Analytic Solution to Generalized Heat Equation

Question

Is this type of PDE known to have a closed form solution? $$ \partial_t(t,x) f = a \Delta f(t,x)\qquad f(0,x)=e^{bx}; $$ where $a,b>0$ and $c \in \mathbb{R}$?

If so, what is that solution and what is the name of this PDE?

Answer 1

Following the Green's function method, the solution to your initial value problem is given by the following convolution $$ u(x,t)=\frac{1}{\sqrt{4\pi at}} \int_{-\infty}^{\infty} \exp\left(-\frac{(x-y)^2}{4at}\right)g(y)\,dy; $$ where in your case, $g(y)=e^{c+bx}$. In this case, your integral is precisely the Laplace transform of a Gaussian density with mean $x$ and variance $2\sqrt{at}$. This integral is well-known, and is called the moment generating function of the normal distribution (with reparameterized with $\lambda =-b$; where $\lambda$ is the input variable of the Laplace transform).

Standard computations then show that $$ \begin{aligned} u(x,t)=&e^{c+xb+\frac1{2} 4at b^2}\\=& e^{c+xb+2at b^2} \end{aligned} $$