Use of the numerical solution on finite interval to describe the behavior of PDE on $Bbb R$

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When we want to simulate the solution of some one dimensional PDE
$$ mathcalL(u) = f,quad texton mathbbR $$
on the real line, why do we use the solution in finite interval $[a,b]$ to describe the behavior of the PDE on $Bbb R$?






pde numerical-methods







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  • In principle you should prove something that says that the system somehow "settles down" outside $[a,b]$, either goes to zero or oscillates in essentially the same manner or something along these lines. This can be hard to do rigorously, so you might just propose a heuristic argument, or run a method on a pair of nested intervals and deduce that settling down has already happened by virtue of the similarity of the two results, or something like this.
    – Ian
    Sep 4 at 20:53











  • Your answer is very helpful Thank you!!
    – Panasun
    Sep 5 at 6:11














up vote
0
down vote

favorite












When we want to simulate the solution of some one dimensional PDE
$$ mathcalL(u) = f,quad texton mathbbR $$
on the real line, why do we use the solution in finite interval $[a,b]$ to describe the behavior of the PDE on $Bbb R$?






pde numerical-methods







share|cite|improve this question























  • In principle you should prove something that says that the system somehow "settles down" outside $[a,b]$, either goes to zero or oscillates in essentially the same manner or something along these lines. This can be hard to do rigorously, so you might just propose a heuristic argument, or run a method on a pair of nested intervals and deduce that settling down has already happened by virtue of the similarity of the two results, or something like this.
    – Ian
    Sep 4 at 20:53











  • Your answer is very helpful Thank you!!
    – Panasun
    Sep 5 at 6:11












up vote
0
down vote

favorite









up vote
0
down vote

favorite











When we want to simulate the solution of some one dimensional PDE
$$ mathcalL(u) = f,quad texton mathbbR $$
on the real line, why do we use the solution in finite interval $[a,b]$ to describe the behavior of the PDE on $Bbb R$?






pde numerical-methods







share|cite|improve this question















When we want to simulate the solution of some one dimensional PDE
$$ mathcalL(u) = f,quad texton mathbbR $$
on the real line, why do we use the solution in finite interval $[a,b]$ to describe the behavior of the PDE on $Bbb R$?






pde numerical-methods




pde numerical-methods






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edited Sep 4 at 20:41









Harry49

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asked Sep 4 at 10:18









Panasun

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  • In principle you should prove something that says that the system somehow "settles down" outside $[a,b]$, either goes to zero or oscillates in essentially the same manner or something along these lines. This can be hard to do rigorously, so you might just propose a heuristic argument, or run a method on a pair of nested intervals and deduce that settling down has already happened by virtue of the similarity of the two results, or something like this.
    – Ian
    Sep 4 at 20:53











  • Your answer is very helpful Thank you!!
    – Panasun
    Sep 5 at 6:11
















  • In principle you should prove something that says that the system somehow "settles down" outside $[a,b]$, either goes to zero or oscillates in essentially the same manner or something along these lines. This can be hard to do rigorously, so you might just propose a heuristic argument, or run a method on a pair of nested intervals and deduce that settling down has already happened by virtue of the similarity of the two results, or something like this.
    – Ian
    Sep 4 at 20:53











  • Your answer is very helpful Thank you!!
    – Panasun
    Sep 5 at 6:11















In principle you should prove something that says that the system somehow "settles down" outside $[a,b]$, either goes to zero or oscillates in essentially the same manner or something along these lines. This can be hard to do rigorously, so you might just propose a heuristic argument, or run a method on a pair of nested intervals and deduce that settling down has already happened by virtue of the similarity of the two results, or something like this.
– Ian
Sep 4 at 20:53





In principle you should prove something that says that the system somehow "settles down" outside $[a,b]$, either goes to zero or oscillates in essentially the same manner or something along these lines. This can be hard to do rigorously, so you might just propose a heuristic argument, or run a method on a pair of nested intervals and deduce that settling down has already happened by virtue of the similarity of the two results, or something like this.
– Ian
Sep 4 at 20:53













Your answer is very helpful Thank you!!
– Panasun
Sep 5 at 6:11




Your answer is very helpful Thank you!!
– Panasun
Sep 5 at 6:11










1 Answer
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0
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One could ask the same question for an ordinary differential equation $dot y=f(y,t)$. Most of the numerical methods proceed by iterating some algorithm in time. Thus, they provide only finite-time solutions (which are already very useful). A few numerical methods provide long-time solutions, such as the harmonic balance method (HBM). However, speaking of the HBM, all the transient information around $t=0$ is lost with this method. Depending on the purpose of the study, one may be interested in the short-time behavior, or rather in the long-time behavior.






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  • Your answer is very helpful Thank you!!
    – Panasun
    Sep 5 at 6:11










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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
0
down vote



accepted










One could ask the same question for an ordinary differential equation $dot y=f(y,t)$. Most of the numerical methods proceed by iterating some algorithm in time. Thus, they provide only finite-time solutions (which are already very useful). A few numerical methods provide long-time solutions, such as the harmonic balance method (HBM). However, speaking of the HBM, all the transient information around $t=0$ is lost with this method. Depending on the purpose of the study, one may be interested in the short-time behavior, or rather in the long-time behavior.






share|cite|improve this answer






















  • Your answer is very helpful Thank you!!
    – Panasun
    Sep 5 at 6:11














up vote
0
down vote



accepted










One could ask the same question for an ordinary differential equation $dot y=f(y,t)$. Most of the numerical methods proceed by iterating some algorithm in time. Thus, they provide only finite-time solutions (which are already very useful). A few numerical methods provide long-time solutions, such as the harmonic balance method (HBM). However, speaking of the HBM, all the transient information around $t=0$ is lost with this method. Depending on the purpose of the study, one may be interested in the short-time behavior, or rather in the long-time behavior.






share|cite|improve this answer






















  • Your answer is very helpful Thank you!!
    – Panasun
    Sep 5 at 6:11












up vote
0
down vote



accepted







up vote
0
down vote



accepted






One could ask the same question for an ordinary differential equation $dot y=f(y,t)$. Most of the numerical methods proceed by iterating some algorithm in time. Thus, they provide only finite-time solutions (which are already very useful). A few numerical methods provide long-time solutions, such as the harmonic balance method (HBM). However, speaking of the HBM, all the transient information around $t=0$ is lost with this method. Depending on the purpose of the study, one may be interested in the short-time behavior, or rather in the long-time behavior.






share|cite|improve this answer














One could ask the same question for an ordinary differential equation $dot y=f(y,t)$. Most of the numerical methods proceed by iterating some algorithm in time. Thus, they provide only finite-time solutions (which are already very useful). A few numerical methods provide long-time solutions, such as the harmonic balance method (HBM). However, speaking of the HBM, all the transient information around $t=0$ is lost with this method. Depending on the purpose of the study, one may be interested in the short-time behavior, or rather in the long-time behavior.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Sep 4 at 20:54

























answered Sep 4 at 20:49









Harry49

5,0052825




5,0052825











  • Your answer is very helpful Thank you!!
    – Panasun
    Sep 5 at 6:11
















  • Your answer is very helpful Thank you!!
    – Panasun
    Sep 5 at 6:11















Your answer is very helpful Thank you!!
– Panasun
Sep 5 at 6:11




Your answer is very helpful Thank you!!
– Panasun
Sep 5 at 6:11

















 

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