convolution of the transient BC with the constant BC version of your problem...