On the convergence of Jacobi‐Gauss collocation method for linear fractional delay differential equations

Abstract

In this paper, we propose a convergent numerical method for solving linear fractional differential equations. We first convert the equation into an equivalent time-dependent equation and then discretize it at the Jacobi-Gauss collocation points. Using this method, we achieve a system of algebraic equations to approximate the solution of the original equation. Here, we gain the solution and its fractional derivative, simultaneously. We fully present the convergence analysis for the suggested method. We finally illustrate the efficiency of our method by solving some numerical examples.

Study of time fractional proportional delayed multi‐pantograph system and integro‐differential equations

“…Delayed volterra integral equations were studied via collocation methods, 27 fractional delay integro‐differential equation by spectral collocation method with Chebyshev polynomials as base functions, 28 and fractional differential transform method (FDTM) 29 . Fractional delayed differential equations have been studied by Taylor wavelets approach, 30 generalized Adams method, 31 Jacobi–Gauss collocation method, 32 integro quadratic spline‐based scheme, 33 Daftardar–Gejji and Jafari transform method for higher order and large time scale, 34 and many more. For more details, the interested readers are referred to previous studies 22,32,35,36 …”

“…Fractional delayed differential equations have been studied by Taylor wavelets approach, 30 generalized Adams method, 31 Jacobi–Gauss collocation method, 32 integro quadratic spline‐based scheme, 33 Daftardar–Gejji and Jafari transform method for higher order and large time scale, 34 and many more. For more details, the interested readers are referred to previous studies 22,32,35,36 …”

“…For more details, the interested readers are referred to previous studies. 22,32,35,36 For instance, the pantograph-type delayed differential equations are beneficial in many rigourous models in the field of biological/physical sciences, and the word pantograph comes from the rigorous research work by Ockendon and Tayler on the accumulation of current by an electric locomotive's pantograph head. 37 In consequence of this, system of multi-pantograph equations is one of the most significant delay differential equations arises in many scientific models functioning in number theory, population studies, dynamical, and electrodynamics systems.…”

A new numerical method to solve pantograph delay differential equations with convergence analysis

“…So far, numerous numerical methods have been proposed for differential equations, including homotopy methods, spectral and pseudospectral methods, tau methods, finite difference methods, finite element methods, and methods utilizing polynomial approximations, in particular, Hermit, Laguerre, Bernstein, Taylor, Bernoulli, and Jacobi approximations can be mentioned. To get acquainted with some of these techniques and methods, the reader can refer to the works [11,15,22,25,[37][38][39][40].…”

Technique to Solve Linear Fractional Differential Equations Using B-Polynomials Bases

“…Nonlinear fractional-order partial differential equations would be considered in future work. In recent years, many authors have used various numerical and analytical procedures to unravel fractional-order differential equations such as the modified simple equation approach [12][13][14][15] the variational iteration procedure [16], Adams-Bashfort-Mowlton Method [17], the Lagrange characteristic approach [18,19], Adomian decomposition method [20], the finite difference procedure [21], the differential transformation method [22], the finite element technique [23], the fractional sub equation procedure [24], the (G /G)-expansion method [25], first integral approach [26], Jacobi-Gauss collocation 2 of 19 method [27], the spectral collocation method (SCM) [28,29], and the fractional complex transform technique [30]. Every method has its own pros and cons.…”