New predictor‐corrector scheme for solving nonlinear differential equations with Caputo‐Fabrizio operator

Abstract

In this paper, we develop a new, simple, and accurate scheme to obtain approximate solution for nonlinear differential equation in the sense of Caputo-Fabrizio operator. To derive this new predictor-corrector scheme, which suits on Caputo-Fabrizio operator, firstly, we obtain the corresponding initial value problem for the differential equation in the Caputo-Fabrizio sense. Hence, by fractional Euler method and fractional trapeziodal rule, we obtain the predictor formula as well as corrector formula. Error analysis for this new method is derived. To test the validity and simplicity of this method, some illustrative examples for nonlinear differential equations are solved. KEYWORDSAdams-Bashforth-Moulton method, Caputo fractional derivative, Caputo-Fabrizio operator, nonlinear differential equation, predictor-corrector scheme INTRODUCTIONIn 2015, Michele Caputo and Mauro Fabrizio introduced a new definition of Caputo derivative, 1 which was called Caputo-Fabrizio fractional derivative. The main characteristics of this new definition are including regular kernel, having two different representations for temporal and spatial variables and as memory operator. Although it was verified that the operator does not fit the usual concepts neither for fractional nor for integer derivative/integral, 2 the new operator was successfully applied in describing the control movement of waves on the area of shallow water, 3 unsteady flows of an incompressible Maxwell fluid, 4 anomalous diffusion phenomena, 5 diffusion and the diffusion-advection equation, 6 epidemiological model for computer viruses, 7 model of circadian rhythms, 8 and many more phenomena. In this research direction, some existing numerical methods have been extended to the problem defined in the sense of this new Caputo-Fabrizio operator. Among them is the combination of the Laplace transform and homotopy analysis method to solve the so-called fractional partial differential equation problem with Caputo-Fabrizio operator. 9 Other methods include multiqaudric (MQ)-RBF collocation method, 10 discretization scheme, 11 fractional Adams-Bashforth method, 12 and three-step fractional Adams-Bashforth method. 13 However, since Caputo-Fabrizio operator is relatively new, there are still relatively limited works that have been done to obtain the simple, reliable, and accurate solution for the problem defined in Caputo-Fabrizio operator.On the other hand, predictor-corrector scheme had been extended to solve fractional differential equations. 14 Since that, some modification was done to obtain a better accurate scheme or more suitable to different problems, such as Abbreviation used: Fractional ordinary differential equation (FODE).

Computation of Stability Criterion for Fractional Shimizu–Morioka System Using Optimal Routh–Hurwitz Conditions

“…This is because most of the system of fractional differential equation or fractional dynamical system do not have analytical solution. Furthermore, we had also modified this approach for solving differential equation in Caputo-Fabrizio sense as in [23].…”

Three-Species Lotka-Volterra Model with Respect to Caputo and Caputo-Fabrizio Fractional Operators

“…Here, we use a numerical method based on Adams-Bashforth methods proposed in [44,45] for solving FDEs with Caputo derivative. In [22], authors have investigated a fractional Adams-Bashforth method for solving FDEs with the CF operator. In this section, we improve the method to solve the LV system of the CF operator and we compare it with the solutions of the counterpart with the Caputo operator.…”

“…Likewise, the Predictor-Corrector methods have been exploited for solving many nonlinear ordinary differential equations. Based on the fractional Euler method and fractional trapeziodal rule, the Predictor-Corrector scheme or the Adams-Bashforth-Moulton method can be extended to efficiently solve a differential equation involving a CF operator [22]. A new form of this method for the solution of differential equations with Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives is proposed in [23] by considering the nonlinearity of the different kernels-the power law for the Riemann-Liouville, the exponential decay law for the Caputo-Fabrizio, and the Mittag-Leffler law for the Atangana-Baleanu fractional derivative.…”

An analysis for heat equations arises in diffusion process using new Yang‐Abdel‐Aty‐Cattani fractional operator

“…Many attempts were allotted to catch many extensible, thought‐provoking, and unique nonsingular fractional operators based on kernels . In 2015, Caputo and Fabrizio discovered a new operator of arbitrary order, namely, Caputo‐Fabrizio (CF) operator with arbitrary order and enforced to the several linear and nonlinear physical problems . In 2016, Atangana and Baleanu introduced another nonsingular derivative based on Miitag‐Leffler kernel and applied to the many problems .…”