I am asking about the stability of numerical methods. Specially pseudo spectral method.
when are solving fractional differential equations or for instance ordinary differential equations using Legendre operational matrices of integration and derivative. Then as clear that highest order derivative is assumed in product of two vectors coefficients vector and function vector, then operational matrices of derivative and integrals are used to obtain the lower order derivatives, and is then substituted in original equation, as a result we get system of algebraic equations,
These algebraic equations are then solved to get approximation of solution differential equations. Clearly the use of large number of orthogonal polynomials will result in more accurate solution (as I experimentally observed over a large variety of problem).
Now if some one is asked that show that the algorithm is stable, and convergent. Then what does the term stable means here. In other words how can we theoretically prove that the algorithm is stable and convergent.
I can suggest you my recently paper in ETNA entitled with "Convergence analysis of operational Tau method for Abel type volterra integral equations".
Dear Dr. Hammad,
Please see the attachment. From
http://www.cscamm.umd.edu/tadmor/pub/spectral/Tadmor.SINUM-89.pdf
and
http://www.math.purdue.edu/~shen/pub/SINUM_lagr.pdf
respectively.
Thanks for sharing useful links. But up know I am not clear on the topic.
There are many definitions for stability, you have to specify (by yourself or by using the help of supervisors) what is your own one.
Dear Hammad,
it's not clear to me whether your question regards time or space fractional derivatives. In the latter, we have recently proposed some Fourier spectral methods for reaction-diffusion problems involving the fractional Laplacian (see attached link). The stability analysis is not provided, but since the decomposition of the fractional Laplacian is equivalent to the standard one, it should be relatively easy to extend all classical results on these types of systems.
Article Fourier spectral methods for fractional-in-space reaction-di...
Thanks to all. For useful comments. The point which I want to discuss is now clear to me. Thanks Alfasno...., Thanks Wiwat....., Also thanks Demetris, Also many thanks Payam
- Badges
- Science topic
- Similar topics
- Mathematical Sciences
- Algebra