|
1
|
Atta AG, Abd‐Elhameed WM, Moatimid GM, Youssri YH. Novel spectral schemes to fractional problems with nonsmooth solutions. Math Methods in App Sciences 2023; 46:14745-14764. [DOI: 10.1002/mma.9343] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/23/2022] [Accepted: 04/19/2023] [Indexed: 09/02/2023]
Abstract
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.
Collapse
Affiliation(s)
- Ahmed G. Atta
- Department of Mathematics Faculty of Education, Ain Shams University Cairo Egypt
| | | | - Galal M. Moatimid
- Department of Mathematics Faculty of Education, Ain Shams University Cairo Egypt
| | - Youssri H. Youssri
- Department of Mathematics Faculty of Science, Cairo University Giza Egypt
- Faculty of Engineering Egypt University of Informatics New Administrative Capital Egypt
| |
Collapse
|
|
2
|
Althubiti S, Mennouni A. A Novel Projection Method for Cauchy-Type Systems of Singular Integro-Differential Equations. Mathematics 2022; 10:2694. [DOI: 10.3390/math10152694] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
This article introduces a new projection method via shifted Legendre polynomials and an efficient procedure for solving a system of integro-differential equations of the Cauchy type. The proposed computational process solves two systems of linear equations. We demonstrate the existence of the solution to the approximate problem and conduct an error analysis. Numerical tests provide theoretical results.
Collapse
|