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ap.evaluation of pdes – When is it helpful to deal the time variable in a evolution downside as a spatial variable? Answer

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ap.evaluation of pdes – When is it helpful to deal the time variable in a evolution downside as a spatial variable?

This query is a bit informally formulated and loosely said, however I await somebody could make sense of it and lead me in some route. With my restricted suffer in PDEs, I’ve at all times seen the time variable being handled in a different way than the house variables.

A little bit of background: As I’m learning a differential equation, 2nd organize in house, 1st organize in time, whose resolution is a duty $u=u(mathbf{x},t)$ the place $mathbf{x}=(x,y)in Omegasubsetmathbb{R}^2$ the place $Omega=[a,b]instances[c,d]$ is an oblong province and $tin [0,infty)$. Using common reasoning in functional analysis, I was looking for solutions in Bochner spaces like $H^1((0,infty),H^2(Omega))$.

However, a certain change of variables $z=y-t $ and $w=x$ (I’m looking for traveling waves) reduced the original equation to the following
$$-Delta phi(w,z)+phi(w,z)-w=F(phi(w,z)),$$
where $u(x,y,t)=phi(w,z)$ and $F$ is an arbitrary (to be determined) function. In the $(w,z)$ variables, the new domain $tilde{Omega}=[a,b]instances (infty,b]$ in query is an innumerable semi divest containing $Omega$ and $phi$ inherits whichever border situations $u$ satisfies. So now I spotted in necessity to search for options in a fresh house $H(tilde{Omega})$ the place $H$ is an arrogate duty house.

Bottom line: To seek the conduct of my resolution $u(x,y,t)$ in time $t$ on a bounded province $Omega$, I now seek the asymptotic conduct of $phi(w,z)$ within the fresh “spatial variable” $z=y-t$.

The query: Is there an edge of learning time $t$ as a spacial variable $w=y-t$?

If the query does not make any sense delight let me know within the observation in order that I make wanted clarifications.

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