Numerical solution for the time-fractional Fokker–Planck equation via shifted Chebyshev polynomials of the fourth kind

Habenom, Haile; Suthar, D. L.

doi:10.1186/s13662-020-02779-7

Cited by 16 publications

(7 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, there are considerable contributions concerned with the first-, second-, third-and fourth-kinds of Chebyshev polynomials. These four kinds of Chebyshev polynomials have played important roles in the numerical solutions of various types of differential equations using the different versions of spectral methods (see, [38][39][40][41][42]). One of the advantages of using Chebyshev polynomials is the good representation of smooth functions by finite Chebyshev expansion.…”

Section: Introductionmentioning

confidence: 99%

Advanced shifted sixth-kind Chebyshev tau approach for solving linear one-dimensional hyperbolic telegraph type problem

et al. 2022

View full text Add to dashboard Cite

A new numerical scheme based on the tau spectral method for solving the linear hyperbolic telegraph type equation is presented and implemented. The derivation of this scheme is based on utilizing certain modified shifted Chebyshev polynomials of the sixth-kind as basis functions. For this purpose, some new formulas concerned with the modified shifted Chebyshev polynomials of the sixth-kind have been stated and proved, and after that, they serve to study the hyperbolic telegraph type equation with our proposed scheme. One advantage of using this scheme is that it reduces the problem into a system of algebraic equations that can be simplified using the Kronecker algebra analysis. The convergence and error estimate of the proposed technique are analyzed in detail. In the end, some numerical tests are presented to demonstrate the efficiency and high accuracy of the proposed scheme.

show abstract

Section: Introductionmentioning

confidence: 99%

Advanced shifted sixth-kind Chebyshev tau approach for solving linear one-dimensional hyperbolic telegraph type problem

et al. 2022

View full text Add to dashboard Cite

show abstract

“…The historical and nonlocal distributed effects are considered via fractional differential coefficients; an outstanding literature on this topic may be found in numerous monographs [12][13][14][15]. For this reason, many authors are attracted to knowing the properties of fractional differential equations and vast applications in modeling and engineering fields [16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

A Comparative Analysis of Fractional Space-Time Advection-Dispersion Equation via Semi-Analytical Methods

Aljahdaly

Shah

Naeem

et al. 2022

Journal of Function Spaces

View full text Add to dashboard Cite

The approximate solutions of the time fractional advection-dispersion equation are presented in this article. The nonlocal nature of solute movement and the nonuniformity of fluid flow velocity in the advection-dispersion process lead to the formation of a heterogeneous system, which can be modeled using a fractional advection-dispersion equation, which generalizes the classical advection-dispersion equation and replaces the time derivative with the fractional Caputo derivative. Researchers use a variety of numerical techniques to study such fractional models, but the nonlocality of the derivative having fractional order leads to high computation complexity and complex calculations, so the task is to find an efficient technique that requires less computation and provides greater accuracy when numerically solving such models. A innovative techniques, homotopy perturbation method and new iteration method, are used in connection with the Elzaki transform to solve the “fractional advection-dispersion equation” which provides the solution in the convergent series form. When the homotopy perturbation method is used with the Elzaki transform, fast convergent series solutions can be obtained with less computation. By solving some cases of time-fractional advection-dispersion equation with varied initial conditions with the help of new iterative transform method and homotopy perturbation transform method demonstrates the usefulness of the proposed methods.

show abstract

“…Moreover, in [10], the authors solved a class of non-linear variable-order fractional reaction-diffusion equation based on using the shifted Chebyshev polynomials of the fifth kind. For some other articles concerned with the different kinds of Chebyshev polynomials, see for example [11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Novel Expressions for the Derivatives of Sixth Kind Chebyshev Polynomials: Spectral Solution of the Non-Linear One-Dimensional Burgers’ Equation

Abd-Elhameed¹

2021

Fractal Fract

View full text Add to dashboard Cite

This paper is concerned with establishing novel expressions that express the derivative of any order of the orthogonal polynomials, namely, Chebyshev polynomials of the sixth kind in terms of Chebyshev polynomials themselves. We will prove that these expressions involve certain terminating hypergeometric functions of the type 4F3(1) that can be reduced in some specific cases. The derived expressions along with the linearization formula of Chebyshev polynomials of the sixth kind serve in obtaining a numerical solution of the non-linear one-dimensional Burgers’ equation based on the application of the spectral tau method. Convergence analysis of the proposed double shifted Chebyshev expansion of the sixth kind is investigated. Numerical results are displayed aiming to show the efficiency and applicability of the proposed algorithm.

show abstract

Numerical solution for the time-fractional Fokker–Planck equation via shifted Chebyshev polynomials of the fourth kind

Cited by 16 publications

References 22 publications

Advanced shifted sixth-kind Chebyshev tau approach for solving linear one-dimensional hyperbolic telegraph type problem

Advanced shifted sixth-kind Chebyshev tau approach for solving linear one-dimensional hyperbolic telegraph type problem

A Comparative Analysis of Fractional Space-Time Advection-Dispersion Equation via Semi-Analytical Methods

Novel Expressions for the Derivatives of Sixth Kind Chebyshev Polynomials: Spectral Solution of the Non-Linear One-Dimensional Burgers’ Equation

Contact Info

Product

Resources

About