Novel spectral schemes to fractional problems with nonsmooth solutions

In this article, we present two numerical methods for treating the fractional initial‐value problem (FIVP) and time‐fractional partial differential problem (FPDP) that caused the error to decay exponentially rapidly. The derivations of the proposed schemes rely on the use of a spectral Galerkin meth...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences 2023-09, Vol.46 (13), p.14745-14764
Main Authors: Atta, Ahmed G., Abd‐Elhameed, Waleed M., Moatimid, Galal M., Youssri, Youssri H.
Format: Article
Language:English
Subjects:
Basis functions
Chebyshev approximation
Chebyshev polynomials of the fifth kind
convergence analysis
Error analysis
fractional initial‐value problem
Galerkin method
Numerical methods
Polynomials
Spectral methods
Thermal expansion
time‐fractional partial differential problem
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Summary:In this article, we present two numerical methods for treating the fractional initial‐value problem (FIVP) and time‐fractional partial differential problem (FPDP) that caused the error to decay exponentially rapidly. The derivations of the proposed schemes rely on the use of a spectral Galerkin method that reduces each of the FIVP and FPDP into an algebraic system of equations in the unknown expansion coefficients. The class of orthogonal polynomials, namely, Chebyshev polynomials of the fifth kind is utilized. In terms of new basis functions called regular shifted Chebyshev poly‐fractionomials of fifth kind, approximate solutions to the FIVP and FPDP are obtained. Moreover, convergence and error analysis of the two problems are investigated in depth. Some numerical examples are presented with some comparisons. In conclusion, our spectral methods are effective and convenient.
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.9343