On the new properties of Caputo–Fabrizio operator and its application in deriving shifted Legendre operational matrix

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).

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“…Proof The proof is following Section in Loh et al Taking Laplace transform of both sides of Equation , we have

\begin{matrix} frakturL [_{C F} D_{0, x}^{α} f false(x false) false] & = frakturL false[g false(x false) false], s > 0, \\ \frac{M false(false{α false} false)}{s false(1 - false{α false} false) + false{α false}} ((), s^{n} F false(s false) - true \sum_{k = 1}^{n} s^{n - k} f^{false(k - 1 false)} false(0 false)) & = G false(s false) . \end{matrix}

Thus,

\begin{matrix} F false(s false) & = \frac{1}{s^{n}} true \sum_{k = 1}^{n} s^{n - k} f^{false(k - 1 false)} false(0 false) + \frac{s false(1 - false{α false} false)}{s^{n} M false(false{α false} false)} G false(s false) + \frac{false{α false}}{s^{n} M false(false{α false} false)} G false(s false) \\ = true \sum_{k = 1}^{n} \frac{1}{s^{k}} f^{false(k - 1 false)} false(0 false) + \frac{false(1 - false{α false} false)}{M false(false{α false} false)} ((), \frac{1}{s^{n - 1}} G false(s false)) + \frac{α}{M false(false{α false} false)} ((), \frac{1}{s}) \end{matrix}

…”

Section: Predictor‐corrector Scheme For Fode Involving Caputo‐fabriziunclassified

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

Toh

Phang

Loh

2018

Math Methods in App Sciences

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“…The past studies on infectious diseases and other related diseases have been studied using Riemann-Liouville fractional order derivative or the Caputo fractional order derivative, but these has been in recent times faulted where it shown that these derivatives pose a little challenge. Present studies have shown that at the end point of the interval where these models are defined their kernels have a singularity which creates discontinuity of the system at such point (Loh et al, 2018;. As a result of these, many new definitions of fractional derivatives have now been proposed in the literature (Shah et Loh et al, 2018).…”

Section: Introductionmentioning

confidence: 99%

A Fractional Order Hiv/Aids Model Using Caputo-Fabrizio Operator

Unaegbu

Onah

Oyesanya³

2021

AJID

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Background: HIV is a virus that is directed at destroying the human immune system thereby exposing the human body to the risk of been affected by other common illnesses and if it is not treated, it generates a more chronic illness called AIDS. Materials and Methods: In this paper, we employed the fixed-point theory in developing the uniqueness and existence of a solution of fractional order HIV/AIDS model having Caputo-Fabrizio operator. This approach adopted in this work is not conventional when solving biological models by fractional derivatives. Results: The results showed that the model has two equilibrium points namely, disease-free, and endemic equilibrium points, respectively. We showed conditions necessitating the existence of the endemic equilibrium point and showed that the disease-free equilibrium point is locally asymptotically stable. We also tested the stability of our solution using the iterative Laplace transform method on our model which was also shown stable agreeing with the disease-free equilibrium. Conclusions: Numerical simulations of our model showed clear comparison with our analytical results. The numerical solutions show that given fractional operator like the Caputo-Fabrizio operator, it is less noisy and plays a major role in making a precise decision and gives room (‘freedom’) to use data of specific patients as the model can be easily adjusted to accommodate this, as it a better fit for the patients’ data and provide meaningful predictions. Finally, the result showed the advantage of using fractional order derivative in the analysis of the dynamics of HIV/AIDS over the classical case.

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“…In this research direction, operational matrix related methods are getting considerable attention for solving fractional calculus problems; among them is the Legendre operational matrix for solving fractional differential equations in the Caputo-Fabrizio sense [18]. Apart from that, for solving fractional differential equations defined in the Atangana-Baleanu fractional derivative, fifth-kind Chebyshev polynomials have been applied to derive operational matrices for multi-variable orders differential equations with non-singular kernels [19].…”

Section: Introductionmentioning

confidence: 99%

An Operational Matrix Method Based on Poly-Bernoulli Polynomials for Solving Fractional Delay Differential Equations

2020

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In this work, we derive the operational matrix using poly-Bernoulli polynomials. These polynomials generalize the Bernoulli polynomials using a generating function involving a polylogarithm function. We first show some new properties for these poly-Bernoulli polynomials; then we derive new operational matrix based on poly-Bernoulli polynomials for the Atangana–Baleanu derivative. A delay operational matrix based on poly-Bernoulli polynomials is derived. The error bound of this new method is shown. We applied this poly-Bernoulli operational matrix for solving fractional delay differential equations with variable coefficients. The numerical examples show that this method is easy to use and yet able to give accurate results.

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On the new properties of Caputo–Fabrizio operator and its application in deriving shifted Legendre operational matrix

Cited by 41 publications

References 14 publications

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

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

A Fractional Order Hiv/Aids Model Using Caputo-Fabrizio Operator

An Operational Matrix Method Based on Poly-Bernoulli Polynomials for Solving Fractional Delay Differential Equations

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