Shifted fractional-order Jacobi orthogonal functions: Application to a system of fractional differential equations

We present the method of lines (MOL), which is based on the spectral collocation method, to solve space‐fractional advection‐diffusion equations (SFADEs) on a finite domain with variable coefficients. We focus on the cases in which the SFADEs consist of both left‐ and right‐sided fractional derivatives. To do so, we begin by introducing a new set of basis functions with some interesting features. The MOL, together with the spectral collocation method based on the new basis functions, are successfully applied to the SFADEs. Finally, four numerical examples, including benchmark problems and a problem with discontinuous advection and diffusion coefficients, are provided to illustrate the efficiency and exponentially accuracy of the proposed method.

Section: Preliminaries On Jacobi Polynomialsmentioning

confidence: 99%

A collocation method of lines for two‐sided space‐fractional advection‐diffusion equations with variable coefficients

Almoaeet

Shamsi

Khosravian-Arab

et al. 2019

“…Recently, Bhrawy and Zaky developed spectral Tau and collocation methods for linear and nonlinear fractional differential equations, where they employed a new family of fractional bases, called fractional‐order Jacobi functions. We introduced this new orthogonal system of fractional functions as the eigenfunctions of the following Sturm–Liouville problem:

\partial_{t} ((), ν^{- 1} {(1 - t^{ν})}^{θ + 1} t^{νϑ + 1} \partial_{t} v (t)) + ν n (n + θ + ϑ + 1) {(1 - t^{ν})}^{θ} t^{νϑ + ν - 1} v (t) = 0, normal‘ t normal′ \in [0, 1],

explicitly given as

J_{i}^{(θ, ϑ, ν)} (t) = P_{1, i}^{(θ, ϑ)} (t^{ν}) = {falsefalse}_{\sum}^{k = 0} \frac{{(- 1)}^{(i + k)} Γ (i + ϑ + 1) Γ (i + k + θ + ϑ + 1)}{Γ (k + ϑ + 1) Γ (i + θ + ϑ + 1) (i - k)! k!} t^{kν}, ν \in (0, 1],

where

J_{i}^{(θ, ϑ, ν)} (0) = {(- 1)}^{i} \frac{Γ (i + ϑ + 1)}{Γ (ϑ + 1) i!}, J_{i}^{(θ, ϑ, ν)}

…”

Section: Fractional‐order Jacobi Functionsmentioning

confidence: 99%

“…Recently, Bhrawy and Zaky [41] developed spectral Tau and collocation methods for linear and nonlinear fractional differential equations, where they employed a new family of fractional bases, called fractional-order Jacobi functions. We introduced this new orthogonal system of fractional functions as the eigenfunctions of the following Sturm-Liouville problem:…”

Section: Fractional-order Jacobi Functionsmentioning

confidence: 99%

A fractional‐order Jacobi Tau method for a class of time‐fractional PDEs with variable coefficients

Bhrawy

Zaky

2015

This paper presents a shifted fractional‐order Jacobi orthogonal function (SFJF) based on the definition of the classical Jacobi polynomial. A new fractional integral operational matrix of the SFJF is presented and derived. We propose the spectral Tau method, in conjunction with the operational matrices of the Riemann–Liouville fractional integral for SFJF and derivative for Jacobi polynomial, to solve a class of time‐fractional partial differential equations with variable coefficients. In this algorithm, the approximate solution is expanded by means of both SFJFs for temporal discretization and Jacobi polynomials for spatial discretization. The proposed tau scheme, both in temporal and spatial discretizations, successfully reduced such problem into a system of algebraic equations, which is far easier to be solved. Numerical results are provided to demonstrate the high accuracy and superiority of the proposed algorithm over existing ones. Copyright © 2015 John Wiley & Sons, Ltd.

“…Recently, in the framework of Laplace transformation, various analytical methods such as Laplace decomposition method [29,30], homotopy perturbation transform method [31,32], and homotopy analysis transform method (HATM) [33,34] have been used to study the solution of linear and nonlinear fractional differential equations. Also, for solving fractional differential equations (FDEs) and partial FDEs, there are new numerical methods available in our open literature [35][36][37][38][39][40].…”

Section: Introductionmentioning

confidence: 99%

A nonlinear fractional model to describe the population dynamics of two interacting species

Kumar

Odibat

2017

In this paper, the approximate analytical solutions of Lotka–Volterra model with fractional derivative have been obtained by using hybrid analytic approach. This approach is amalgamation of homotopy analysis method, Laplace transform, and homotopy polynomials. First, we present an alternative framework of the method that can be used simply and effectively to handle nonlinear problems arising in several physical phenomena. Then, existence and uniqueness of solutions for the fractional Lotka–Volterra equations are discussed. We also carry out a detailed analysis on the stability of equilibrium. Further, we have derived the approximate solutions of predator and prey populations for different particular cases by using initial values. The numerical simulations of the result are depicted through different graphical representations showing that this hybrid analytic method is reliable and powerful method to solve linear and nonlinear fractional models arising in science and engineering. Copyright © 2017 John Wiley & Sons, Ltd.