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

Abstract

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.

Fractional Differential Equations in the Model of Vocational Education and Teaching Practice Environment

“…Try to find the angular velocity of the camera's rotation when the drone reaches the sky above the scenic spot. The problem can be easily solved after learning the function's derivative and its related rate of change [10]. The detailed process is given below: Assume that the horizontal distance between the drone and the scenic area is the angle between ( ),…”

A novel shifted Jacobi operational matrix method for nonlinear multi‐terms delay differential equations of fractional variable‐order with periodic and anti‐periodic conditions

“…And in the endpoint values are given as $$$$ {\displaystyle \begin{array}{c}{P}_{T,i}^{\left(\alpha, \beta \right)}(0)={\left(-1\right)}^i\frac{\Gamma \left(\alpha +i+1\right)}{\Gamma \left(\alpha +1\right)\Gamma \left(k+1\right)},\\ {}{P}_{T,i}^{\left(\alpha, \beta \right)}(T)=\frac{\Gamma \left(\beta +i+1\right)}{\Gamma \left(\beta +1\right)\Gamma \left(k+1\right)}.\end{array}} $$$$ Note that we can estimate the $$$ {P}_{T,i}^{\left(\alpha, \beta \right)}(t) $$$ by the recurrence formula; we refer the interested reader to El‐Sayed et al 33 …”

“…Now, the goal of current work is to generalize the orthogonal polynomials in the base of solution. In fact, this technique is introduced in El‐Sayed et al, 33 and we present a novel shifted Jacobi operational matrix for the fractional derivatives to solve a class of nonlinear multi‐terms delay $$$ DE $$$s of fractional variable‐order with periodic condition which as follow: $$$$ {\displaystyle \begin{array}{cc}\hfill \sum \limits_{j=1}^n{\alpha}_j{D}^{\eta_j(t)}w(t)& +{\alpha}_{n+1}w\left(t-\tau \right)\hfill \\ {}\hfill & =F\kern2.05482pt \left(t,w(t),{D}^{\eta_1(t...$$…”

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

“…(see, for instance, [9,18,[26][27][28]). The ordinary/partial differential equations that contain fractional-order derivatives provide more flexible models compared with the classical ones that are characterized by integer-orders [2,7,10]. To understand the idea of the fractional derivative more clearly, see the example of the Lane-Emden-type equations of the fractional-order derivatives that include aspects of a stellar structure, the thermal history of a spherical cloud of the gas, isothermal gas spheres, and thermionic currents [19].…”