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 methodWerner1984 ; 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 includingDiethelm1998 ; 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 .
The –order Riemann–Liouville fractional integral of a function is defined as
where and is the Gamma function defined as
The –order Caputo fractional derivative of a function is defined as
The Caputo-Fabrizio fractional integral of a function is defined as
where , and is a normalization function satisfying .
Let , , and . The Caputo-Fabrizio fractional derivative of a function is defined as
The Atangana-Baleanu fractional integral of a function is defined as
where , and
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
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
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
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