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?
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.
According to the conditions of problems
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
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.
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
No, it is not possible to create a single general algorithm for the exact numerical solution of all partial boundary value problems (BVPs) of any nature. Here's why:
- Diversity of PDEs: (Equation Types, Nonlinearities, Dimensions, Boundary Conditions).
- Complexity of Solutions: (
Analytical Solutions, Numerical Challenges: Stability, Accuracy).
And so on.
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...
https://sciresol.s3.us-east-2.amazonaws.com/IJST/Articles/2016/Issue-45/Article4.pdf
Article Solving Parabolic and Hyperbolic Equations with Variable Coe...
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.
Greetings.
I want only to renew a link was indicated in my previous answer. The link does not correct. Please use the following link
https://research-journal.org/media/articles/13978.pdf.
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