How to solve a differential equation in the form $frac{partial}{partial t}g(x,t)=g(x-Delta,t)+frac{partial^2}{partial x^2} g(x,t)$?

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?

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

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?

MathOverflow Asked on November 22, 2021

1 Answers

One Answer

Fourier transform $G(k,t)=int_{-infty}^infty e^{ikx} g(x,t)dx$ with respect to $x$, then $$frac{partial}{partial t}G(k,t)=e^{ikDelta}G(k,t)-k^2 G(k,t),$$ hence $$G(k,t)=expleft(te^{ikDelta}-tk^2right)G(k,0).$$ For the second differential equation you would similarly Fourier transform with respect to $t$.

Answered by Carlo Beenakker on November 22, 2021

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