On the formulation of a predictor–corrector method to model IVPs with variable‐order Liouville–Caputo‐type derivatives

Abstract

Variable‐order fractional calculus operators can be used as a valuable tool to simulate many nonlinear models with a memory property in fractional calculus applications. In this paper, we developed a novel predictor–corrector method for solving numerically nonlinear differential equations with variable‐order Liouville–Caputo‐type fractional derivatives. First, we found an inversion integral formula that is equivalent to the studied problem, and then we used it to formulate our numerical algorithm. Numerical approximate solutions of some variable‐order fractional models have been presented to display the efficiency and accuracy of the proposed algorithm. The influence of the variable‐order on the dynamic behavior of the considered problems is described.

Numerical solutions of linear time-fractional advection-diffusion equations with modified Mittag-Leffler operator in a bounded domain

“…The reader is recommended to review the work presented in [37] for the existence problem and the numerical simulation issue for solutions of IVPs involving MABC fractional derivatives. Various predictor-corrector methods have been successfully developed to deal with IVPs involving different Caputo-type fractional derivative models [42][43][44][45][46][47][48].…”

A study on variable-order delay fractional differential equations: existence, uniqueness, and numerical simulation via a predictor corrector algorithm

“…In the literature, some P-C methods have been effectively developed to numerically simulate Caputo-types FDEs [45][46][47][48][49]. In the current section, we propose a P-C scheme by modifying the P-C method developed in [37] to numerically simulate the VO-DFDE…”

“…In principle, some of the presented techniques for dealing with VO-FDEs depend on applying an inexact inversion formula that describes the connection between the VO fractional derivative and integral operators, and thus, in this case, the presented technique is generally invalid. Therefore, recently, a convenient and effective predictor-corrector (P-C) scheme was proposed in [37] to provide stable and accurate approximate solutions for VO-FDEs. This novel P-C scheme was applied to an equivalent integral equation of Volterra type based on an appropriate and valid inversion formula.…”

“…The present paper is motivated by recent configurations and developments of differential systems involving VO fractional derivatives such as VO delay fractional differential equations (DFDEs), the existence of invalid techniques and the difficulty of solving such models. Therefore, in the current work, we modify the P-C scheme developed in [37] to numerically simulate VO-DFDEs and study the unique existence of solutions of such models. This manuscript is organized as follows.…”

“…Mainly, in section 3, we discussed the existence and uniqueness of solutions for VO-DFDEs. In section 4, we utilized the P-C scheme developed in [37] to design a numerical approach for solving VO-DFDEs. Next, in section 5, numerical simulations of some illustrative examples of VO-DFDEs are presented to illustrate the performance of the suggested approach.…”