A Newton interpolation based predictor-corrector numerical method for fractional differential equations with an activator-inhibitor case study

by   Redouane Douaifia, et al.

This paper presents a new predictor-corrector numerical scheme suitable for fractional differential equations. An improved explicit Atangana-Seda formula is obtained by considering the neglected terms and used as the predictor stage of the proposed method. Numerical formulas are presented that approximate the classical first derivative as well as the Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives. Simulation results are used to assess the approximation error of the new method for various differential equations. In addition, a case study is considered where the proposed scheme is used to obtained numerical solutions of the Gierer-Meinhardt activator-inhibitor model with the aim of assessing the system's dynamics.



There are no comments yet.


page 1

page 2

page 3

page 4


The Wynn identity as the long sought criterion for the choice of the optimal Padé approximant

The performed numerical analysis reveals that Wynn's identity for the co...

An Accurate Numerical Method and Algorithm for Constructing Solutions of Chaotic Systems

In various fields of natural science, the chaotic systems of differentia...

Efficient and fast predictor-corrector method for solving nonlinear fractional differential equations with non-singular kernel

Efficient and fast predictor-corrector methods are proposed to deal with...

Efficient computation of the Wright function and its applications to fractional diffusion-wave equations

In this article, we deal with the efficient computation of the Wright fu...

Approximation of fractional harmonic maps

This paper addresses the approximation of fractional harmonic maps. Besi...

HPC optimal parallel communication algorithm for the simulation of fractional-order systems

A parallel numerical simulation algorithm is presented for fractional-or...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

Over the last century, ordinary and partial differential equations have been shown to produce accurate models of real life phenomena spanning a range of different scientific and engineering disciplines. Based on these models, researchers are able to infer the characteristics of these phenomena and devise effective control strategies. Such characteristics include the existence and boundedness of solutions, blow-up time, asymptotic behavior, and more. Since these models can be quite complicated and analytical solutions are not always attainable, numerical analysis became a useful tool that helps obtain approximate solutions and give indications on the behavior of these models. The simplest numerical methods reported in the literature and suitable for linear systems are based on linear interpolation, which has been around for over 2000 years. For the nonlinear case, well established interpolation techniques include Newton’s method, Lagrange interpolation polynomials, Gaussian elimination, and Euler’s method

Werner1984 ; Fred1970 ; Yang2015 ; Dimitrov1994 .

In recent years, an apparent shift has been observed from classic models involving integer-order derivatives to fractional ones. This shift may be attributed to the many benefits associated with fractional derivatives including their infinite memory and wider dynamical range. Numerical methods had to evolve in order for researchers to investigate these fractional models. Several numerical schemes have been proposed for solving fractional ordinary differential equations, especially nonlinear ones including

Diethelm1998 ; Ford2001 ; Odibat2008 ; Moghaddam2016 ; Asl2017 ; Patricio2019 . To the best of the authors’ knowledge, the most widely accepted scheme is the Adams-Bashforth method developed with a Lagrange interpolation polynomial basis Zhang2018 ; Jain2018 . In recent years, studies have shown that on average, Newton’s method is superior to Lagrange polynomials taking into consideration a wide range of polynomial functions Srivastava2012a ; Srivastava2012b . A numerical method suitable for both integer and fractional ordinary differential systems was proposed by Atangana and Seda by replacing the Lagrange polynomial interpolation of the Adams-Bashforth scheme with Newton quadratic interpolation in Atangana2020 ; Atangana2020Corrigendum . The authors derived iterative numerical formulas for the standard and fractal versions of the Caputo, Caputo-Fabrizio, and Atangana-Baleanu fractional derivatives. This method was applied to chaotic systems and showed promising results Alkahtani2019 ; Atangana2020a ; Atangana2020c . The method was also extended to partial differential equations with integer and non-integer orders Atangana2020b .

Over the last few decades a class of numerical methods called predictor-corrector emerged and became the center of attention for many researchers Gragg1964 ; Marciniak2017 ; Butcher2016 . It is well known that numerical methods are generally divided into implicit and explicit types and that the implicit type is more stable and efficient but difficult to solve due to the fact that the unknown appears on both sides of the formula. Predictor-corrector methods work in two steps. An initial explicit approximation (predictor) of the solution is obtained and substituted into right side of the implicit formula (corrector). A predictor-corrector Adams-Bashforth method was introduced in Diethelm2002 . In this method, the explicit one-step Adams–Bashforth rule and the implicit one-step Adams-Moulton method are used as predictor and corrector, respectively. Other more recent works include Nguyen2017 ; Douaifia2019 ; Kumar2019 ; Heris2019 . In this paper, we propose a new predictor-corrector method where an improved version of the Atangana-Seda method of Atangana2020 ; Atangana2020Corrigendum is used as the predictor. We derive iterative formulas for the classical as well as the Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivative scenarios. Numerical examples are presented to evaluate the effectiveness of the proposed methods.

2 Important Definitions

Before we delve into the main concern of the paper, let us describe the fractional integrals and derivatives that will be used in our work. For more on these definitions, the reader may wish to refer to Atangana2020 ; Podlubny ; Kilbas2006 ; CaputoFabrizio2015 ; AtanganaBaleanu2016 .

Definition 1

The –order Riemann–Liouville fractional integral of a function is defined as


where and  is the Gamma function defined as


for .

Definition 2

The –order Caputo fractional derivative of a function is defined as

Definition 3

The Caputo-Fabrizio fractional integral of a function is defined as


where , and is a normalization function satisfying .

Definition 4

Let , , and . The Caputo-Fabrizio fractional derivative of a function is defined as

Definition 5

The Atangana-Baleanu fractional integral of a function is defined as


where , and

Definition 6

Let , , and . The Atangana-Baleanu fractional derivative in the Caputo sense of a function is defined as


where is the Mittag-Leffler kernel function of order defined as


for and.

3 The Proposed Predictor-Corrector Method

3.1 Classical Derivative

We start with the simple classical initial-value problem given by


where is a smooth nonlinear function guaranteeing a unique solution . In order to develop a numerical formula approximating the solution of (10), we convert the differential equation into the integral


In an iterative approximation, we may choose two distinct points in time  and . Substituting these points into (11) yields


respectively. Taking the difference yields


Hence, the function  may be approximated over the interval  by means of Newton’s second order interpolation polynomial given by


Substitution into (12) leads to the difference formula


Given that




formula (14) reduces to the implicit form


The term appears on both sides of the formula. The predictor-corrector scheme works by first producing an approximation of denoted by , and then using (17) to correct the approximation. The correction formula is, thus, given by


where the predictor is obtained by means of the Atangana-Seda scheme (cf. Atangana2020 ), i.e.


3.2 Caputo Fractional Derivative

Let us now move to the fractional derivative case. Various derivatives have been proposed throughout the years. However, the most commonly used is the Caputo one. We consider the initial-value problem


with , and being a smooth nonlinear function such that (20) admits a unique solution . Following the same procedure of the standard case, we start with the integral


At the single point , we have the following


with . Function can be approximated over the sub-interval as a polynomial by means of






Using the Newton polynomial (23), formula (22) becomes


Simplifying and rearranging the terms leads to


The four different integrals in (27) can be calculated as




respectively. By substituting these calculations into (27), we obtain


In order to simplify the formulas to come, let us define the expresion


with the convention


Using this notation, (32) can be rewritten in the form


Formula (35) will serve as our implicit part, i.e. the corrector. The terms on the right hand side will be replaced by the predictor , which will be an improved version of the Atangana-Seda scheme derived for the Caputo fractional derivative in Atangana2020 . To obtain our predictor formula, let us go back to (21) and use the predictor notation , which yields

and, consequently, at , we have


The function can be approximated over each sub-interval using a delayed version of the Newton’s polynomial seen earlier in (23) and given by






Substituting the interpolated approximation of into (36) yields the predictor


We can calculate the integrals as




Substituting these calculations into (40) produces the improved Atangana-Seda scheme predictor


In each iteration, the predictor (44) is calculated and then corrected by means of the implicit formula


3.3 Caputo-Fabrizio Fractional Derivative

In this section, we will follow the same steps to derive a predictor-corrector numertical scheme for the Caputo-Fabrizio fractional initial-value problem


where the fractional order and is a nonlinear smooth function chosen such that system (46) admits a unique solution . Similar to the previous section, we start with the difference formula

which when evaluated at two points in time and yields