# On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo
fractional derivative to model chaotic problems11footnotemark: 1

Kolade M. Owolabi222E-mail addresses<EMAIL_ADDRESS>(K.M. Owolabi);
<EMAIL_ADDRESS>(A. Atangana) and Abdon Atangana33footnotemark: 3
222E-mail addresses<EMAIL_ADDRESS>(K.M. Owolabi);
<EMAIL_ADDRESS>(A. Atangana) 33footnotemark: 3 Institute for
Groundwater Studies, Faculty of Natural and Agricultural Sciences
University of the Free State, Bloemfontein 9300, South Africa
222E-mail addresses<EMAIL_ADDRESS>(K.M. Owolabi);
<EMAIL_ADDRESS>(A. Atangana) Department of Mathematical Sciences,
Federal University of Technology, PMB 704, Akure,
Ondo State, Nigeria

###### Abstract

Mathematical analysis with numerical application of the newly formulated
fractional version of the Adams-Bashforth method using the Atangana-Baleanu
derivative which has nonlocal and nonsingular properties is considered in this
paper. We adopt the fixed point theory and approximation method to prove the
existence and uniqueness of the solution via a general two-component time
fractional differential equations. The method is tested with three nonlinear
chaotic dynamical system in which the integer-order derivative is modelled
with the proposed fractional-order case. Simulation result for different
$\alpha$ values in $(0,1]$ is presented.

2010 Mathematics Subject Classification: 26A33, 35A05, 49M25, 65L05, 65M06

Keywords: Atangana-Baleanu-Caputo derivative; Chaotic systems; Fractional
Adams-Bashforth method; Numerical simulations.

## Leading paragraph

A range of chaotic and hyperchaotic processes were modeled with the Atangana-
Baleanu fractional derivative which has both nonlocal and nonsingular
properties in the sense of Caputo. The standard or local derivative in such
systems are replaced in time with the fractional derivative versions. A two-
step fractional Adams-Bashforth scheme is proposed to numerically approximate
the Atangana-Baleanu operator. We apply the fixed point theory and
approximation method to show the existence and uniqueness of solutions when
applied to a two-component dynamical system. Numerical results are given for
different instances of fractional power $\alpha$.

##  1 Introduction

Fractional calculus is regarded as applicable mathematics. The theories and
properties of these fractional operators have generated a lot of interest, and
become an active subjects of study in the last few decades. Recently,
scientists, engineers and applied mathematicians have found the fractional
calculus concept useful in various fields: quantitative biology, rheology,
electro-chemistry, diffusion process, scattering theory, transport theory,
elasticity, probability and potential theory [11, 13, 34, 35]. In-fact, it has
been regarded as the future of mathematical modelling of real-life phenomena
in groundwater studies, geo-hydrology and fractals [1, 2, 5, 6, 7, 8].

However, many engineers, technologists, mathematicians and other scientists
are still not familiar with this area of research possibly because they have
not been used to its applications. Thus, while the theory of fractional
calculus has been developed, its use is still poorly understood or lagged
behind. So, another achievement of this paper is to encourage researchers to
find and apply the new numerical scheme [9, 23, 24, 25, 26, 27, 28, 29, 30,
31] with the Atangana-Baleanu (AB) derivative in the Caputo sense, which
represents some chaotic phenomena with mathematical equations that can be
treated with the robustness of fractional calculus.

Chaotic processes are best described as nonlinear dynamical problems which are
more sensitive to initial data, and dense periodic orbits. In recent years,
many chaotic systems such as Zhou-system, Zhu-system, Li-system, Wei-Yang
system, Sundarapandian-system, Vaidyanathan-system and Pham-system to mention
a few. A list of old and new chaotic systems are is well classified in [11,
33, 35]. Chaos and control dynamics have a lot of application in almost every
areas of applied science and engineering [1, 6, 10]. Mostly known applications
are biology, chemical reactions, ecology, groundwater treatment, lasers,
oscillators and finance [1, 2, 3, 14, 15, 16, 19, 20, 21, 20], and references
therein.

Specifically speaking, we consider a time-fractional $\alpha-$th order
nonlinear system

$\begin{split}\mathcal{D}_{0,t}^{\alpha}u_{i}(t)&=f_{i}(u_{1}(t),u_{2}(t),\ldots,u_{n}(t),t)\\\
u_{i}(0)&=\Phi_{i},\;\;\;i=1,2,\ldots,n,\end{split}$ (1)

where $\alpha\in(0,1]$ determines the order of the differential system,
$\Phi_{i}$ is the initial functions, and nonlinear (coupled) functions
$f_{i}(u_{1},u_{2},\ldots,u_{n},t)$ for $i=1,2,\ldots,n$ which account for the
local kinetics of the system. The fractional derivative
$\mathcal{D}_{0,t}^{\alpha}u_{i}(t)$ is defined by the Atangana-Baleanu
derivative

${}^{ABC}_{0}\mathcal{D}_{t}^{\alpha}u_{i}(t)=\frac{1-\alpha}{AB(\alpha)}f_{i}(u_{1},u_{2},\ldots,u_{n},t)+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t}f_{i}(u_{1},u_{2},\ldots,u_{n},\tau)(t-\tau)^{\alpha-1}d\tau.$
(2)

The rest of this paper is broken into different sections. A quick tour of some
basic properties of fractional differentiation and integration is reviewed in
Section 2. By following the work reported in [9], we present the fractional
version of the proposed method in the sense of Caputo in Section 3, and show
that the non-integer derivative is well-defined and Lipschitz continuous.
Numerical experiment via three chaotic fractional differential equations is
considered in Section 4. Finally, we conclude with the last section.

##  2 Basic properties

Here, we quickly have a review of some necessary and useful properties of
fractional calculus, based on fractional integration and differentiation for a
continuous function $u\in C^{n}[a,b]$. Let $u:[a,b]\rightarrow\mathbb{R}$ be a
function, $\alpha>0$ a non-integer value and $n\in\mathbb{N}$ be such that
$\alpha\in(n-1,n)$. We assumed that all the necessary conditions for the
fractional operator to be well defined is satisfied by $u$.

The left-right Riemann-Liouville (RL) integrals of fractional-order
$\alpha\in(0,1]$ is respectively defined as

${}^{RL}\mathcal{I}_{a,t}^{\alpha}u(t)=\frac{1}{\Gamma(\alpha)}\int_{a}^{t}(t-\xi)^{\alpha-1}u(\xi)d\xi$

and

${}^{RL}\mathcal{I}_{t,b}^{\alpha}u(t)=\frac{1}{\Gamma(\alpha)}\int_{t}^{b}(\xi-t)^{\alpha-1}u(\xi)d\xi.$

The left-right RL derivatives of fractional-order $\alpha\in(0,]$ is
respectively defined as

${}^{RL}\mathcal{D}_{a,t}^{\alpha}u(t)=\frac{1}{\Gamma(n-\alpha)}\frac{d^{n}}{dt^{n}}\int_{a}^{t}(t-\xi)^{n-\alpha-1}u(\xi)d\xi$

and

${}^{RL}\mathcal{D}_{t,b}^{\alpha}u(t)=\frac{1}{\Gamma(n-\alpha)}\frac{d^{n}}{dt^{n}}\int_{t}^{b}(\xi-t)^{n-\alpha-1}u(\xi)d\xi.$

Likewise, the left-right Caputo fractional derivatives are defined by

${}^{C}\mathcal{D}_{a,t}u(t)=\frac{1}{\Gamma(n-\alpha)}\int_{a}^{t}(t-\xi)^{n-\alpha-1}u^{(n)}(\xi)d\xi$

and

${}^{C}\mathcal{D}_{t,b}u(t)=\frac{1}{\Gamma(n-\alpha)}\int_{a}^{t}(\xi-t)^{n-\alpha-1}u^{(n)}(\xi)d\xi$

respectively, where $n>0$ is an integer not less than $\alpha$. The
connections between the left-right RL and the Caputo operators of fractional
order $0<\alpha\leq 1$ is respectively given by

${}^{C}\mathcal{D}_{a,t}u(t)={{}^{RL}\mathcal{D}_{a,t}u(t)}-\sum_{s=0}^{n-1}\frac{u^{(s)(a)}}{\Gamma(s-\alpha+1)}(t-a)^{s-\alpha},$

and

${}^{C}\mathcal{D}_{t,b}u(t)={{}^{RL}\mathcal{D}_{t,b}u(t)}-\sum_{s=0}^{n-1}\frac{u^{(s)(b)}}{\Gamma(s-\alpha+1)}(b-t)^{s-\alpha}.$

Hence, if $u(a)=u^{\prime}(a)=\cdots=u^{(n-1)}(a)=0$, then
${}^{C}\mathcal{D}_{a,t}u(t)={{}^{RL}\mathcal{D}_{a,t}u(t)}$. Similarly, if
$u(b)=u^{\prime}(b)=\cdots=u^{(n-1)}(b)=0$, then
${}^{C}\mathcal{D}_{t,b}u(t)={{}^{RL}\mathcal{D}_{t,b}u(t)}$, and $u^{(s)}$
integrable on both $[a,t]$ and $[t,b]$, see [17, 34] for details.

In-line with the Caputo, a derivative with nonsingular and nonlocal properties
which was proposed by Atangana and Baleanu in 2016 [4] as

$^{ABC}\mathcal{D}_{a,t}^{\alpha}[u(t)]=\frac{M(\alpha)}{1-\alpha}\int_{a}^{t}u^{\prime}(\xi)E_{\alpha}\left[-\alpha\frac{(t-\xi)^{\alpha}}{1-\alpha}\right]d\xi$
(3)

where $M(\alpha)$ is the usual Caputo-Fabrizio normalized function [12, 13],
and $E_{\alpha}$ is a one-parameter Mittag-Leffler function expressed in terms
of power series

$u(z)=E_{\alpha}(z)=\sum_{s=0}^{\infty}\frac{z^{s}}{\Gamma(\alpha
s+1)},\;\;\;\alpha>0,\;\;\alpha\in\mathbb{R},\;\;z\in\mathbb{C}.$ (4)

##  3 Approximation of the Atangana-Baleanu fractional derivative

We examine the following differential equation

$_{0}^{ABC}\mathsf{D}_{t}^{\alpha}u(t)=f(t,u(t)).$ (5)

By applying the fundamental theorem, we have

$u(t)-u(0)=\frac{1-\alpha}{ABC(\alpha)}f(t,u(t))+\frac{\alpha}{ABC(\alpha)\Gamma(\alpha)}\int_{0}^{t}(t-\tau)^{\alpha-1}f(\tau,u(\tau))d\tau.$
(6)

At $t_{n+1}$, we have

$u(t_{n+1})-u(0)=\frac{1-\alpha}{ABC(\alpha)}f(t_{n},u_{n})+\frac{\alpha}{ABC(\alpha)\Gamma(\alpha)}\int_{0}^{t_{n+1}}(t_{n+1}-\tau)^{\alpha-1}f(t,u(t))dt$

and at $t_{n}$ we have

$u(t_{n})-u(0)=\frac{1-\alpha}{ABC(\alpha)}f(t_{n-1},u_{n-1})+\frac{\alpha}{ABC(\alpha)\Gamma(\alpha)}\int_{0}^{t_{n}}(t_{n}-\tau)^{\alpha-1}f(t,u(t))dt$

which on subtraction yields

$\displaystyle u(t_{n+1})-u(t_{n})$ $\displaystyle=$
$\displaystyle\frac{1-\alpha}{ABC(\alpha)}\left\\{f(t_{n},u_{n})-f(t_{n-1},u_{n-1})\right\\}+\frac{\alpha}{ABC(\alpha)\Gamma(\alpha)}\int_{0}^{t_{n+1}}(t_{n+1}-t)^{\alpha-1}$
(7) $\displaystyle\times
f(t,u(t))dt-\frac{\alpha}{ABC(\alpha)\Gamma(\alpha)}\int_{0}^{t_{n}}(t_{n}-t)^{\alpha-1}f(t,u(t))dt.$

Therefore,

$u(t_{n+1})-u(t_{n})=\frac{1-\alpha}{ABC(\alpha)}\left\\{f(t_{n},u_{n})-f(t_{n-1},u_{n-1})\right\\}+A_{\alpha,1}-A_{\alpha,2}.$

Next, we consider

$A_{\alpha,1}=\frac{\alpha}{ABC(\alpha)\Gamma(\alpha)}\int_{0}^{t_{n+1}}(t_{n+1}-t)^{\alpha-1}f(t,u(t))dt$

Again we use the approximation

$p(t)=\frac{t-t_{n-1}}{t_{n}-t_{n-1}}f(t_{n},u_{n})+\frac{t-t_{n-1}}{t_{n-1}-t_{n}}f(t_{n-1},u_{n-1})$
(8)

thus

$\displaystyle A_{\alpha,1}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{ABC(\alpha)\Gamma(\alpha)}\int_{0}^{t_{n+1}}(t_{n+1}-t)^{\alpha-1}\left\\{\frac{t-t_{n-1}}{h}f(t_{n},u_{n})-\frac{t-t_{n}}{h}f(t_{n},u_{n})\right\\}$
(9) $\displaystyle=$ $\displaystyle\frac{\alpha
f(t_{n},u_{n})}{ABC(\alpha)\Gamma(\alpha)h}\left\\{\int_{0}^{t_{n+1}}(t_{n+1}-t)^{\alpha-1}f(t-t_{n-1})\right\\}dt$
$\displaystyle-\frac{\alpha
f(t_{n-1},u_{n-1})}{ABC(\alpha)\Gamma(\alpha)h}\left\\{\int_{0}^{t_{n+1}}(t_{n+1}-t)^{\alpha-1}f(t-t_{n-1})\right\\}dt$
$\displaystyle=$ $\displaystyle\frac{\alpha
f(t_{n},u_{n})}{ABC(\alpha)\Gamma(\alpha)h}\left\\{\frac{2ht_{n+1}^{\alpha}}{\alpha}-\frac{t^{\alpha+1}_{n+1}}{\alpha+1}\right\\}-\frac{\alpha
f(t_{n-1},u_{n-1})}{ABC(\alpha)\Gamma(\alpha)h}\left\\{\frac{ht_{n+1}^{\alpha}}{\alpha}-\frac{t^{\alpha+1}_{n+1}}{\alpha+1}\right\\}.$

Similarly, we obtain

$A_{\alpha,2}=\frac{\alpha
f(t_{n},u_{n})}{ABC(\alpha)\Gamma(\alpha)h}\left\\{\frac{ht_{n}^{\alpha}}{\alpha}-\frac{t^{\alpha+1}_{n}}{\alpha+1}\right\\}-\frac{f(t_{n-1},u_{n-1})}{ABC(\alpha)\Gamma(\alpha)h}$
(10)

thus

$\displaystyle u(t_{n+1})-u(t_{n})$ $\displaystyle=$
$\displaystyle\frac{1-\alpha}{ABC(\alpha)}\left\\{f(t_{n},u_{n})-f(t_{n-1},u_{n-1})\right\\}+\frac{\alpha
f(t_{n},u_{n})}{ABC(\alpha)\Gamma(\alpha)h}\left\\{\frac{2ht_{n+1}^{\alpha}}{\alpha}-\frac{t^{\alpha+1}_{n+1}}{\alpha+1}\right\\}$
(11) $\displaystyle-\frac{\alpha
f(t_{n-1},u_{n-1})}{ABC(\alpha)\Gamma(\alpha)h}\left\\{\frac{ht_{n+1}^{\alpha}}{\alpha}-\frac{t^{\alpha+1}_{n+1}}{\alpha+1}\right\\}-\frac{\alpha
f(t_{n},u_{n})}{ABC(\alpha)\Gamma(\alpha)h}\left\\{\frac{ht_{n}^{\alpha}}{\alpha}-\frac{t^{\alpha+1}_{n}}{\alpha+1}\right\\}$
$\displaystyle+\frac{f(t_{n-1},u_{n-1})}{ABC(\alpha)\Gamma(\alpha)}t_{n}^{\alpha+1}.$

$\displaystyle u(_{n+1})$ $\displaystyle=$ $\displaystyle
u(t_{n})+f(t_{n},u_{n})\left\\{\frac{1-\alpha}{AB(\alpha)}+\frac{\alpha}{ABC(\alpha)h}\left[\frac{2ht_{n+1}^{\alpha}}{\alpha}-\frac{t^{\alpha+1}_{n+1}}{\alpha+1}\right]\right.$
$\displaystyle\left.-\frac{\alpha}{AB(\alpha)\Gamma(\alpha)h}\left[\frac{ht^{\alpha}_{n}}{\alpha}-\frac{t^{\alpha+1}_{n}}{\alpha+1}\right]\right\\}+f(t_{n-1},u_{n-1})$
$\displaystyle\times\left\\{\frac{\alpha-1}{AB(\alpha)}-\frac{\alpha}{h\Gamma(\alpha)AB(\alpha)}\left[\frac{ht_{n+1}^{\alpha}}{\alpha}-\frac{t_{n+1}^{\alpha+1}}{\alpha+1}+\frac{t^{\alpha+1}}{h\Gamma(\alpha)AB(\alpha)}\right]\right\\}.$

This equation is known as the two-step fractional Adams-Bashforth scheme for
Atangana-Baleanu derivative in Caputo sense.

Further, we simplify the last equation by putting $t_{n}=nh$ and
$t_{n+1}=(n+1)h$, and collect the resulting equation in powers of $h$ to yield

$\displaystyle u_{n+1}$ $\displaystyle=$ $\displaystyle
u_{n}+f(t_{n},u_{n})\left\\{\frac{1-\alpha}{AB(\alpha)}-\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}h^{\alpha}\left[\frac{2(n+1)^{\alpha}}{\alpha}-\frac{(n+1)^{\alpha+1}}{\alpha+1}\right]\right.$
$\displaystyle\left.-\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}h^{\alpha}\left[\frac{n^{\alpha}}{\alpha}-\frac{n^{\alpha+1}}{\alpha+1}\right]\right\\}+f(t_{n-1},u_{n-1})$
$\displaystyle\times\left\\{\frac{\alpha-1}{AB(\alpha)}-\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}h^{\alpha}\left[\frac{(n+1)^{\alpha}}{\alpha}-\frac{(n+1)^{\alpha+1}}{\alpha+1}+\frac{n^{\alpha+1}}{AB(\alpha)\Gamma(\alpha)h}\right]\right\\}.$

In the followings we briefly outline the convergence and stability results
previously obtained in [9].

###### Theorem 3.1

(Convergence result [9]) Let $u(t)$ be a solution of

${}^{ABC}_{0}\mathcal{D}_{t}^{\alpha}u(t)=f(t,u(t))$

with function $f$ continuous and bounded, the solution of u(t) is defined as

$\displaystyle u_{n+1}$ $\displaystyle=$ $\displaystyle
u_{n}+f(t_{n},u_{n})\left\\{\frac{1-\alpha}{AB(\alpha)}+\frac{\alpha}{AB(\alpha)h}\left[\frac{2ht_{n+1}^{\alpha}}{\alpha}-\frac{t^{\alpha+1}_{n+1}}{\alpha+1}\right]\right.$
$\displaystyle\left.-\frac{\alpha}{AB(\alpha)\Gamma(\alpha)h}\left[\frac{ht^{\alpha}_{n}}{\alpha}-\frac{t^{\alpha+1}_{n}}{\alpha+1}\right]\right\\}+f(t_{n-1},u_{n-1})$
$\displaystyle\times\left\\{\frac{\alpha-1}{AB(\alpha)}-\frac{\alpha}{h\Gamma(\alpha)AB(\alpha)}\left[\frac{ht_{n+1}^{\alpha}}{\alpha}-\frac{t_{n+1}^{\alpha+1}}{\alpha+1}+\frac{t^{\alpha+1}}{h\Gamma(\alpha)AB(\alpha)}\right]\right\\}+R_{\alpha}$

where $\|R_{\alpha}\|_{\infty}<M$

###### Theorem 3.2

(Condition for stability [9]) If function $f$ is Lipschitz continuous, then
the stability condition for fractional version of the Adams-Bashforth scheme
when applied to approximate the ABC derivative is obtained if

$\left\|f(t_{n},u_{n})-f(t_{n-1},u_{n-1})\right\|_{\infty}\rightarrow 0$

as $n\rightarrow\infty$.

##  4 Uniqueness and existence of solution via chaotic process

In this section, we choose a chaotic system to examine the uniqueness and
existence of the new method when applied to approximate the ABC fractional
derivative. To start with, we consider the general two component system

$\begin{split}{}_{0}^{ABC}\mathcal{D}_{t}^{\alpha}x(t)&=f(x,y,t),\\\
{}_{0}^{ABC}\mathcal{D}_{t}^{\alpha}y(t)&=g(x,y,t),\end{split}$ (14)

By adopting the fundamental calculus theorem along $x$and $y$ components, we
have

$\begin{split}x(t)-x(0)&=\frac{1-\alpha}{AB(\alpha)}f(x,y,t)+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t}(t-\tau)^{\alpha-1}f(x,y,\tau)d\tau\\\
y(t)-y(0)&=\frac{1-\alpha}{AB(\alpha)}g(x,y,t)+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t}(t-\tau)^{\alpha-1}g(x,y,\tau)d\tau\end{split}$
(15)

Next, we require to create a compact $G_{a,b}$, which means

$G_{a,b}=I_{a}(t_{0})\times\mathcal{B}_{b}(\xi)$ (16)

where

$\xi=\min\\{x_{0},x_{0}\\}$

and

$I_{a}(t_{0})=[t_{0}-a,t_{0}+a],\;\;\;\mathcal{B}_{0}(\xi)=[\xi-b,\xi+b].$

Let

$M=\max_{G_{a,b}}\left\\{\sup_{G_{a,b}}\|f\|,\;\sup_{G_{a,b}}\|g\|\right\\}.$

By adopting the infinite norm, we get

$\|\varPhi\|_{\infty}=\sup_{t\in I_{a}}||\varPhi(t)|.$

Also, we create a function, say

$\Gamma:G_{a,b}\rightarrow G_{a,b}$

so that

$\begin{split}\Gamma
x(t)&=x_{0}+\frac{1-\alpha}{AB(\alpha)}f(x,y,t)+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t}f(x,y,t)(t-\tau)^{\alpha-1}d\tau\\\
\Gamma
y(t)&=y_{0}+\frac{1-\alpha}{AB(\alpha)}g(x,y,t)+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t}g(x,y,t)(t-\tau)^{\alpha-1}d\tau\end{split}$
(17)

Next to prove that the fractional operator is well-defined, we evaluate the
condition for which

$\begin{split}\left\|\Gamma_{1}x(t)-x_{0}\right\|_{\infty}&<b,\\\
\left\|\Gamma_{2}y(t)-y_{0}\right\|_{\infty}&<b.\end{split}$ (18)

So, beginning with the $x$ component, we get

$\begin{split}\left\|\Gamma_{1}x(t)-x_{0}\right\|_{\infty}=&\left\|\frac{1-\alpha}{AB(\alpha)}f(x,y,t)+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t}f(x,y,t)(t-\tau)^{\alpha-1}d\tau\right\|_{\infty}\\\
\leq&\frac{1-\alpha}{AB(\alpha)}\|f(x,y,t)\|_{\infty}+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\|f(x,y,t)\int_{0}^{t}(t-\tau)d\tau\\\
\leq&\frac{(1-\alpha)M}{AB(\alpha)}+\frac{\alpha
M}{AB(\alpha)\Gamma(\alpha)}\cdot{a^{\alpha}}<b\end{split}$ (19)

This implies that

$a=\left(\frac{b-\frac{(1-\alpha)M}{AB(\alpha)}}{\frac{\alpha
M}{AB(\alpha)\Gamma(\alpha)}}\right)^{\frac{1}{\alpha}}.$

Next, we show that the functions $x(t)$ and $y(t)$ satisfy a Lipschitz
condition. Which implies

$\|\Gamma x_{1}-\Gamma x_{2}\|_{\infty}\leq K\|x_{1}-x_{2}\|_{\infty}$ (20)

which implies,

$\begin{split}\Gamma(x_{1})=&\frac{1-\alpha}{AB(\alpha)}f(x_{1},y,t)+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t}f(x_{1},y,\tau)(t-\tau)^{\alpha-1}d\tau,\\\
\Gamma(x_{2})=&\frac{1-\alpha}{AB(\alpha)}f(x_{2},y,t)+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t}f(x_{2},y,\tau)(t-\tau)^{\alpha-1}d\tau,\end{split}$
(21)

so that

$\displaystyle\begin{split}\|\Gamma x_{1}-\Gamma
x_{2}\|_{\infty}=&\frac{1-\alpha}{AB(\alpha)}\|f(x_{1},y,t)-f(x_{2},y,t)\|_{\infty}+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\|f(x_{1},y,t)\\\
&-f(x_{2},y,t)\|_{\infty}\int_{0}^{t}(t-\tau)d\tau\\\
\leq&\|f(x_{1},y,t)-f(x_{2},y,t)\|_{\infty}\left(\frac{1-\alpha}{AB(\alpha)}+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\cdot\frac{a^{\alpha}}{\alpha}\right)\\\
\leq&\|f(x_{1},y,t)-f(x_{2},y,t)\|_{\infty}\left(\frac{1-\alpha}{AB(\alpha)}+\frac{a^{\alpha}}{AB(\alpha)\Gamma(\alpha)}\right)\end{split}$
(22)

In other words, if $f$ is Lipschitz continuous with respect to $x$, then

$\begin{split}\|\Gamma x_{1}-\Gamma
x_{2}\|_{\infty}\leq&K\|x_{1}-x_{2}\|_{\infty}\left\\{\frac{1-\alpha}{AB(\alpha)}+\frac{a^{\alpha}}{AB(\alpha)\Gamma(\alpha)}\right\\}\\\
\leq&L\|x_{1}-x_{2}\|_{\infty}.\end{split}$ (23)

Similarly, $g$ is Lipschitz with respect to $y$ if

$\begin{split}\|\Gamma y_{1}-\Gamma
y_{2}\|_{\infty}\leq&L\|y_{1}-y_{2}\|_{\infty}.\end{split}$ (24)

The procedure described above can be generalized to multicomponent case of
fractional differential equations.

##  5 Numerical experiments

Under this segment, we apply the numerical scheme illustrated in the above
section to some chaotic systems. The classical time-derivative is replaced
with the Atangana-Baleanu fractional derivative of order $\alpha\in(0,1]$ in
the sense of Caputo.

From (3), we have

$u_{n+1}=u_{n}+\omega_{1}(n,\alpha,h)f(t_{n},u_{n})+\omega_{2}(n,\alpha,h)f(t_{n-1},u_{n-1})$
(25)

where

$\displaystyle\omega_{1}(n,\alpha,h)$ $\displaystyle=$
$\displaystyle\left\\{\frac{1-\alpha}{AB(\alpha)}-\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}h^{\alpha}\left[\frac{2(n+1)^{\alpha}}{\alpha}-\frac{(n+1)^{\alpha+1}}{\alpha+1}\right]\right.$
$\displaystyle\left.-\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}h^{\alpha}\left[\frac{n^{\alpha}}{\alpha}-\frac{n^{\alpha+1}}{\alpha+1}\right]\right\\}$

and

$\displaystyle\omega_{2}(n,\alpha,h)$ $\displaystyle=$ $\displaystyle
f(t_{n-1},u_{n-1})\left\\{\frac{\alpha-1}{AB(\alpha)}\right.$
$\displaystyle\left.-\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}h^{\alpha}\left[\frac{(n+1)^{\alpha}}{\alpha}-\frac{(n+1)^{\alpha+1}}{\alpha+1}+\frac{n^{\alpha+1}}{AB(\alpha)\Gamma(\alpha)h}\right]\right\\}.$

###  5.1 Example 1

The following chaotic system [11] is described by the AB fractional derivative
in the sense of Caputo as

$\begin{split}{}^{ABC}_{0}\mathcal{D}_{t}^{\alpha}x_{1}(t)=&f_{1}(x_{1},x_{2},x_{3})=\phi(x_{2}(t)-x_{1}(t))+\sigma
x_{2}(t)x_{3}(t),\\\
{}^{ABC}_{0}\mathcal{D}_{t}^{\alpha}x_{2}(t)=&f_{2}(x_{1},x_{2},x_{3})=\varphi
x_{1}(t)-x_{1}(t)x_{3}(t),\\\
{}^{ABC}_{0}\mathcal{D}_{t}^{\alpha}x_{3}(t)=&f_{3}(x_{1},x_{2},x_{3})=x_{1}(t)x_{2}(t)-\psi
x_{3}(t)+\delta x_{2}^{2}(t),\end{split}$ (26)

where $x_{1}(t),x_{2}(t),x_{3}(t)$ are the densities or states, and
$\phi,\varphi,\psi,\sigma,\delta$ are positive parameters. From (25), we have

$\displaystyle x_{1,n+1}$ $\displaystyle=$ $\displaystyle
x_{1,n}+\omega_{1}(n,\alpha,h)f_{1}(t_{n},x_{1,n},x_{2,n},x_{3,n})+\omega_{2}(n,\alpha,h)f_{1}(t_{n-1},x_{1,n-1},x_{2,n-1},x_{3,n-1})$
$\displaystyle x_{2,n+1}$ $\displaystyle=$ $\displaystyle
x_{2,n}+\omega_{1}(n,\alpha,h)f_{2}(t_{n},x_{1,n},x_{2,n},x_{3,n})+\omega_{2}(n,\alpha,h)f_{2}(t_{n-1},x_{1,n-1},x_{2,n-1},x_{3,n-1})$
$\displaystyle x_{3,n+1}$ $\displaystyle=$ $\displaystyle
x_{3,n}+\omega_{1}(n,\alpha,h)f_{3}(t_{n},x_{1,n},x_{2,n},x_{3,n})+\omega_{2}(n,\alpha,h)f_{3}(t_{n-1},x_{1,n-1},x_{2,n-1},x_{3,n-1})$

---
Figure 1: Numerical experiment results for chaotic system (26) obtained at
$\alpha=0.75$.
---
Figure 2: Numerical experiment results for chaotic system (26) obtained at
instance of $\alpha=0.95$.
---
Figure 3: Time-series result for system (26). The three species interact
chaotically in phase as a function of time at $\alpha=0.89$.

The novel three dimensional system (26) shows a highly chaotic and oscillatory
attractor for the choice of parameters values
$\phi=12,\varphi=16,\psi=5,\sigma=96,\delta=10$. The 3-D phase portrait and
2-D projections with the initial data $x_{1}(0)=0.2,x_{2}(0)=0.1,x_{3}(0)=0.2$
for two instances of fractional index $\alpha=0.75$ and $\alpha=0.95$ as
displayed in Figures 1 and 2, respectively. Numerical simulation is performed
with time-step $h=0.01$ and final computational time $t=100$. The chaotic and
spatiotemporal oscillation of $(x_{1},x_{2},x_{3})$ at different instances of
time $(t=1,2,4)$ is shown in Figure 3.

###  5.2 Example 2

For the second example, consideration is given to a six-term novel 3-D
dissipative chaotic system [11] which is illustrated by the fractional 3-D
dynamics

$\begin{split}{}^{ABC}_{0}\mathcal{D}_{t}^{\alpha}x_{1}(t)=&f_{1}(x_{1},x_{2},x_{3})=\phi(x_{2}(t)-x_{1}(t)),\\\
{}^{ABC}_{0}\mathcal{D}_{t}^{\alpha}x_{2}(t)=&f_{2}(x_{1},x_{2},x_{3})=x_{1}(t)-x_{1}(t)x_{3}(t),\\\
{}^{ABC}_{0}\mathcal{D}_{t}^{\alpha}x_{3}(t)=&f_{3}(x_{1},x_{2},x_{3})=50-\varphi
x_{1}^{2}(t)-\psi x_{3}(t),\end{split}$ (27)

where $x_{1}(t),x_{2}(t),x_{3}(t)$ are the densities and
$\phi>0,\varphi>0,\psi>0$ are parameters. The simulation results in Figures 4
and 5 are obtained with $\alpha=0.75$ and $\alpha=0.95$, respectively. We
utilize parameter values $\phi=2.6,\varphi=0.5,\psi=0.4$, subject to the
initial conditions $x_{1}(0)=0.6,x_{2}(0)=0.5$ and $x_{3}(0)=0.4$. The Initial
formation of chaotic and spatiotemporal patterns obtained in Figure 6
corresponds to $\alpha=0.68$ and time $t=5$. It should be noted that the
species oscillate chaotically in phase regardless of the variation in $\alpha$
and computational time $t$.

---
Figure 4: Numerical simulation results for chaotic system (27) obtained at
instance of $\alpha=0.75$. Plots depict the species strange attractor in
$\mathbb{R}^{3}$, and 2-D phase portraits.
---
Figure 5: Numerical simulation results for chaotic system (27) obtained at
instance of $\alpha=0.95$. Results represent the species strange attractor in
$\mathbb{R}^{3}$, and 2-D phase portraits.
---
Figure 6: Emergence of some strange chaotic attractors for system (27) at
$t=5$ and $\alpha=0.68$. Other parameters are as given above.
---
Figure 7: Numerical simulation of fractional system (28) showing hyperchaotic
oscillations in 3-D $(upper-row)$ and 2-D $(lower-row)$ projections in phase.
The lower-row corresponds to time series solutions at $t=(1,10)$ for
$\alpha=0.93$.

###  5.3 Example 3

We extend our numerical experiment to a four-component system, by considering
the novel 4-D hyper chaotic four-wing fractional equations

$\begin{split}{}^{ABC}_{0}\mathcal{D}_{t}^{\alpha}x_{1}(t)=&f_{1}(x_{1},x_{2},x_{3},x_{4})=\phi
x_{1}(t)-x_{2}(t)x_{3}(t)+x_{4}(t),\\\
{}^{ABC}_{0}\mathcal{D}_{t}^{\alpha}x_{2}(t)=&f_{2}(x_{1},x_{2},x_{3},x_{4})=-\varphi
x_{2}(t)+x_{1}(t)x_{3}(t)+x_{4}(t),\\\
{}^{ABC}_{0}\mathcal{D}_{t}^{\alpha}x_{3}(t)=&f_{3}(x_{1},x_{2},x_{3},x_{4})=-\psi
x_{3}(t)+x-1(t)x_{1}^{2}(t)+x_{1}(t),\\\
{}^{ABC}_{0}\mathcal{D}_{t}^{\alpha}x_{4}(t)=&f_{1}(x_{1},x_{2},x_{3},x_{4})=\sigma
x_{1}(t),\end{split}$ (28)

where $x_{1}(t),x_{2}(t),x_{3}(t)$ and $x_{4}(t)$ are the densities and
$\phi>0,\varphi>0,\psi>0,\sigma>0$ are mere constant parameters. By applying
(25), equation (27) can be written in the form

$\displaystyle x_{1,n+1}$ $\displaystyle=$ $\displaystyle
x_{1,n}+\omega_{1}(n,\alpha,h)f_{1}(t_{n},x_{1,n},x_{2,n},x_{3,n},x_{4,n})+\omega_{2}(n,\alpha,h)$
$\displaystyle\times f_{1}(t_{n-1},x_{1,n-1},x_{2,n-1},x_{3,n-1},x_{4,n-1})$
$\displaystyle x_{2,n+1}$ $\displaystyle=$ $\displaystyle
x_{2,n}+\omega_{1}(n,\alpha,h)f_{2}(t_{n},x_{1,n},x_{2,n},x_{3,n},x_{4,n})+\omega_{2}(n,\alpha,h)$
$\displaystyle\times f_{2}(t_{n-1},x_{1,n-1},x_{2,n-1},x_{3,n-1},x_{4,n-1})$
$\displaystyle x_{3,n+1}$ $\displaystyle=$ $\displaystyle
x_{3,n}+\omega_{1}(n,\alpha,h)f_{3}(t_{n},x_{1,n},x_{2,n},x_{3,n},x_{4,n})+\omega_{2}(n,\alpha,h)$
$\displaystyle\times f_{3}(t_{n-1},x_{1,n-1},x_{2,n-1},x_{3,n-1},x_{4,n-1})$
$\displaystyle x_{4,n+1}$ $\displaystyle=$ $\displaystyle
x_{4,n}+\omega_{1}(n,\alpha,h)f_{4}(t_{n},x_{1,n},x_{2,n},x_{3,n},x_{4,n})+\omega_{2}(n,\alpha,h)$
$\displaystyle\times f_{4}(t_{n-1},x_{1,n-1},x_{2,n-1},x_{3,n-1},,x_{4,n-1})$

The 4-D chaotic results given in Figure 7 is obtained with the parameter
values

$\phi=8,\;\;\varphi=33,\;\;\psi=16,\;\;\sigma=1.25.$

We use the initial data $x_{1}(0)=0.2,x_{2}(0)=0.4,x_{3}(0)=0.2$ and
$x_{4}(0)=0.7$ for the numerical simulation at $\alpha=0.93$ with final
computational time $t=150$ for the upper- and middle-rows. The 3-D
hyperchaotic projections is obtained on
$(x_{1}(t),x_{2}(t),x_{3}(t)),(x_{1}(t),x_{3}(t),x_{4}(t))$ and
$(x_{2}(t),x_{3}(t),x_{4}(t))$ respectively. The 3-D and 2-D projections have
the shape of four-wind attractors. The result displayed in the bottom-row
depicts time series solution, obtained at instances of time $t=(1,10)$.

##  6 Conclusion

Mathematical analysis and simulation of the newly proposed Adams-Bashforth
scheme with the Atangana-Baleanu in the sense of Caputo, when applied to study
different chaotic and hyperchaotic systems that are of current and recurrent
interests are considered in this paper. We considered a general two-component
fractional differential system to study the existence and uniqueness of
solutions. Numerical results which reveal some strange attractors at some
instances of fractional power $\alpha$ are given to address the points and
queries therein. In the future work, the existence and uniqueness of solution
reported for a general two-component differential equations will be extended
to multi-dimensional problems.

## Correction to: Chaos https://doi.org/10.1063/1.5085490.

## Introduction

Following the publication of an article in [32], we provide corrections to
mistakes therein as pointed out by some anonymous reviewers. Details of the
original article can be found online at https://doi.org/10.1063/1.5085490.

In abstract
Lines 2-3 $\rightarrow$ We adopt the Banach fixed point theorem…

Correction to equations (10) and (11).
Equations (10) and (11) in [32] are corrected as follows.

$\begin{split}A_{\alpha,1}&=\frac{\alpha}{\Gamma(\alpha)ABC(\alpha)}\int_{0}^{t_{n+1}}(t_{n+1}-t)^{\alpha-1}\left\\{\frac{t-t_{n-1}}{h}f(t_{n},u_{n})-\frac{t-t_{n}}{h}f(t_{n-1},u_{n-1})\right\\}d\tau\\\
&=\frac{f(t_{n},u_{n})}{\Gamma(\alpha)ABC(\alpha)h}\left\\{\frac{t^{\alpha+1}_{n+1}}{\alpha+1}-t_{n-1}t^{\alpha}_{n+1}\right\\}-\frac{f(t_{n-1},u_{n-1})}{h\Gamma(\alpha)ABC(\alpha)}\left\\{\frac{t^{\alpha+1}_{n+1}}{\alpha+1}-t_{n}t^{\alpha}_{n+1}\right\\}.\end{split}$
(29)

Also,

$\begin{split}A_{\alpha,2}&=\frac{f(t_{n},u_{n})}{h\Gamma(\alpha)ABC(\alpha)}\left\\{\frac{t^{\alpha+1}_{n}}{\alpha+1}-t_{n-1}t^{\alpha}_{n}\right\\}-\frac{f(t_{n-1},u_{n-1})}{h\Gamma(\alpha)ABC(\alpha)}\left\\{\frac{t^{\alpha+1}_{n}}{\alpha+1}-t_{n}^{\alpha+1}\right\\}.\end{split}$
(30)

Equation (12) is corrected as

$\begin{split}&u(t_{n+1})-u(t_{n})=\frac{1-\alpha}{AB(\alpha)}\left\\{f(t_{n},u_{n})-f(t_{n-1},u_{n-1})\right\\}+\frac{f(t_{n},u_{n})}{h\Gamma(\alpha)AB(\alpha)}\left\\{t^{\alpha+1}_{n+1}-t_{n}t^{\alpha}_{n+1}\right\\}\\\
&-\frac{f(t_{n-1},u_{n-1})}{h\Gamma(\alpha)AB(\alpha)}\left\\{\frac{t_{n+1}^{\alpha+1}}{\alpha+1}-t_{n}t_{n+1}^{\alpha}\right\\}-\frac{f(t_{n},u_{n})}{h\Gamma(\alpha)AB(\alpha)}\left\\{\frac{t_{n}^{\alpha+1}}{\alpha+1}-t_{n-1}t_{n}^{\alpha}\right\\}\\\
&+\frac{f(t_{n-1},u_{n-1})}{h\Gamma(\alpha)AB(\alpha)}\left\\{\frac{t^{\alpha+1}_{n}}{\alpha+1}-t_{n}^{\alpha+1}\right\\}\end{split}$

where $n\geq 1$ and the $u(0)=u_{0}$ is given by initial condition, where
$u_{1}$ can be obtained using the Runge-Kutta method.

Under equation (15) in [32], we assume that the function $f(x,y,t)$ and
$g(x,y,t)$ are uniformly Lipschitz continuous in $x$ and $y$ respectively,
also $g$ and $f$ are continuous in $t$.

To avoid confusion on $\Gamma$ as gamma function, and $\Gamma$ as defined
mapping, we suggest that $\Gamma$ should be replaces by $\Pi$

$\Pi:G_{a,b}\longrightarrow G_{a,b}$ $\Pi
x(t)=x_{0}+\frac{1-\alpha}{AB(\alpha)}f(x,y,t)+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t}f(x,y,\tau)(t-\tau)^{\alpha-1}d\tau,$

and

$\Pi
y(t)=y_{0}+\frac{1-\alpha}{AB(\alpha)}g(x,y,t)+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t}g(x,y,\tau)(t-\tau)^{\alpha-1}d\tau.$

In page 8, the misprint $\xi=\min\\{x_{0},x_{0}\\}$ should read
$\xi=\min\\{u_{0},v_{0}\\}$.

In equations (19) and (20), $\Gamma_{1},\Gamma_{2}$ should simply be $\Pi$,
$n\geq 1$.

Uniqueness

###### Theorem 6.1

Consider a general Cauchy problem

$\begin{split}{}^{ABC}_{0}D^{\alpha}_{t}y(t)&=f(t,y),\;\;\;\;t\in[0,c],[0,c]\subset\mathbb{R}\\\
y(0)&=0,\hskip 42.67912pty\in\subset([0,c])\end{split}$

where $f$ is continuous bounded function on $[0,c]$. The above problem has a
unique solution if the function $f(t,y(t))$ is Lipschitz with respect to $y$.

Proof
To achieve the proof, we define the following mapping

$\Pi
y=x_{0}+\frac{1-\alpha}{AB(\alpha)\Gamma(\alpha)}f(t,y(t))+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t}f(\tau,y(\tau))(t-\tau)^{\alpha-1}d\tau$
$\begin{split}&\overline{I_{c}}=[0,c],\\\
&\overline{B_{b}(y_{0})}=[y_{0}-b,y_{0}+b]\\\
&C_{c,b}=[0,c]\times\overline{B_{b}(y_{0})}.\end{split}$

We adopt the following norm

$\|y\|_{\infty}=\sup_{t\in[0,c]}|y(t)|,\;\;C_{c,b}\longmapsto C_{c,b}$

We shall first proof that $(C_{c,b},\|\cdot\|_{\infty})$ is Banach space,
$C_{c,b}$ is a space of bounded continuous functions. It is well known that
the space of bounded functions together with $\|\cdot\|_{\infty}$ is a Banach
space. However, for the sake of readers, we present the proof.

We have that $(I_{c},d)$ is a compact metric space, where
$d(v,\omega)=\|v-\omega\|$.

For simplicity, let $C_{c,b}=A$, we assume that $\\{f_{n}\\}$ is a Cauchy
sequence in $A$, thus $\forall\epsilon>0$ there exists $N>0$ such that
$\|f_{m}-f_{n}\|_{\infty}<\epsilon$, $\forall t\in[0,c]$

$|f_{n}(t)-f_{m}(t)|\leq\|f_{n}-f_{m}\|_{\infty}<\epsilon$

therefore $\forall t\in[0,c]$ the real numbers $\left\\{f_{n}(t)\right\\}$ is
a Cauchy sequence. Since $\overline{B_{b}(y_{0})}$ is complete,
$\\{f_{n}(t)\\}$ is convergent.

Let

$f(t)=\lim\limits_{n\rightarrow\infty}f_{n}(t).$

Here we have $f(t)\in\overline{B_{b}(y_{0})}$. On the other hand
$\|f_{m}\|_{\infty}<M$, since for every $t\in[0,c]$

$|f_{n}(t)|\leq\|f_{n}\|_{\infty}\leq M.$

Now

$|f(t)|\leq\lim\limits_{n\rightarrow\infty}|f_{n}(t)|<M,\;\;\forall t\in[0,c]$

this shows that $f(t)$ is bounded on $[0,c]$ thus $f\in A$.

$\forall t\in[0,c]$ and $\forall n\geq N$

$|f_{n}(t)-f(t)|=\lim\limits_{m\rightarrow 0}|f_{n}(t)-f_{m}(t)|<\epsilon$

this shows that $\forall n\geq N$, $\|f_{n}-f\|_{\infty}<\epsilon$, we have
shown that $\\{f_{n}\\}$ is convergent in $A$, this proves that $C_{c,b}=A$
with $\|\cdot\|_{\infty}$ is a Banach space.

The above proof can be found in some textbooks. We now have to show that the
defined mapping is a contraction under some conditions

$\begin{array}[]{l}\left|\Pi
y-y_{0}\right|=\left|\frac{1-\alpha}{AB(\alpha)}f(t,y(t))+\frac{\alpha}{AB(\alpha)\Gamma\left(\alpha\right)}\int\limits_{0}^{t}f(\tau,y(\tau))\left(t-\tau\right)^{\alpha-1}d\tau\right|\\\
\hskip
42.67912pt\leq\frac{1-\alpha}{AB(\alpha)}\left|f(t,y(t))\right|+\frac{\alpha}{AB(\alpha)\Gamma\left(\alpha\right)}\int\limits_{0}^{t}\left|f(\tau,y(\tau))\right|\left(t-\tau\right)^{\alpha-1}d\tau\end{array}$

$\begin{array}[]{l}\left\|\Pi
y-y_{0}\right\|_{\infty}<\frac{1-\alpha}{AB(\alpha)}\sup\limits_{t\in\left[0,t\right]}\left|f(t,y(t))\right|+\frac{\alpha}{AB(\alpha)\Gamma\left(\alpha\right)}\int\limits_{0}^{t}\sup\limits_{t\in\left[0,t\right]}\left|f(\tau,y(\tau))\right|\left(t-\tau\right)^{\alpha-1}d\tau\\\
\hskip 42.67912pt<\frac{1-\alpha}{AB(\alpha)}M+\frac{\alpha
M}{AB(\alpha)\Gamma\left(\alpha\right)}\int\limits_{0}^{t}\left(t-\tau\right)^{\alpha-1}d\tau\\\
\hskip 42.67912pt<\frac{1-\alpha}{AB(\alpha)}M+\frac{\alpha
M}{AB(\alpha)\Gamma\left(\alpha+1\right)}t^{\alpha}<\frac{M}{AB(\alpha)}\left\\{1-\alpha+\frac{c^{\alpha}}{\Gamma\left(\alpha\right)}\right\\}\end{array}$

$M>\sup\limits_{t\in\left[0,t\right]}\left|f(t,y(t))\right|$ Thus,

$\left\|\Pi
y-x_{0}\right\|_{\infty}<\frac{M}{AB(\alpha)}\left\\{1-\alpha+\frac{c^{\alpha}}{\Gamma\left(\alpha\right)}\right\\}.$

Here we need

$\left\|\Pi y-x_{0}\right\|_{\infty}<b$

Thus

$c<\left\\{\left(\frac{AB(\alpha)b}{M}+\alpha-1\right)\Gamma\left(\alpha\right)\right\\}^{\frac{1}{\alpha}}.$

Next we shall show $\Pi$ is a contraction; $y_{1},y_{2}$ $\in C_{c,b}$

$\begin{array}[]{l}\left|\Pi y_{1}-\Pi
y_{2}\right|=\left|\frac{1-\alpha}{AB(\alpha)}\left(f(t,y_{1})-f(t,y_{2})\right)+\frac{\alpha}{AB(\alpha)\Gamma\left(\alpha\right)}\int\limits_{0}^{t}\left(f(\tau,y_{1})-f(\tau,y_{2})\right)\left(t-\tau\right)^{\alpha-1}d\tau\right|\\\
\hskip
42.67912pt\leq\frac{1-\alpha}{AB(\alpha)}\left|f(t,y_{1})-f(t,y_{2})\right|+\frac{\alpha}{AB(\alpha)\Gamma\left(\alpha\right)}\int\limits_{0}^{t}\left|f(\tau,y_{1})-f(\tau,y_{2})\right|\left(t-\tau\right)^{\alpha-1}d\tau\end{array}$

$\left\|\Pi y_{1}-\Pi
y_{2}\right\|_{\infty}\leq\frac{1-\alpha}{AB(\alpha)}L\left\|y_{1}-y_{2}\right\|_{\infty}+\frac{\alpha}{AB(\alpha)\Gamma\left(\alpha\right)}Lt^{\alpha}\left\|y_{1}-y_{2}\right\|_{\infty}$

since $f$ is a Lipschitz with respect to $y$

$\left\|\Pi y_{1}-\Pi
y_{2}\right\|_{\infty}\leq\frac{L}{AB(\alpha)}\left(1-\alpha+\frac{c^{\alpha}}{\Gamma\left(\alpha\right)}\right)\left\|y_{1}-y_{2}\right\|_{\infty}$

To have a contraction, we need

$\frac{L}{AB(\alpha)}\left(1-\alpha+\frac{c^{\alpha}}{\Gamma\left(\alpha\right)}\right)<1$

that implies

$c<\left\\{\left(\frac{AB(\alpha)}{L}+\alpha-1\right)\Gamma\left(\alpha\right)\right\\}^{\frac{1}{\alpha}}.$

Therefore the equation has unique solution if

$c<\min\left\\{\left\\{\left(\frac{AB(\alpha)}{L}+\alpha-1\right)\Gamma\left(\alpha\right)\right\\}^{\frac{1}{\alpha}},\left\\{\left(\frac{AB(\alpha)b}{M}+\alpha-1\right)\Gamma\left(\alpha\right)\right\\}^{\frac{1}{\alpha}}\right\\}$

where $AB(\alpha)=1-\alpha+\frac{\alpha}{\Gamma\left(\alpha\right)}$ if
$\alpha=1,AB(\alpha)=1.$

In Theorem 1,
We present a detailed proof of error analysis for approximating the nonlinear
function $f(t,x(t),y(t))$ using the Lagrange polynomial interpolation.
Finally, we shall consider the case when the fractional order is 1 to show
that the function $\Phi(n,\alpha)$ is indeed bounded regardless of the natural
number $n\geq 1$ chosen.

The error $R^{\alpha}_{n}(\zeta)$ is calculated as first integral minus the
second integral, that is,

$\begin{split}R^{\alpha}_{n}(\zeta)&=\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t_{n+1}}(\tau-
t_{n})(\tau-
t_{n-1})(t_{n+1}-\tau)^{\alpha-1}\left.\frac{\partial^{f}(\tau,y(\tau))}{\partial\tau^{2}}\right|_{\tau=\zeta}d\tau\\\
&\;\;\;-\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t_{n}}(\tau-
t_{n})(\tau-
t_{n-1})(t_{n+1}-\tau)^{\alpha-1}\left.\frac{\partial^{f}(\tau,y(\tau))}{\partial\tau^{2}}\right|_{\tau=\zeta}d\tau\\\
\end{split}$

In what follows we evaluate the first part of the integral, and also the
second part. Bear in mind that $(\tau-t_{n})(\tau-
t_{n-1})=\tau^{2}-(t_{n}+t_{n-1})\tau+t_{n}t_{n-1}$, we now evaluate the error
as follows:

$\begin{split}&\frac{\alpha}{2\Gamma(\alpha)AB(\alpha)}\int_{0}^{n+1}\left[\tau^{2}-(t_{n}+t_{n-1})\tau+t_{n}t_{n-1}\right](t_{n+1}-\tau)^{\alpha-1}\frac{\partial^{2}}{\partial\tau^{2}}f(\tau,y(\tau))|_{\tau=\zeta}d\tau\\\
&\leq\frac{\alpha}{2\Gamma(\alpha)AB(\alpha)}\int_{0}^{t_{n+1}}\left[\tau^{2}-(t_{n}+t_{n-1})\tau+t_{n}t_{n-1}\right](t_{n+1}-\tau)^{\alpha-1}\sup_{\ell\in[0,\tau]}\left|\frac{\partial^{2}}{\partial\ell^{2}}f(\ell,y(\ell))\right|_{\tau=\zeta}d\tau\\\
&\leq\frac{\alpha}{2\Gamma(\alpha)AB(\alpha)}\sup_{t\in[0,t_{n+1}]}\left|\frac{\partial^{2}}{\partial
t^{2}}f(t,y(t))\right|_{t=\zeta}\int_{0}^{t_{n+1}}\left[\tau^{2}-(t_{n}+t_{n-1})\tau+t_{n}t_{n-1}\right](t_{n+1}-\tau)^{\alpha-1}d\tau\\\
&\leq\frac{\sup_{t\in[0,t_{n+1}]}\left|\frac{\partial^{2}}{\partial
t^{2}}f(t,y(t))\right|_{t=\zeta}}{2AB(\alpha)(1+\alpha)(2+\alpha)\Gamma(\alpha)}\left\\{t_{n+1}^{\alpha}[(1+\alpha)(2+\alpha)t_{n}t_{n-1}-(2+\alpha)(t_{n}+t_{n-1})t_{n+1}+2t^{2}_{n+1}]\right\\}\\\
&\leq\frac{\mathcal{M}}{2AB(\alpha)(1+\alpha)(2+\alpha)\Gamma(\alpha)}t_{n+1}^{\alpha}\left\\{(1+\alpha)(2+\alpha)t_{n}t_{n-1}-(2+\alpha)(t_{n}+t_{n-1})t_{n+1}+2t^{2}_{n+1}]\right\\}\end{split}$

We are evaluating the second part of the error error as follows

$\begin{split}&\frac{\alpha}{2\Gamma(\alpha)AB(\alpha)}\int_{0}^{t_{n}}[\tau^{2}-(t_{n}+t_{n-1})\tau+t_{n}t_{n-1}]\frac{\partial}{\partial\tau^{2}}f(\tau,y(\tau))|_{\tau=\zeta}(t_{n}-\tau)^{\alpha-1}d\tau\\\
&\frac{\sup_{t\in[0,t_{n}]}\left|\frac{\partial}{\partial
t^{2}}f(t,y(t))\right|_{t=\zeta}}{2\Gamma(\alpha)AB(\alpha)}\int_{0}^{t_{n}}[\tau^{2}-(t_{n}+t_{n-1})\tau+t_{n}t_{n-1}](t_{n}-\tau)^{\alpha-1}d\tau\\\
&\leq\frac{\mathcal{M}\alpha
t_{n}^{\alpha+1}}{2AB(\alpha)(1+\alpha)(2+\alpha)\Gamma(\alpha)}\\{(2+\alpha)t_{n-1}-t_{n}\\}\end{split}$

Therefore,

$\begin{split}&|R^{\alpha}_{n}(\zeta)|\leq\frac{\mathcal{M}}{2(1+\alpha)(2+\alpha)\Gamma(\alpha)AB(\alpha)}\left|\left\\{2\alpha
t^{2+\alpha}_{n}+t_{n+1}^{1+\alpha}[2t_{n+1}-(2+\alpha)t_{n}]+(2+\alpha)t_{n-1}\right.\right.\\\
&\hskip 56.9055pt\left.\left.\left[-2\alpha
t^{\alpha+1}_{n}+((1+\alpha)t_{n}-t_{n+1})t^{\alpha}_{n+1}\right]\right\\}\right|\end{split}$

$\begin{split}&|R^{\alpha}_{n}(\zeta)|\leq\frac{\mathcal{M}\Delta
t^{\alpha+2}}{2(1+\alpha)(2+\alpha)\Gamma(\alpha)AB(\alpha)}\left|\left\\{2\alpha
n^{2+\alpha}+(n+1)^{\alpha+1}(2(n+1)\right.\right.\\\ &\hskip
42.67912pt\left.\left.-(2+\alpha)n)+(2+\alpha)(n-1)[-2\alpha
n^{\alpha+1}((1+\alpha)n-(n+1))(n+1)^{\alpha}]\right\\}\right|\end{split}$

$\begin{split}&|R^{\alpha}_{n}(\zeta)|\leq\frac{\mathcal{M}\Delta
t^{\alpha+2}_{\alpha}}{2\Gamma(\alpha+3)}\left|\left\\{2\alpha
n^{2+\alpha}+(n+1)^{\alpha+2}(2-\alpha n)\right.\right.\\\ &\hskip
42.67912pt\left.\left.+(2+\alpha)(n-1)[-2\alpha n^{\alpha+1}+(\alpha
n-1)(n+1)^{\alpha}]\right\\}\right|\end{split}$

$|R^{\alpha}_{n}(\zeta)|<\frac{\mathcal{M}\Delta
t^{\alpha}}{2\Gamma(\alpha+3)}\Phi(n,\alpha)$

where

$\begin{split}&|\Phi(n,\alpha)|\\\ &\;\;\;=\left|\alpha\left\\{2\alpha
n^{2+\alpha}+(n+1)^{\alpha+2}(2-\alpha n)+(2+\alpha)(n-1)[-2\alpha
n^{\alpha+1}+(\alpha n-1)(n+1)^{\alpha}]\right\\}\right|.\end{split}$

Indeed the error depends on $\alpha$, therefore for a given $\alpha$ we get an
associate error, which is different for the case of classical integral as the
order is fixed to $\alpha=1$, the error depends only on $\Delta h$. It should
be noted that a fractional integral has memory which is depicted by
$(t_{n+1}-\tau)^{\alpha-1}$. In what follows, we shall give some examples

## Example 1

We consider the following Cauchy problem with ABC derivative

$\begin{split}{}^{ABC}_{0}D^{\alpha}_{t}y(t)&=t^{\beta},\;\;\;\;[0,10],\\\
y(0)&=0\end{split}$ (31)

and

$y(t)=\frac{1-\alpha}{AB(\alpha)}t^{\beta}+\frac{\alpha
t^{\alpha+\beta}\Gamma(\beta+1)}{AB(\alpha)\Gamma(\alpha+\beta+1)}.$

Although the above problem has the exact solution, we will verify the
convolution

$\Pi
y(t)=\frac{(1-\alpha)}{AB(\alpha)}f(t,y(t))+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t}f(\tau,y(\tau))(t-\tau)^{\alpha-1}d\tau$
$\begin{split}f(t,y(t))&=t^{\beta}\\\ |f(t,y(t))|&=|t^{\beta}|\leq
10^{\beta}\end{split}$

$\begin{split}|\Pi
y(t)-y(0)|&<\frac{(1-\alpha)10^{\beta}}{AB(\alpha)}+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t}t^{\beta}(t-\tau)^{\alpha-1}d\tau\\\
&<\frac{(1-\alpha)10^{\beta}}{AB(\alpha)}+\frac{\alpha
10^{\beta}}{AB(\alpha)\Gamma(\alpha)}\frac{t^{\alpha}\Gamma(\alpha)}{\Gamma(\alpha+1)}\\\
&<\frac{(1-\alpha)10^{\beta}}{AB(\alpha)}+\frac{\alpha
10^{\beta+\alpha}}{AB(\alpha)\Gamma(\alpha+1)}<10\end{split}$

Since $0<\alpha\leq 1$, then

$|\Pi y_{1}-\Pi y_{2}|=0<\frac{1}{1000}|y_{1}-y_{2}|$

$k=\frac{1}{1000}<1$ which is a contraction.
We can conclude that the solution exists and unique.

The numerical results showing comparison between the exact and approximate
solutions for different instances of fractional index $\alpha$

and $\beta$ for $t=2$ is displayed in Figure 8.

---
Figure 8: Numerical solutions for test problem 1 for different values of
$\alpha,\beta$ and $h$ with $t=2$.

## Example 2

We consider the following

$\begin{split}{}^{ABC}_{0}D^{\alpha}_{t}y(t)&=\frac{t}{1000}y(t),\;\;\;\;[0,10],\\\
y(0)&=0\end{split}$ (32) $\Pi
y(t)=\frac{1-\alpha}{AB(\alpha)}\frac{ty(t)}{1000}+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t}\tau
y(\tau)(t-\tau)^{\alpha-1}d\tau$ $|\Pi y_{1}-\Pi
y_{2}|=\left|\frac{t}{1000}(y_{1}-y_{2})(1-\alpha)+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t}\frac{\tau(t-\tau)^{\alpha-1}}{1000}(y_{1}-y_{2})d\tau\right|$

Nothing that $f(t,y(t))=\frac{ty(t)}{1000}$

$\begin{split}|f(t,y_{1})-f(t,y_{2})|&=\frac{t}{1000}|y_{1}-y_{2}|\\\
&\leq\frac{t}{100}|y_{1}-y_{2}|\end{split}$

where $f$ is Lipschitz with $k=1/100$

$\begin{split}\|\Pi y_{1}-\Pi
y_{2}\|_{\infty}&\leq\frac{t}{1000}\|y_{1}-y_{2}\|_{\infty}\frac{(1-\alpha)}{AB(\alpha)}+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\|y_{1}-y_{2}\|_{\infty}\frac{1}{100}\int_{0}^{t}(t-\tau)^{\alpha-1}d\tau\\\
|\Pi y_{1}-\Pi
y_{2}\|_{\infty}&\leq\frac{(1-\alpha)}{100AB(\alpha)}\|y_{1}-y_{2}\|_{\infty}+\frac{\alpha}{100AB(\alpha)}\|y_{1}-y_{2}\|_{\infty}\frac{10^{\alpha}}{\Gamma(\alpha+1)}\\\
&<\frac{1}{100}\left\\{\frac{1-\alpha}{AB(\alpha)}+\frac{\alpha
10^{\alpha}}{AB(\alpha)\Gamma(\alpha+1)}\right\\}\|y_{1}-y_{2}\|_{\infty}\\\
&\leq k\|y_{1}-y_{2}\|_{\infty}\end{split}$

$k<1$, which is a contraction.

$\begin{split}\Pi
y-y(0)&=\frac{1-\alpha}{AB(\alpha)}\frac{ty}{1000}+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t}\tau
y(\tau)\frac{(t-\tau)^{\alpha-1}}{1000}d\tau\\\ \|\Pi
y-y(0)\|_{\infty}&\leq\frac{(1-\alpha)}{AB(\alpha)100}\|y\|_{\infty}+\frac{\alpha\|y\|_{\infty}10^{\alpha}}{AB(\alpha)1000\Gamma(\alpha+1)}\\\
&\leq\left(\frac{1-\alpha}{AB(\alpha)}+\frac{10^{\alpha-1}}{\Gamma(\alpha+1)AB(\alpha)}\right)\frac{\|y\|_{\infty}}{100}\end{split}$
(33)

Hence, we can conclude that the solution exists and unique.

###  Error for classical- vs fractional-derivatives in Caputo sense

In this section, we present a derivation of the error analysis for the case of
$\alpha=1$. This will help the reader to see the difference between the Cauchy
problem with fractional derivative and that with classical derivative. It is
worth noting the error for the Caputo case is the same as the Atangana-Baleanu
derivative case since the fractional integral associated with the Atangana-
Baleanu operator is the same with an ${\alpha}/{AB(\alpha)}$ coefficient.

Assuming that a function $f(t,y(t))$ is two time differentiable, such that
$\frac{\partial^{2}f(t,y(t))}{\partial t^{2}}$ is bounded.

$\begin{split}\frac{dy(t)}{dt}=f(t,y(t)),\;\;[0,T]\\\ y(0)=y_{0}\end{split}$
(34)

From (34) we apply the integral to get

$y(t)-y(0)=\int_{0}^{t}f(\tau,y(\tau))d\tau,$

$t_{n+1}=(n+1)\Delta t$ and $t_{n}=n\Delta t$ So that

$y(t_{n+1})-y(0)=\int_{0}^{t}f(\tau,y(\tau))d\tau$ (35)

and

$y(t_{n})-y(0)=\int_{0}^{t_{n}}f(\tau,y(\tau))d\tau$ (36)

Taking (35)-(36) yields

$y(t_{n+1})-y(t_{n})=\int_{t_{n}}^{t_{n+1}}f(\tau,y(\tau))d\tau$ (37)

where $f(\tau,y(\tau))$is approximate within $[t_{n},t_{n+1}]$ with Lagrange
polynomial

$\begin{split}f(\tau,y(\tau))=P_{n}(\tau)&=\frac{\tau-t_{n-1}}{\Delta
t}f(t_{n},y(t_{n}))-\frac{\tau-t_{n}}{\Delta t}f(t_{n-1},y(t_{n-1}))\\\
&\;\;\;\;+\left.\frac{\partial^{2}f(\tau,y(\tau))}{\partial\tau^{2}}\right|_{\tau=\zeta}\frac{(\tau-
t_{n-1})}{2!}.\end{split}$ (38)

The error is given as

$R_{n}(\zeta)=\int_{t_{n}}^{t_{n+1}}\left.\frac{\partial^{2}f(\tau,y(\tau))}{\partial\tau^{2}}\right|_{\tau=\zeta}\frac{(\tau-
t_{n})(\tau-t_{n-1})}{2!}d\tau$

$\begin{split}|R_{n}(\zeta)|&=\left|\int_{t_{n}}^{t_{n+1}}\left.\frac{\partial^{2}f(\tau,y(\tau))}{\partial\tau^{2}}\right|_{\tau=\zeta}\frac{(\tau-
t_{n})(\tau-t_{n-1})}{2!}d\tau\right|\\\
&\leq\int_{t_{n}}^{t_{n+1}}\left|\left.\frac{\partial^{2}f(\tau,y(\tau))}{\partial\tau^{2}}\right|_{\tau=\zeta}\right|\frac{(\tau-
t_{n})(\tau-t_{n-1})}{2}d\tau\\\
&<\int_{t_{n}}^{t_{n-1}}\sup_{\ell\in[t_{n},\tau]}\left|\frac{\partial^{2}f(\tau,y(\tau))}{\partial\tau^{2}}\right|_{\tau=\zeta}\frac{(\tau-
t_{n})(\tau-t_{n-1})}{2}d\tau\end{split}$

Since we assume that $\frac{\partial^{2}}{\partial t^{2}}$ is bounded

$\begin{split}|R_{n}(\zeta)|&<\sup_{t\in[t_{n},t_{n+1}]}\left|\frac{\partial^{2}f(t,y(t))}{\partial\tau^{2}}\right|_{t=\zeta}\int_{t_{n}}^{t_{n-1}}\frac{(\tau-
t_{n})(\tau-t_{n-1})}{2!}d\tau\\\
&<\frac{M}{2}\left[\frac{\tau^{3}}{3}-(t_{n-1}+t_{n})\frac{\tau^{2}}{2}+t_{n-1}t_{n}\tau\right]_{t_{n}}^{t{n-1}}\\\
&<\frac{M}{2}\left\\{\frac{t_{n+1}^{3}}{3}-\frac{t_{n}^{3}}{3}-(t_{n-1}+t_{n})\left[\frac{t_{n+1}^{2}}{2}-\frac{t^{2}_{n}}{2}\right]+t_{n-1}t_{n}[t_{n+1}-t_{n}]\right\\}\\\
&<\frac{M}{2}\Delta
t^{3}\left\\{\frac{(n+1)^{3}-n^{3}}{3}-(2n-1)\left(\frac{(n+1)^{2}}{2}-\frac{n^{2}}{2}\right)+(n-1)n\right\\}\\\
&<\frac{M}{2}\Delta
t^{3}\left\\{\frac{3n^{2}+3n+1}{3}-\frac{(4n^{2}-1)}{2}+(n-1)n\right\\}\\\
&<\frac{M}{2}\Delta t^{3}\left\\{\frac{1}{3}+\frac{1}{2}\right\\}\\\
&<\frac{5}{12}M\Delta t^{3}.\end{split}$

The above error depends on $\Delta t$ only because of integer order. So the
numerical scheme for this is

$y_{n+1}=y_{n}+\frac{3}{2}\Delta tf(t_{n},y_{n})-\frac{\Delta
t}{2}f(t_{n-1},y_{n-1}).$

This method is well-known to be accurate and is called Adams-Bashforth scheme.
What is the difference with the fractional case?

$\begin{split}y(t_{n+1})-y(t_{n})&=\frac{1}{\Gamma(\alpha)}\int_{0}^{t_{n+1}}(t_{n+1}-\tau)^{\alpha-1}f(\tau,y(\tau))d\tau\\\
&\;\;-\frac{1}{\Gamma(\alpha)}\int_{0}^{t_{n}}(t_{n}-\tau)^{\alpha-1}f(\tau,y(\tau))d\tau.\end{split}$

We have new component which is the power law contribution
$(t_{n+1}-\tau)^{\alpha-1}$ and $(t_{n}-\tau)^{\alpha-1}$, by approximating
$f(t,y(t))$

$\begin{split}y(t_{n+1})&=y(t_{n})+\left[\int_{0}^{t_{n+1}}(t_{n+1}-\tau)^{\alpha-1}\left\\{\frac{(\tau-
t_{n-1})}{\Delta t}f(t_{n},y(t_{n}))\right.\right.\\\
&\;\;\left.\left.-\frac{\tau-t_{n}}{\Delta
t}f(t_{n-1},y(t_{n-1}))\right\\}d\tau-\int_{0}^{t_{n}}(\tau-t_{n})(\tau-
t_{n-1})(t_{n}-\tau)^{\alpha-1}d\tau\right]\frac{1}{\Gamma(\alpha)}\\\
&\;\;+\left\\{\int_{0}^{t_{n+1}}(t_{n+1}-\tau)^{\alpha-1}\frac{\partial^{2}f(\tau,y(\tau))}{\partial\tau^{2}}|_{\tau=\zeta}(\tau-
t_{n})(\tau-t_{n-1})d\tau\right.\\\
&\;\;\left.-\int_{0}^{t_{n}}(t_{n}-\tau)^{\alpha-1}\frac{\partial^{2}f(\tau,y(\tau))}{\partial\tau^{2}}|_{\tau=\zeta}(\tau-
t_{n})\frac{\tau-t_{n-1}}{2!}d\tau\right\\}\end{split}$

These $(t_{n+1}-\tau)^{\alpha-1}$ and $(t_{n}-\tau)^{\alpha-1}$ are nonlinear
if $\alpha=1$, indeed

$y(t_{n+1})-y(t_{n})=\int_{t_{n}}^{t_{n+1}}(\tau-t_{n})(\tau-
t_{n-1})\frac{\partial^{2}f(\tau,y(\tau))}{\partial\tau^{2}}|_{t=\tau}d\tau+\int_{t_{n}}^{t_{n+1}}f(\tau,y(\tau))d\tau$

Then, we have

$|R_{n}(\zeta)|<\frac{5M\Delta t}{12}.$

But if $\alpha\neq 1$, $0<\alpha<1$ then

$\begin{split}R_{n}(\zeta)&=\frac{1}{\Gamma(\alpha)}\left\\{\int_{0}^{t_{n+1}}(t_{n+1}-\tau)^{\alpha-1}\frac{(\tau-
t_{n})(\tau-
t_{n-1})}{2!}\frac{\partial^{2}f}{\partial\tau^{2}}(\tau,y(\tau))|_{\tau=\zeta}d\tau\right.\\\
&\;\;\left.-\int_{0}^{t_{n}}(t_{n+1}-\tau)^{\alpha-1}\frac{(\tau-t_{n})(\tau-
t_{n-1})}{2!}\frac{\partial^{2}f}{\partial\tau^{2}}(\tau,y(\tau))|_{\tau=\zeta}d\tau\right\\}\\\
|R_{n}(\zeta)|&<\frac{Mh^{\alpha+2}}{2!}\Phi(n,\alpha)\end{split}$

Indeed, if $\alpha=1$, then $\Phi(n,\alpha)=\frac{5}{12}$.

In Figure 9 below, we have presented the plots of
$\Phi(n,\alpha)\cdot\frac{h^{\alpha+2}}{2}$, first as a function of $n$ for
different values of alpha from $0.1$ to $1$, then as function of $n$ and
$\alpha$. The results show that when $\Phi(n,\alpha)<\alpha(n,1)$.

A
---
B
Figure 9: A: shows plot of $\Phi(n,\alpha)$ as a function of $n$ for different
values of $\alpha$. B: denotes $\Phi(n,\alpha)$ as a function of $n$ and
$\alpha$.

Alternatively, one can proceed as follows:

$\begin{split}u(t_{n+1})-u(t_{n})&=\frac{1-\alpha}{AB(\alpha)}\left[f(t_{n},u(t_{n}))-f(t_{n-1},u(t_{n-1}))\right]\\\
&+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\sum_{j=1}^{n}\int_{t_{j}}^{t_{j+1}}f(\tau,u(\tau))(t_{n+1}-\tau)^{\alpha-1}d\tau\\\
&-\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\sum_{j=1}^{n-1}\int_{t_{j}}^{t_{j+1}}f(\tau,u(\tau))(t_{n}-\tau)^{\alpha-1}d\tau\end{split}$
(39)

such that in $[t_{j},t_{j+1}]$ the function $f(\tau,u(\tau))$ can be
interpolated using the Lagrange polynomial interpolation for the first
integral. Also, the function can be interpolated within $[t_{k},t_{k+1}]$ for
the second integral.

## Acknowledgment

The authors are grateful to all of the anonymous reviewers and the Editor-in-
Chief for their professional support and valuable comments.

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