Operational matrix-based technique treating mixed type fractional differential equations via shifted fifth-kind Chebyshev polynomials

Obeid, Mohamed A.; Salam, Mohamed A. Abd El; Younis, Jihad

doi:10.1080/27690911.2023.2187388

Cited by 5 publications

(1 citation statement)

References 24 publications

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“…The third kind of CPs was employed in [44]. Other types of Chebyshev polynomials, namely CPs of the fifthand sixth-kinds, were used in other articles to treat different DEs; see, for example, [45,46].…”

Section: Introductionmentioning

confidence: 99%

A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices

Ahmed,

Hafez,

Abd-Elhameed

2024

Phys. Scr.

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This paper uses a new method to numerically solve the nonlinear time-fractional generalized Kawahara equations (NTFGKE) with uniform initial boundary conditions (IBCs). A class of modified shifted eighth-kind Chebyshev polynomials (MSEKCPs) is defined to satisfy the given IBCs. The proposed method is based on using operational matrices (OMs) for the ordinary derivatives (ODs) and fractional derivatives (FDs) of MSEKCPs. These OMs are employed together with the spectral collocation method (SCM). Our presented algorithm enables the extraction of efficient and accurate numerical solutions. The convergence of the suggested method and the error analysis have been developed. Three numerical examples are presented to demonstrate the applicability and accuracy of our algorithm. Some comparisons of the presented numerical results with other existing ones are offered to validate the efficiency and superiority of our approach. Tables and graphs demonstrate that the proposed approach produces approximate solutions with high accuracy.

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Section: Introductionmentioning

confidence: 99%

A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices

Ahmed,

Hafez,

Abd-Elhameed

2024

Phys. Scr.

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Adopted spectral tau approach for the time-fractional diffusion equation via seventh-kind Chebyshev polynomials

Abd-Elhameed,

Youssri,

Atta

2024

Bound Value Probl

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This study utilizes a spectral tau method to acquire an accurate numerical solution of the time-fractional diffusion equation. The central point of this approach is to use double basis functions in terms of certain Chebyshev polynomials, namely Chebyshev polynomials of the seventh-kind and their shifted ones. Some new formulas concerned with these polynomials are derived in this study. A rigorous error analysis of the proposed double expansion further corroborates our research. This analysis is based on establishing some inequalities regarding the selected basis functions. Several numerical examples validate the precision and effectiveness of the suggested method.

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A New Generalized Chebyshev Matrix Algorithm for Solving Second-Order and Telegraph Partial Differential Equations

Abd-Elhameed,

Hafez,

Napoli

et al. 2024

Algorithms

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This article proposes numerical algorithms for solving second-order and telegraph linear partial differential equations using a matrix approach that employs certain generalized Chebyshev polynomials as basis functions. This approach uses the operational matrix of derivatives of the generalized Chebyshev polynomials and applies the collocation method to convert the equations with their underlying conditions into algebraic systems of equations that can be numerically treated. The convergence and error bounds are examined deeply. Some numerical examples are shown to demonstrate the efficiency and applicability of the proposed algorithms.

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Operational matrix-based technique treating mixed type fractional differential equations via shifted fifth-kind Chebyshev polynomials

Cited by 5 publications

References 24 publications

A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices

A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices

Adopted spectral tau approach for the time-fractional diffusion equation via seventh-kind Chebyshev polynomials

A New Generalized Chebyshev Matrix Algorithm for Solving Second-Order and Telegraph Partial Differential Equations

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