Numerical approach for solving space fractional orderdiffusion equations using shifted Chebyshev polynomials of the fourth kind

El‐Sayed

2020

Adv Differ Equ

In this article, a fractional-order mathematical physics model, advection–dispersion equation (FADE), will be solved numerically through a new approximative technique. Shifted Vieta–Lucas orthogonal polynomials will be considered as the main base for the desired numerical solution. These polynomials are used for transforming the FADE into an ordinary differential equations system (ODES). The nonstandard finite difference method coincidence with the spectral collocation method will be used for converting the ODES into an equivalence system of algebraic equations that can be solved numerically. The Caputo fractional derivative will be used. Moreover, the error analysis and the upper bound of the derived formula error will be investigated. Lastly, the accuracy and efficiency of the proposed method will be demonstrated through some numerical applications.

Section: Introductionsupporting

confidence: 55%

Vieta–Lucas polynomials for solving a fractional-order mathematical physics model

El‐Sayed

2020

Adv Differ Equ

“…The fractional differential equations model areal problem in life that needs a solution. Therefore, there are many different numerical methods that solve these equations, such as the predictor‐corrector method, Legendre wavelets, Legendre spectral method, Legendre collocation method, pseudo‐spectral scheme, Haar wavelet collocation method, Chebyshev spectral methods,() other techniques,() and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…() This mathematical phenomena, fractional calculus, enable us to characterize and model a real object more accurately than the classical “integer” methods. () Therefore, various works study and present the fractional modeling and fractional order differential equations; for example, see Pinto and Carvalho and Sweilam et al However, not all systems are best described by constant fractional order; for this reason, the variable‐order differential calculus appeared (see, for instance, previous studies()).…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of multiterm variable‐order fractional differential equations via shifted Legendre polynomials

El‐Sayed

Math Methods in App Sciences

2019

In this paper, shifted Legendre polynomials will be used for constructing the numerical solution for a class of multiterm variable‐order fractional differential equations. In the proposed method, the shifted Legendre operational matrix of the fractional variable‐order derivatives will be investigated. The fundamental problem is reduced to an algebraic system of equations using the constructed matrix and the collocation technique, which can be solved numerically. The error estimate of the proposed method is investigated. Some numerical examples are presented to prove the applicability, generality, and accuracy of the suggested method.

“…These applications are used in several fields of engineering and science such as biology [8], dynamics [9], physics [10,11], medicine [12], fluid [13] and others [14][15][16]. For these reasons, there have been a great number of essential works for studying the fractional-order differential equations, see [17][18][19][20][21][22][23][24][25][26][27]. Moreover, the fractional variable-order calculus is considered as a regular filter to supply an influential mathematical building for the description of the dynamical problems in the complex form [28,29].…”

Section: Introductionmentioning

confidence: 99%

A novel Jacobi operational matrix for numerical solution of multi-term variable-order fractional differential equations

Elsayed

Băleanu

Journal of Taibah University for Science

2020

In this article, we introduce a numerical technique for solving a class of multi-term variable-order fractional differential equation. The method depends on establishing a shifted Jacobi operational matrix (SJOM) of fractional variable-order derivatives. By using the constructed (SJOM) in combination with the collocation technique, the main problem is reduced to an algebraic system of equations that can be solved numerically. The bound of the error estimate for the suggested method is investigated. Numerical examples are introduced to illustrate the applicability, generality, and accuracy of the proposed technique. Moreover, many physical applications problems that have the multi-term variable-order fractional differential equation formulae such as the damped mechanical oscillator problem and Bagley-Torvik equation can be solved via the presented method. Furthermore, the proposed method will be considered as a generalization of many numerical techniques.