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(x−Delta,t)+fracpartial2partialx2g(x,t)
where Delta>0. And what about the following?
fracpartialpartialtg(x,t)=g(x,t−Delta)+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?
Fourier transform G(k,t)=inti−inftynftyeikxg(x,t)dx with respect to x, then fracpartialpartialtG(k,t)=eikDeltaG(k,t)−k2G(k,t), hence G(k,t)=expleft(teikDelta−tk2right)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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