How to find the general solution of a differential equation with a shift, in the following form?


$$\frac{\partial}{\partial t}g(x,t)=g(x-\Delta,t)+\frac{\partial^2}{\partial x^2} g(x,t)$$


where $\Delta > 0$. And what about the following?


$$\frac{\partial}{\partial t}g(x,t)=g(x,t-\Delta)+\frac{\partial^2}{\partial x^2} g(x,t)$$




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Edit1: Here are few follow-up details about my question. Is there a ``nice" way to represent the solution in $x$-space, as opposed to e.g., Fourier? Is the solution real + positive + normalizable? Does it have the correct properties of a probability density function?



How to solve a differential equation in the form $\frac{\partial}{\partial t}g(x,t)=g(x-\Delta,t)+\frac{\partial^2}{\partial x^2} g(x,t)$?
Fourier transform $G(k,t)=\int\_{-\infty}^\infty e^{ikx} g(x,t)dx$ with respect to $x$, then
$$\frac{\partial}{\partial t}G(k,t)=e^{ik\Delta}G(k,t)-k^2 G(k,t),$$
hence
$$G(k,t)=\exp\left(te^{ik\Delta}-tk^2\right)G(k,0).$$
For the second differential equation you would similarly Fourier transform with respect to $t$.


