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How to solve a differential equation in the form fracpartialpartialtg(x,t)=g(xDelta,t)+fracpartial2partialx2g(x,t)?

MathOverflow Asked on November 22, 2021

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

fracpartialpartialtg(x,t)=g(xDelta,t)+fracpartial2partialx2g(x,t)

where Delta>0. And what about the following?

fracpartialpartialtg(x,t)=g(x,tDelta)+fracpartial2partialx2g(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?

One Answer

Fourier transform G(k,t)=intiinftynftyeikxg(x,t)dx with respect to x, then fracpartialpartialtG(k,t)=eikDeltaG(k,t)k2G(k,t), hence G(k,t)=expleft(teikDeltatk2right)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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