Numerical Treatment of Multi-Term Fractional Differential Equations via New Kind of Generalized Chebyshev Polynomials

Abd-Elhameed, W. M.

doi:10.3390/fractalfract7010074

Cited by 20 publications

(9 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The outcomes are encouraging. In the near future, we want to expand the existing methods to handle the same PDEs in two and three dimensions, for instance, [38][39]. Also, we plan to generalize the method to handle nonlinear fractional PDEs.…”

Section: Discussionmentioning

confidence: 99%

Jacobi Rational Operational Approach for Time-Fractional Sub-Diffusion Equation on a Semi-Infinite Domain

Hafez,

Youssri,

Atta

2023

Contemp. Math.

View full text Add to dashboard Cite

In this study, we employ a rational Jacobi collocation technique to effectively address linear time-fractional subdiffusion and reaction sub-diffusion equations. The semi-analytic approximation solution, in this case, represents the spatial and temporal variables as a series of rational Jacobi polynomials. Subsequently, we apply the operational collocation method to convert the target equations into a system of algebraic equations. A comprehensive investigation into the convergence properties of the dual series expansion employed in this approximation is conducted, demonstrating the robustness of the numerical method put forth. To illustrate the method's accuracy and practicality, we present several numerical examples. The advantages of this method are: high accuracy, efficiency, applicability, and high rate of convergence.

show abstract

Section: Discussionmentioning

confidence: 99%

Jacobi Rational Operational Approach for Time-Fractional Sub-Diffusion Equation on a Semi-Infinite Domain

Hafez,

Youssri,

Atta

2023

Contemp. Math.

View full text Add to dashboard Cite

show abstract

“…Abd-Elhameed and Alsuyuti [14] generalized the class of Chebyshev polynomials of the first kind by introducing a new class of orthogonal polynomials. They established some basic properties of the generalized Chebyshev polynomials and their shifted ones, and, additionally, they found, for these generalized polynomials, some new formulas.…”

Section: Introductionmentioning

confidence: 99%

“…Abd-Elhameed et al [15] obtained numerical solutions of the nonlinear time-fractional generalized Kawahara Equation (NTFGKE) by giving an innovative approach involving a spectral collocation algorithm. They introduced the "Eighth-kind Chebyshev polynomials (CPs)" which are a new set of orthogonal polynomials (OPs) and represent special cases of generalized Gegenbauer polynomials.…”

Section: Introductionmentioning

confidence: 99%

Discrete Entropies of Chebyshev Polynomials

Sfetcu,

Preda

2024

Mathematics

View full text Add to dashboard Cite

Because of its flexibility and multiple meanings, the concept of information entropy in its continuous or discrete form has proven to be very relevant in numerous scientific branches. For example, it is used as a measure of disorder in thermodynamics, as a measure of uncertainty in statistical mechanics as well as in classical and quantum information science, as a measure of diversity in ecological structures and as a criterion for the classification of races and species in population dynamics. Orthogonal polynomials are a useful tool in solving and interpreting differential equations. Lately, this subject has been intensively studied in many areas. For example, in statistics, by using orthogonal polynomials to fit the desired model to the data, we are able to eliminate collinearity and to seek the same information as simple polynomials. In this paper, we consider the Tsallis, Kaniadakis and Varma entropies of Chebyshev polynomials of the first kind and obtain asymptotic expansions. In the particular case of quadratic entropies, there are given concrete computations.

show abstract

“…In a recent article [1], we used expansions in fractional powers to solve, in an elementary way, several multi-term fractional differential equations, which appeared in the literature (see, e.g., [2][3][4][5][6][7][8][9]). The fractional derivative is a critical concept for innumerable applications in the most diverse fields of applied sciences.…”

Section: Introductionmentioning

confidence: 99%