University of Ilorin
Question
Asked 6 December 2024
Is it possible to create only one general algorithm for exact numerical solution of any partial boundary value problem of any nature?
Hello dear professors, students and all researchers of mathematical sciences, my question is whether it is possible to define only one algorithm by considering all innovative numerical methods for solving partial boundary value problems And the nature of the problem, with the best numerical method that is more consistent and stable with the least error to get the numerical solution of the problem?
Most recent answer
Hello Profs,
Yes, it is possible.
I have been working on some analytical methods combined with a particular integral transform lately. I believe it can handle PDE of any kinds.
Popular answers (1)
Ziane Achour University of Djelfa
A small and direct answer is 'No'.
It depends on several parameters. The PDE is linear or nonlinear? The boundary conditions are Dirichlet, Neumann or mixed type? The dimension of the problem? Is it time-depends or no?....
If such algorithm exists, it will be a revolution in the field of numerical methods. 'if'
Best regards
4 Recommendations
All Answers (12)
University of Cincinnati
Not sure what partial boundary value problem is. Do you mean IBVP? As far as I know, the FDTD method absolute error depends on the grid resolution. So if you make the grid resolution very high, you can make absolute error very small. However, your computational complexity will also become very high. In the limit, the computational complexity will approach infinity and the absolute error will approach zero. Perhaps there are more accurate methods, however I think that there is a tradeoff in other characteristics.
1 Recommendation
Ziane Achour University of Djelfa
A small and direct answer is 'No'.
It depends on several parameters. The PDE is linear or nonlinear? The boundary conditions are Dirichlet, Neumann or mixed type? The dimension of the problem? Is it time-depends or no?....
If such algorithm exists, it will be a revolution in the field of numerical methods. 'if'
Best regards
4 Recommendations
Ganja State University
In the paper "On boundary problem in Bohr space of multivarate almost periodic functions" of N. Allahyarova (https://research-journal.org/) is considered a problem, in which the partial differential equation reduced to family of ordinary differential equations and a solution of initial problem can be represented as a limit of almost periodic solutions. This method is possible to apply to wide class of boundary problems. So, the answer to the question partially is positive.
1 Recommendation
Oles Honchar Dnipro National University
No - the choosing (or design) of the num method for any PDE is illposed problem (it has infinity set of solution). So it has no an unique solution, only the "best" one. But the sense of the "bestness" is different for the different researchers (optimality criterion), so you can choose/develop the num method which will be best for you (maybe for your labmates), but not for all community
Ziane Achour University of Djelfa
Hi researchers...
Even my answer was 'No'. The radial basis collocation method (and the space-time one) can handle the thing. The PDE must be linear and all things will be clear. See files in attachment. We note that the numerical solution can be obtained by the leat square method (even we have an infinity of solutions). Good luck.... Best regards...
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