# A high-order predictor-corrector method for initial value problems with
fractional derivative involving Mittag-Leffler kernel: epidemic model case
study

Sami Aljhania, Mohd Salmi Md Noorania, Redouane Douaifiab, Salem Abdelmalekb,c
$(a)$ Department of Mathematical Sciences, Faculty of Science and Technology,
Universiti Kebangsaan Malaysia 43600 Bangi, Selangor, Malaysia
$(b)$ Laboratory of Mathematics, Informatics and Systems (LAMIS), Larbi
Tebessi University, Tebessa, Algeria
$(c)$ Department of Mathematics and Computer Science, Larbi Tebessi
University, Tebessa, Algeria
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we propose a numerical scheme of the predictor-corrector type
for solving nonlinear fractional initial value problems, the chosen fractional
derivative is called the Atangana-Baleanu derivative defined in Caputo sense
(ABC). This proposed method is based on Lagrangian quadratic polynomials to
approximate the nonlinearity implied in the Volterra integral which is
obtained by reducing the given fractional differential equation via the
properties of the ABC-fractional derivative. Through this technique, we get
corrector formula with high accuracy which is implicit as well as predictor
formula which is explicit and has the same precision order as the corrective
formula. On the other hand, the so-called memory term is computed only once
for both prediction and correction phases, which indicates the low cost of the
proposed method. Also, the error bound of the proposed numerical scheme is
offered.
Furthermore, numerical experiments are presented in order to assess the
accuracy of the new method on two differential equations. Moreover, a case
study is considered where the proposed predictor-corrector scheme is used to
obtained approximate solutions of ABC-fractional generalized SI (susceptible-
infectious) epidemic model for the purpose of analyzing dynamics of the
suggested system as well as demonstrating the effectiveness of the new method
to solve systems dealing with real-world problems.

###### keywords:

Fractional differential equations; Atangana-Baleanu derivative; Numerical
method; Predictor-corrector; Epidemic model.

††journal: a Q1 journal

##  1 Introduction

Despite the fact that fractional calculus has a lengthy history in
mathematics, it has only recently seen a considerable number of real-world
applications. Fractional derivatives are important due to their attractive
characteristics (e.g. memory effect), including their wide dynamical range [1,
2, 3]. Many definitions of fractional derivatives and integrals exist,
including Riemann-Liouville, Caputo, Grunwald-Letnikov [4]. In 2015, Caputo
and Fabrizio (CF) have suggested a new idea of fractional derivatives based on
the exponential decay [5]. Following that, Atangana and Baleanu (AB) proposed
a novel concept of fractional derivatives with non-singular and non-local
kernel based on the Mittag-Leffler function [6]. Due to the difficulty or
impossibility of obtaining to obtain exact solution of fractional differential
equations, numerical techniques are required to provide approximate solutions.
Over the past few decades, numerical methods to solve initial value problems
have attracted great interest from the research community, which support
various disciplines, including medicine, chemistry, biology, physics,
business, and the arts, etc. Where computational algorithms are implemented.
Moreover, numerical analysis techniques are the perfect tools to evaluate the
behavior of real-life complicated models. Therefore, the implementation of
accurate numerical methods is suibtle in the problems representing real world
phenomena [6, 7, 8, 9]. Over 2000 years ago, the simplest method in the
literature was introduced with the concept of interpolation. In contrast,
several other techniques were introduced for nonlinear systems, such as
Newton’s method, Euler’s method, Gaussian elimination, and Lagrange
interpolation [10, 11, 12, 13]. Over the past few years, several numerical
tools have been developed for solving fractional ordinary differential
equations with general nonlinearity [14, 15, 16, 17, 18, 19, 20]. Such as the
fractional Adams-Bashforth method, which is the most popular scheme in which
the basis of Lagrange interpolation is used [22, 23] and the fractional Adams-
Bashforth/Moulton method developed with Newton linear/quadratic interpolations
[24, 25]. Baleanu et al. [17] used the definition of the Atangana-Baleanu
derivative in the Caputo sense to produce a predictor-corrector approach for
fractional differential equation as well as to prove the existence and
uniqueness of such problem. They have converted the ABC-fractional ordinary
differential equation into a Volterra Integral equation, then they have used
the rectangle rule for the predictor and the trapezoid rule for the corrector.
On the other hand, based on the techniques and approximations appeared in [18,
19] for finding approximate solutions of fractional differential equations
with Caputo and Caputo-Fabrizio fractional derivatives, respectively. We can
to propose a new and efficient predictor-corrector method for solving ABC-
fractional initial value problems by using quadratic lagrange interpolation,
memory term, and second-order Taylor’s expansion.

The following is a breakdown of the paper’s structure. Basic definitions and
notations, including the Atangana-Baleanu fractional derivatives and integral,
are introduced in section 2. In section 3, a new numerical method is proposed
for solving fractional initial value problems involving the Atangana-Baleanu
fractional derivative in Caputo sense. The schemes’ error estimates are
obtained in section 4. Finally, in section 5, we present numerical examples to
demonstrate the efficacy of the proposed scheme, and we also compare the
solutions with other methods [8, 17]. Also, the stability analysis of the ABC-
fractional-order SI (Susceptible-Infection) epidemiological model and its
comparison with the numerical results obtained via the proposed scheme, which
investigates the significance of disease prevalence in the community, is
discussed in the last section.

##  2 Preliminaries

The Atangana-Baleanu fractional derivative in the Caputo sense (ABC) is
defined as (cf. [6]):

${}^{ABC}_{\quad\,0}D^{\alpha}_{t}f(t)=\displaystyle\frac{AB(\alpha)}{1-\alpha}\int^{t}_{0}f^{\prime}(s)\textbf{E}_{\alpha}\left(\frac{-\alpha}{1-\alpha}(t-s)^{\alpha}\right)ds,$
(1)

where $\alpha\in(0,1)$, $AB(\alpha)>0$ is a normalization function obeying
$AB(0)=AB(1)=1$
$\left(\text{e.g.
}AB(\alpha)=1-\alpha+\displaystyle\frac{\alpha}{\Gamma(\alpha)}\right)$,
$\textbf{E}_{\alpha}(.)$ denotes the Mittag-Leffler function of order $\alpha$
defined by

$\begin{matrix}\textbf{E}_{\alpha}(z)=\displaystyle\sum^{\infty}_{k=0}\frac{{z}^{k}}{\Gamma(\alpha
k+1)},&z,\alpha\in\mathbb{C},&\text{and }Re(\alpha)>0,\end{matrix}$ (2)

and $\Gamma(.)$ denotes Gamma function, defined as

$\begin{matrix}\Gamma(\alpha)=\displaystyle\int^{+\infty}_{0}t^{\alpha-1}e^{-t}dt,&\qquad
Re(\alpha)>0.\end{matrix}$ (3)

The fractional integral for the ABC, which is newly defined with a nonlocal
kernel and does not have singularities at $t=s$, is defined as follows (cf.
[27]):

$\displaystyle{}^{ABC}_{\quad\,0}I^{\alpha}_{t}f(t)=\frac{1-\alpha}{AB(\alpha)}f(t)+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int^{t}_{0}f(s)(t-s)^{\alpha-1}ds.$
(4)

We consider the initial-value problem with Atangana-Baleanu derivative

$\begin{cases}&{}^{ABC}_{\quad\,0}D^{\alpha}_{t}y\left(t\right)=f\left(t,y\left(t\right)\right),\qquad\qquad\qquad
0<t<T<\infty,\\\ \\\ &\qquad\quad\,y\left(0\right)=y_{0},\end{cases}$ (5)

where $f$ is a smooth nonlinear function that guarantees the existence of a
unique solution for (5), with the fractional order
$\alpha\in\left(0,1\right)$, and $y_{0}\in\mathbb{R}$. A continuous function
$y(t)$ is the solution of (5) if and only if it is the solution of following
Voltera-integral equation:

$\begin{matrix}y(t)&=\displaystyle
y_{0}+\frac{1-\alpha}{AB(\alpha)}f(t,y(t))+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t}f\left(s,y(s)\right)\left(t-s\right)^{\alpha-1}ds.\end{matrix}$
(6)

At point $t_{n+1}=h(n+1)$, $n=0,1,\dots,N\in\mathbb{N}$ with $h=\frac{T}{N}$,
we get

$\begin{split}y_{n+1}&=y_{0}+\frac{1-\alpha}{AB(\alpha)}f_{n+1}+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t_{n+1}}f\left(s,y(s)\right)\left(t_{n+1}-s\right)^{\alpha-1}ds\\\
&=y_{0}+\frac{1-\alpha}{AB(\alpha)}f_{n+1}+\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\left(\int_{0}^{t_{n}}f\left(s,y(s)\right)\left(t_{n+1}-s\right)^{\alpha-1}ds+\int_{t_{n}}^{t_{n+1}}f\left(s,y(s)\right)\left(t_{n+1}-s\right)^{\alpha-1}ds\right)\\\
&=y_{0}+\frac{1-\alpha}{AB(\alpha)}f_{n+1}+y^{lag}_{n+1}+y^{inc}_{n+1},\end{split}$
(7)

where, $y^{lag}_{n+1}$ and $y^{inc}_{n+1}$ are called lag term and increment
term respectively, which take the following forms:

$y^{lag}_{n+1}:=\displaystyle\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t_{n}}(t_{n+1}-s)^{\alpha-1}f(s,y(s))ds,$
(8)

and

$y^{inc}_{n+1}:=\displaystyle\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{t_{n}}^{t_{n+1}}(t_{n+1}-s)^{\alpha-1}f(s,y(s))ds.$
(9)

##  3 Predictor-corrector scheme with quadratic interpolation

To start presenting the proposed numerical method to solve the initial-value
problem (5), we need the following lemma:

###### Lemma 1

([18]) Assume that $\psi$ $\in$ $\mathbb{P}_{2}([0,T])$, where
$\mathbb{P}_{2}([0,T])$ is the space of all polynomials of degree less than or
equal to two. Let $\psi_{k}$, $k=0,....,N$ be the restricted value of
$\psi(t)$ on $t_{k}$ ( $0\leq k\leq N$ ). Then there exist reals
$b^{0}_{n+1},b^{1}_{n+1}$ and $b^{2}_{n+1}$, such that

$\int_{t_{n}}^{t_{n+1}}(t_{n+1}-s)^{\alpha-1}\psi(s)ds=B\sum_{j=0}^{2}b^{j}_{n+1}\psi_{n+j-2}.$
(10)

Where,
$B=\displaystyle\frac{h^{\alpha}}{\alpha(\alpha+1)(\alpha+2)},b^{0}_{n+1}=\frac{\alpha+4}{2},b^{1}_{n+1}=-2(\alpha+3),b^{2}_{n+1}=\frac{2\alpha^{2}+9\alpha+12}{2}$.

Now, we use quadratic Lagrange polynomial of $f(s,y(s))$ over the intervals
$[t_{i-1},t_{i+1}]$ , $1\leq i\leq N-1$:

$\displaystyle f(s,y(s))\approx\sum_{l=i-1}^{i+1}f_{l}q^{i}_{l}(s),$ (11)

where

$\displaystyle q^{i}_{l}(s)=\prod_{\begin{subarray}{c}k=i-1\\\ l\neq
k\end{subarray}}^{i+1}\frac{s-t_{k}}{t_{l}-t_{k}}.$ (12)

But, on $[t_{0},t_{1}]$ we can interpolate $f(s,y(s))$ with the points
$\big{\\{}{t_{0},t_{\frac{1}{2}},t_{1}}\big{\\}}$, then we get

$\displaystyle f(s,y(s))\approx
f_{0}q^{0}_{0}(s)+f_{\frac{1}{2}}q^{0}_{\frac{1}{2}}(s)+f_{1}q^{0}_{1}(s),$
(13)

where

$\displaystyle
q^{0}_{0}(s)=\displaystyle\frac{(s-t_{0})(s-t_{\frac{1}{2}})}{(t_{1}-t_{0})(t_{1}-t_{\frac{1}{2}})},\
q^{0}_{\frac{1}{2}}(s)=\frac{(s-t_{0})(s-t_{1})}{(t_{\frac{1}{2}}-t_{1})(t_{\frac{1}{2}}-t_{1})},\
q^{0}_{1}(s)=\frac{(s-t_{0})(s-t_{\frac{1}{2}})}{(t_{1}-t_{0})(t_{1}-t_{\frac{1}{2}})}.$
(14)

The approximation of $y(t_{n+1})$ denoting by $\widetilde{y}_{n+1}$, according
to (11)-(14), $\widetilde{y}_{n+1}$ takes the following form:

$eq216\widetilde{y}_{n+1}=\displaystyle
y_{0}+{}^{AB}\widetilde{f}^{P}_{n+1}+\widetilde{y}^{lag}_{n+1}+\widetilde{y}^{inc}_{n+1},$
(15)

where

${}^{AB}\widetilde{f}^{P}_{n+1}=\displaystyle\frac{1-\alpha}{AB(\alpha)}f(t_{n+1},\widetilde{y}^{P}_{n+1}),$
(16)

$\widetilde{y}^{lag}_{n+1}=\displaystyle\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\left(\sum_{j=0}^{2}a^{j,0}_{n+1}\widetilde{f}_{\frac{j}{2}}+\sum_{j=0}^{2}\sum_{k=1}^{n-1}a^{j,k}_{n+1}\widetilde{f}_{k+j-1}\right)$
(17)

and

$\widetilde{y}^{inc}_{n+1}=\displaystyle\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\left(\sum_{j=0}^{1}a^{j,n}_{n+1}\widetilde{f}_{n+j-1}+a^{2,n}_{n+1}\widetilde{f}^{P}_{n+1}\right)$
(18)

with

$\begin{matrix}a^{j,k}_{n+1}=\displaystyle\int_{t_{k}}^{t_{k+1}}(t_{n+1}-s)^{\alpha-1}q^{k}_{\widehat{j}}(s)ds,&\textit{for}&j\in\left\\{0,1,2\right\\},\end{matrix}$
(19)

where,

$\widehat{j}=\begin{cases}\displaystyle\frac{j}{2}&\text{if }k=0,\\\
j+k-1&\text{if }1\leq k\leq n.\end{cases}$ (20)

The predictor term can be approximated as follows:

$\widetilde{y}^{P}_{n+1}=y_{0}+{}^{AB}\widetilde{f}^{apprx}_{n+1}+\widetilde{y}^{lag}_{n+1}+\frac{\alpha
h^{\alpha}}{AB(\alpha)\Gamma(\alpha+3)}\sum_{j=0}^{2}b^{j}_{n+1}\widetilde{f}_{n+j-2},$
(21)

where

${}^{AB}\widetilde{f}^{apprx}_{n+1}=\frac{1-\alpha}{AB(\alpha)}\left(f_{n-2}-3f_{n-1}-3f_{n}\right),\hskip
15.0ptn\geq 2.$ (22)

##  4 Error analysis

Throughout this section, we need the following lemmas:

###### Lemma 2

Let $f$ $\in$ $C^{n+1}([a,b])$ and $P_{n}$ $\in$ $\mathbb{P}_{n}([a,b])$,
(where $[a,b]\subset\mathbb{R}$ and $\mathbb{P}_{n}([a,b])$ is the space of
all polynomials of degree less than or equal to $n$) interpolate the function
$f$ at $t_{k}$, $0\leq k\leq N$, with $t_{0}=a,t_{N}=b,$, then there exists
$\xi$ $\in(a,b)$ such that, for any $s$ $\in[a,b]$

$\displaystyle
f(s)-P_{n}(s)=\frac{f^{n+1}(\xi)}{(n+1)!}\prod_{k=0}^{n}(s-t_{k}).$ (23)

###### Lemma 3

([18]) For $\delta>0$, we have

$\sum_{k=0}^{n-1}\int_{t_{k}}^{t_{k+1}}(t_{n+1}-s)^{\delta-1}ds\leq\frac{T^{\delta}}{\delta}.$
(24)

###### Lemma 4

([21]) (Discrete Gronwall’s Inequality) Let
$\left\\{a_{n}\right\\}^{N}_{n=0},\left\\{b_{n}\right\\}^{N}_{n=0}$ be non-
negative sequences with second one is monotonic increasing and satisfy that

$\begin{matrix}a_{n}\leq\displaystyle
b_{n}+Mh^{\theta}\sum_{k=0}^{n-1}(n-k)^{\theta-1}a_{k},&0\leq n\leq
N,\end{matrix}$ (25)

where, $M>0$ is independent of $h>0$, and $0<\theta\leq 1$. Then,

$a_{n}\leq
b_{n}\textbf{E}_{\theta}\big{(}M\Gamma(\theta)(nh)^{\theta}\big{)}.$ (26)

###### Lemma 5

([18]) There exist $K_{1},K_{2}>0$ such that for $\alpha\in(0,1)$, and
$j=0,1,2$, we have

$\Bigl{|}a^{j,k}_{n+1}\Bigl{|}\leq\begin{cases}\displaystyle
K_{1}(n-k)^{\alpha-1}h^{\alpha}&\text{if }0\leq k\leq n-1,\\\ K_{2}&\text{if
}k=n.\end{cases}$ (27)

Let $T^{P}_{n+1}$ be the truncation error of prediction at point $t_{n+1}$,
defined by

$\displaystyle T^{P}_{n+1}=$
$\displaystyle\Biggl{|}\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\int_{0}^{t_{n+1}}(t_{n+1}-s)^{\alpha-1}f(s,y(s))ds-\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\sum_{j=0}^{2}a^{j,0}_{n+1}f_{\frac{j}{2}}$
(28)
$\displaystyle-\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\sum_{j=0}^{2}\sum_{k=1}^{n-1}a^{j,k}_{n+1}f_{k+j-1}-\frac{\alpha
h^{\alpha}}{AB(\alpha)\Gamma(\alpha+3)}\sum_{j=0}^{2}b^{j}_{n+1}f_{n+j-2}\Biggl{|}.$

###### Theorem 1

Assume that $f(.,y(.))\in C^{3}([0,T]).$ Then, there exists $C>0$ (independent
of all grid parameters) such that:

$\displaystyle T^{P}_{n+1}\leq Ch^{3}.$ (29)

###### Proof 1

We set the notation,
$AB_{g}=\displaystyle\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}$, then

$T^{P}_{n+1}\leq\displaystyle\sum_{j=1}^{3}\mathcal{I}_{j},$ (30)

where,

$\displaystyle\mathcal{I}_{1}$
$\displaystyle:=AB_{g}\int_{0}^{t_{1}}(t_{n+1}-s)^{\alpha-1}\Bigl{|}f(s,y(s))-\sum_{j=0}^{2}f_{\frac{j}{2}}q^{0}_{\frac{j}{2}}(s)\Bigl{|}ds,$
(31) $\displaystyle\mathcal{I}_{2}$
$\displaystyle:=AB_{g}\sum_{k=1}^{n-1}\int_{t_{k}}^{t_{k+1}}(t_{n+1}-s)^{\alpha-1}\Bigl{|}f(s,y(s))-\sum_{j=0}^{2}f_{k+j-1}q^{k}_{k+j-1}(s)\Bigl{|}ds,$
(32) $\displaystyle\mathcal{I}_{3}$
$\displaystyle:=\Bigl{|}AB_{g}\int_{t_{n}}^{t_{n+1}}(t_{n+1}-s)^{\alpha-1}f(s,y(s))-\Gamma(\alpha)AB_{g}\frac{h^{\alpha}}{\Gamma(\alpha+3)}\sum_{j=0}^{2}b^{j}_{n+1}f_{n+j-2}\Bigl{|}.$
(33)

Thanks to lemma 2, there exists $C_{1}>0$, such that

$\begin{matrix}\mathcal{I}_{j}\leq C_{1}h^{3},&j=1,2.\end{matrix}$ (34)

The Taylor expansion of $f$ around $t_{n}$ gives:

$\begin{matrix}f(t)=\displaystyle
P_{2}(t)+\frac{1}{3!}f^{\prime\prime\prime}(\xi_{n})(t-t_{n})^{3}+O(h^{3}),&\xi_{n}\in(t_{n},t),\end{matrix}$
(35)

where,

$\begin{matrix}P_{2}(t)=\displaystyle
f_{n}+f^{\prime}(t_{n})(t-t_{n})+\frac{1}{2}f^{\prime\prime}(t_{n})(t-t_{n})^{2},&(P_{2}\in\mathbb{P}_{2}([0,T])).\end{matrix}$
(36)

According to lemma 2 and (33), we have

$\displaystyle\mathcal{I}_{3}=$
$\displaystyle\Bigl{|}AB_{g}\int_{t_{n}}^{t_{n+1}}(t_{n+1}-s)^{\alpha-1}(P_{2}(s)+\frac{1}{3!}f^{\prime}(\xi_{n})(s-t_{n})^{3}+O(h^{3}))ds$
$\displaystyle-\Gamma(\alpha)AB_{g}\frac{h^{\alpha}}{\Gamma(\alpha+3)}\sum_{j=0}^{2}b^{j}_{n+1}f_{n+j-2}\Bigl{|},$
(37)

it follows that

$\displaystyle\mathcal{I}_{3}$
$\displaystyle\leq\Gamma(\alpha)AB_{g}\frac{h^{\alpha}}{\Gamma(\alpha+3)}\Big{(}\mid
b^{0}_{n+1}\mid\mid P_{2}(t_{n-2})-f_{n-2}-f_{n-2}\mid+\mid
b^{1}_{n+1}\mid\mid P_{2}(t_{n-1})-f_{n-1}\mid\Big{)}$ $\displaystyle\ \ \
+AB_{g}\left|\int_{t_{n}}^{t_{n+1}}(t_{n+1}-s)^{\alpha-1}\Big{(}\frac{1}{6}f^{\prime\prime\prime}(\xi_{n}){(s-t_{n})^{3}}\Big{)}ds\right|+O(h^{3}).$
(38)

According to (30) and (35), we have

$\begin{matrix}\Bigl{|}P_{2}(t_{n-2})-f_{n-2}\Bigl{|}\leq\displaystyle\frac{4Mh^{3}}{3},&\text{where
}M:=\max\left\\{\mid f^{\prime\prime\prime}(\xi_{k})\mid\text{ : }0\leq k\leq
N\right\\},\end{matrix}$ (39)

and

$\Bigl{|}P_{2}(t_{n-1})-f_{n-1}\Bigl{|}\leq\frac{M}{6}h^{3}.$ (40)

We thus obtain the estimate

$\mathcal{I}_{3}\leq\Gamma(\alpha)AB_{g}\frac{h^{\alpha}}{\Gamma(\alpha+3)}\Big{(}b^{0}_{n+1}\frac{4Mh^{3}}{3}+\left|b^{1}_{n+1}\right|\frac{M}{6}h^{3}\Big{)}+\frac{Mh^{3+\alpha}}{6AB(\alpha)\Gamma(\alpha)}+O(h^{3}).$
(41)

Consequently, there exists $C_{2}>0$ such that

$\mathcal{I}_{3}\leq C_{2}h^{3}.$ (42)

According to (30), (34) and (42), proof of theorem is achieved.

###### Theorem 2

(Global Predictor Error) Assume that $f(.,y(.))\in C^{3}([0,T])$ and is
Lipschitz continuous in its second argument, i.e.

$\begin{matrix}\exists L>0,\textit{ such that
}\left|f(t,y_{1})-f(t,y_{2})\right|\leq L\left|y_{1}-y_{2}\right|,&\forall
y_{1},y_{2}\in\mathbb{R}.\end{matrix}$ (43)

Then, there exist $K_{1},K_{2}>0$ such that, the global predictor error
satisfies

$E^{P}_{n+1}:=\left|y_{n+1}-\widetilde{y}^{P}_{n+1}\right|\leq
T^{P}_{n+1}+K_{1}AB_{g}Lh^{\alpha}E_{\frac{1}{2}}+K_{2}AB_{g}Lh^{\alpha}\sum_{k=1}^{n}(n-k+1)^{\alpha-1}E_{k}+O(h^{3}).$
(44)

Where, $AB_{g}:=\displaystyle\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}$.

###### Proof 2

We set
${}^{AB}{f}_{n+1}:=\displaystyle\frac{1-\alpha}{AB(\alpha)}f(t_{n+1},y_{n+1})$.
Thus

$E^{P}_{n+1}=\left|{}^{AB}{f}_{n+1}+AB_{g}\int_{0}^{n+1}(t_{n+1}-s)^{\alpha-1}f(s,y(s))ds-{}^{AB}\widetilde{f}^{apprx}_{n+1}-\widetilde{y}^{lag}_{n+1}-\displaystyle\frac{\Gamma(\alpha)AB_{g}h^{\alpha}}{\Gamma(\alpha+3)}\sum_{j=0}^{2}b^{j}_{n+1}\widetilde{f}_{n+j-2}\right|,$
(45)

therefore

$\displaystyle E^{P}_{n+1}=\Biggl{|}$
$\displaystyle{}^{AB}{f}_{n+1}+AB_{g}\int_{0}^{t_{n+1}}(t_{n+1}-s)^{\alpha-1}f(s,y(s))ds-
AB_{g}\sum_{j=0}^{2}a^{j,0}_{n+1}f_{\frac{j}{2}}$ $\displaystyle-
AB_{g}\sum_{j=0}^{2}\sum_{k=1}^{n-1}a^{j,k}_{n+1}f_{k+j-1}-AB_{g}h^{\alpha}\sum_{j=0}^{2}b^{j}_{n+1}f_{n+j-2}+AB_{g}\sum_{j=0}^{2}a^{j,0}_{n+1}f_{\frac{j}{2}}$
$\displaystyle+AB_{g}\sum_{j=0}^{2}\sum_{k=1}^{n-1}a^{j,k}_{n+1}f_{k+j-1}-\displaystyle\frac{\Gamma(\alpha)AB_{g}h^{\alpha}}{\Gamma(\alpha+3)}\sum_{j=0}^{2}b^{j}_{n+1}f_{n+j-2}-{}^{AB}\widetilde{f}^{apprx}_{n+1}-\widetilde{y}^{lag}_{n+1}$
$\displaystyle-\displaystyle\frac{\Gamma(\alpha)AB_{g}h^{\alpha}}{\Gamma(\alpha+3)}\sum_{j=0}^{2}b^{j}_{n+1}\widetilde{f}_{n+j-2}\Biggl{|},$
(46)

it follows that

$\displaystyle E^{P}_{n+1}$ $\displaystyle\leq
T^{P}_{n+1}+\left|{}^{AB}{f}_{n+1}-{}^{AB}{\widetilde{f}}_{n+1}^{apprx}\right|$
(47) $\displaystyle\ \ \
+AB_{g}\sum_{j=0}^{2}\left|a^{j,0}_{n+1}\right|\left|f_{\frac{j}{2}}-\widetilde{f}_{\frac{j}{2}}\right|+AB_{g}\sum_{j=0}^{2}\sum_{k=1}^{n-1}\left|a^{j,k}_{n+1}\right|\left|f_{k+j-1}-\widetilde{f}_{k+j-1}\right|$
$\displaystyle\ \ \
+\frac{\Gamma(\alpha)AB_{g}h^{\alpha}}{\Gamma(\alpha+3)}\sum_{j=0}^{2}\left|b^{j}_{n+1}\right|\left|f_{n+j-2}-\widetilde{f}_{n+j-2}\right|.$

Hence, according to the lemma 3 there exist $C_{1},K_{1}>0$, such that

$\displaystyle E^{P}_{n+1}$ $\displaystyle\leq
T^{P}_{n+1}+C_{1}\sum_{j=0}^{2}E_{n+j-2}+K_{1}AB_{g}Lh^{\alpha}n^{\alpha-1}(E_{\frac{1}{2}}+E_{1})$
$\displaystyle\ \ \
+K_{1}AB_{g}Lh^{\alpha}\sum_{j=0}^{2}\sum_{k=1}^{n-1}(n-k)^{\alpha-1}E_{k+j-1}+\frac{\Gamma(\alpha)AB_{g}Lh^{\alpha}}{\Gamma(\alpha+3)}\sum_{j=0}^{2}\left|b^{j}_{n+1}\right|E_{n+j-2}+O(h^{3}).$
(48)

On the other hand, for $\alpha\in(0,1)$ we have

$\begin{matrix}(n+1-i)^{\alpha-1}+(n-1)^{\alpha-1}+(n-i-1)^{\alpha-1}\leq
6(n+1-i)^{\alpha-1},&1\leq i\leq n-2.\end{matrix}$ (49)

Consequently, there exists $K_{2}>0$ such that the estimate (2), becomes

$E^{P}_{n+1}\leq
T^{P}_{n+1}+K_{1}AB_{g}Lh^{\alpha}E_{\frac{1}{2}}+K_{2}AB_{g}Lh^{\alpha}\sum_{k=1}^{n}(n+1-k)^{\alpha-1}E_{k}+O(h^{3}).$
(50)

###### Theorem 3

Assume that $f(.,y(.))\in C^{3}([0,T]).$ Then there exists positive constants
$C_{1}$ and $C_{2}$ (independent of grid parameters) such that:

$\displaystyle T^{C}_{n+1}\leq C_{1}h^{3}+C_{2}E^{P}_{n+1},$ (51)

where,

$\displaystyle T^{C}_{n+1}$
$\displaystyle=\Biggl{|}AB_{g}\int_{0}^{t_{n+1}}(t_{n+1}-s)^{\alpha-1}f(s,y(s))ds-
AB_{g}\sum_{j=0}^{2}a^{j,0}_{n+1}f_{\frac{j}{2}}$ $\displaystyle\ \ \ \
-AB_{g}\left(\sum_{j=0}^{2}\sum_{k=1}^{n-1}a^{j,k}_{n+1}f_{k+j-1}+\sum_{j=0}^{1}a^{j,n}_{n+1}f_{n+j-1}+a^{2,n}_{n+1}f^{P}_{n+1}\right)\Biggl{|},$
(52)

with, $AB_{g}=\displaystyle\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}$.

###### Proof 3

It follows immediately that,

$\displaystyle T^{C}_{n+1}$ $\displaystyle\leq
AB_{g}\int_{t_{0}}^{t_{1}}(t_{n+1}-s)^{\alpha-1}\Bigl{|}f(s,y(s))-\sum_{j=0}^{2}q^{0}_{\frac{j}{2}}(s)f_{\frac{j}{2}}\Bigl{|}ds$
$\displaystyle\ \ \
+AB_{g}\sum_{k=1}^{n-1}\int_{t_{k}}^{t_{k+1}}(t_{n+1}-s)^{\alpha-1}\Bigl{|}f(s,y(s))-\sum_{j=0}^{2}q^{k}_{k+j-1}(s)f_{k+j-1}\Bigl{|}ds$
$\displaystyle\ \ \
+AB_{g}\int_{t_{n}}^{t_{n+1}}(t_{n+1}-s)^{\alpha-1}\Bigl{|}f(s,y(s))-\sum_{j=0}^{1}q^{n}_{n+j-1}(s)f_{n+j-1}-q^{n}_{n+1}(s)f^{P}_{n+1}\Bigl{|}ds.$
(53)

Thus, according to the lemma 2, there exist $\widehat{C},\widetilde{C}>0$ such
that

$T^{C}_{n+1}\leq\frac{\widetilde{C}T^{\alpha}}{6AB(\alpha)\Gamma(\alpha)}h^{3}+\frac{\widehat{C}h^{\alpha}}{AB(\alpha)\Gamma(\alpha)}E^{P}_{n+1}+O(h^{3}).$
(54)

The desired estimate (3) can be derived by direct consideration of the last
inequality (54).

###### Theorem 4

(Global Error of the proposed method) With the same assumptions as those of
Theorem 2. Then, we have

$E_{n+1}:=\left|y_{n+1}-\widetilde{y}_{n+1}\right|\leq Ch^{3},$ (55)

where $C>0$ (independent of grid parameters), given
$E_{1},E_{2}\leq\widehat{C}_{1}h^{3}$, and
$E_{\frac{1}{2}}\leq\widehat{C}_{2}h^{3-\alpha}$, with
$\widehat{C}_{1},\widehat{C}_{2}>0$.

###### Proof 4

By taking into account the previous results, there exist $C_{j}>0$ (with
$j=1,\dots,10$), such that

$\displaystyle E_{n+1}$ $\displaystyle\leq
T^{C}_{n+1}+AB_{g}\left|f^{P}_{n+1}-\widetilde{f}^{P}_{n+1}\right|+\frac{L(1-\alpha)}{AB(\alpha)}E^{P}_{n+1}+AB_{g}\sum_{j=0}^{2}\left|a^{j,0}_{n+1}\right|\left|f_{\frac{j}{2}}-\widetilde{f}_{\frac{j}{2}}\right|$
$\displaystyle\ \ \
+AB_{g}\sum_{j=0}^{2}\sum_{k=1}^{n-1}\left|a^{j,k}_{n+1}\right|\left|f_{k+j-1}-\widetilde{f}_{k+j-1}\right|+AB_{g}\sum_{j=1}^{1}\left|a^{j,n}_{n+1}\right|\left|f_{n+j-1}-\widetilde{f}_{n+j-1}\right|$
$\displaystyle\leq
T^{C}_{n+1}+\frac{L(1-\alpha)}{AB(\alpha)}E^{P}_{n+1}+AB_{g}Lh^{\alpha}E_{\frac{1}{2}}+\frac{\widetilde{C}_{1}h^{\alpha}}{AB(\alpha)\Gamma(\alpha)}E^{P}_{n+1}$
$\displaystyle\ \ \
+AB_{g}\widetilde{C}_{2}Lh^{\alpha}\sum_{k=1}^{n}(n-k+1)^{\alpha-1}E_{k}+O(h^{3})$
$\displaystyle\leq\widetilde{C}_{3}h^{3}+\widetilde{C}_{4}E^{P}_{n+1}+\frac{L(1-\alpha)}{AB(\alpha)}E^{P}_{n+1}+AB_{g}Lh^{\alpha}E_{\frac{1}{2}}$
$\displaystyle\ \ \
+\frac{\widetilde{C}_{5}h^{\alpha}}{AB(\alpha)}E^{p}_{n+1}+AB_{g}\widetilde{C}_{6}Lh^{\alpha}\sum_{k=1}^{n}(n-k+1)^{\alpha-1}E_{k}$
$\displaystyle\leq\widetilde{C}_{7}h^{3}+\widetilde{C}_{8}T^{P}_{n+1}+\widetilde{C}_{9}h^{\alpha}E_{\frac{1}{2}}+\widetilde{C}_{10}h^{\alpha}\sum_{k=1}^{n}(n-k+1)^{\alpha-1}E_{k}.$
(56)

Consequently, by the discrete Gronwall’s inequality (i.e. lemma 4) the desired
result holds.

###### Remark 1

Since the global error indicated in the Theorem 4 is dependent on that of the
start-up ($E_{\frac{1}{2}},E_{1}$, and $E_{2}$). We suggest employing the
start-up scheme described in Appendix A to generate approximate solutions for
the first stages (i.e. $\widetilde{y}_{\frac{1}{2}},\widetilde{y}_{1}$, and
$\widetilde{y}_{2}$).

##  5 Numerical Illustrations and Simulation

In this section, we give some numerical experiments through the proposed
predictor-corrector scheme (PPC) (15)-(21), the predictor-corrector method
introduced by Baleanu-Jajarmi-Hajipour (BJH-PC) [17], and the explicit
numerical scheme introduced by Toufik-Atangana (TAE) [8], to show the
efficiency and accuracy of our new method. The experimental order of
convergence (EOC) is computed by

$EOC=log_{2}\frac{AE\left(\frac{N}{2}\right)}{AE\left(N\right)},$ (57)

where $AE$ is the absolute error, which takes the following form:

$AE:=AE(N)=\underset{1\leqslant k\leqslant
N}{\max}\left|y\left(t_{k}\right)-\widetilde{y}_{k}\right|.$ (58)

###### Example 1

Consider the following fractional Initial value problem (IVP):

$\left\\{\begin{matrix}{}^{ABC}_{\quad\,0}D^{\alpha}_{t}y\left(t\right)=t^{n},&0<t\leq
2,\\\ \\\ y\left(0\right)=1,&\end{matrix}\right.$ (59)

where $\alpha\in(0,1)$, and $n\in\mathbb{N}$. On account of [17], the exact
solution of (59) is given by

$\displaystyle
y\left(t\right)=1+\frac{1-\alpha}{AB\left(\alpha\right)}t^{n}+\frac{\alpha\Gamma\left(n+1\right)}{AB\left(\alpha\right)\Gamma\left(\alpha+n+1\right)}t^{\alpha+n}.$
(60)

In addition, the problem (59) is numerically solved. Table 1 and table 2 show
the comparison of absolute error of different numerical methods ((PPC)
(15)-(21), BJH-PC [17], and TAE [8]) for (59) with $n=2,3$,
$t\in\left[0,2\right]$ and various values of fractional order
$\alpha\in\left\\{0.5,0.7,0.9,0.99\right\\}$, where we notice that our method
is superior to them in terms of accuracy (i.e. PPC (15)-(21) achieves a lower
error than TAE and BJH-PC). Moreover, table 1 and table 2 offer that EOC for
TAE are roughly $1$, for BJH-PC are roughly $2$, and for PPC (15)-(21) are
roughly $3$. Also, we note that the approximate solutions obtained by our
proposed scheme get closer to the exact solutions by the increase in $\alpha$.
From Figure 1 and Figure 2, we note that the approximate solution obtained by
PPC (15)-(21) almost matches with the exact solution with small step size
compared to its counterparts. These figures and tables, indicate the efficacy
of the current predictor-corrector numerical method (PPC) (15)-(21).

Table 1: The absoluate error of various numerical methods for problem (59) with $n=2$, N=$40$, and various values of $\alpha$. Methods |  $\alpha=0.5$ |  $\alpha=0.7$ |  $\alpha=0.9$ |  $\alpha=0.99$
---|---|---|---|---
PPC (15)-(21) |  $5.3e{-15}$ |  $1.8e{-15}$ |  $3.6e{-15}$ |  $9.0e{-16}$
BJH-PC [17] |  $4.1e{-4}$ |  $6.2e{-4}$ |  $7.7e{-4}$ |  $8.3e{-4}$
TAE [8] |  $1.3e{-1}$ |  $7.4e{-2}$ |  $2.5e{-2}$ |  $6.1e{-3}$
Table 2: The absolute error, experimental order of convergence, and CPU time in seconds (CTs) of various numerical methods for problem (59) with $n=3$. Methods |  | $\alpha=0.5$ | $\alpha=0.7$ | $\alpha=0.9$
---|---|---|---|---
| N | AE | EOC | CTs | AE | EOC | CTs | AE | EOC | CTs
PPC (15)-(21) | $10$ | $1.8e{-3}$ | $-$ | $3.8e{-2}$ | $2.7e{-3}$ | $-$ | $2.4e{-3}$ | $3.4e{-3}$ | $-$ | $2.2e{-3}$
| $20$ | $2.4e{-4}$ | $2.94$ | $7.7e{-3}$ | $3.6e{-4}$ | $2.94$ | $7.3e{-3}$ | $4.4e{-4}$ | $2.93$ | $6.4e{-3}$
| $40$ | $3.1e{-5}$ | $2.97$ | $2.0e{-2}$ | $4.6e{-5}$ | $2.97$ | $2.1e{-2}$ | $5.7e{-5}$ | $2.97$ | $2.1e{-2}$
| $80$ | $3.9e{-6}$ | $2.98$ | $7.5e{-2}$ | $5.8e{-6}$ | $2.99$ | $7.6e{-2}$ | $7.2e{-6}$ | $2.98$ | $7.5e{-2}$
| $160$ | $4.9e{-7}$ | $2.99$ | $0.31$ | $7.2e{-7}$ | $2.99$ | $0.29$ | $9.0e{-7}$ | $2.99$ | $0.29$
| $320$ | $6.2e{-8}$ | $2.99$ | $1.12$ | $9.1e{-8}$ | $3.00$ | $1.14$ | $1.1e{-7}$ | $3.00$ | $1.15$
BJH-PC [17] | $10$ | $2.4e{-2}$ | $-$ | $2.7e{-2}$ | $3.4e{-2}$ | $-$ | $1.8e{-3}$ | $3.9e{-2}$ | $-$ | $9.0e{-4}$
| $20$ | $6.3e{-3}$ | $1.95$ | $3.5e{-3}$ | $8.6e{-3}$ | $1.98$ | $1.7e{-3}$ | $9.7e{-3}$ | $2.00$ | $1.9e{-3}$
| $40$ | $1.6e{-3}$ | $1.97$ | $3.7e{-3}$ | $2.2e{-3}$ | $1.99$ | $3.9e{-3}$ | $2.4e{-3}$ | $2.00$ | $4.0e{-3}$
| $80$ | $4.1e{-4}$ | $1.98$ | $9.3e{-3}$ | $5.4e{-4}$ | $1.99$ | $1.0e{-2}$ | $6.1e{-4}$ | $2.00$ | $1.0e{-2}$
| $160$ | $1.0e{-4}$ | $1.99$ | $2.6e{-2}$ | $1.4e{-4}$ | $2.00$ | $3.1e{-2}$ | $1.5e{-4}$ | $2.00$ | $3.1e{-2}$
| $320$ | $2.6e{-5}$ | $1.99$ | $8.4e{-2}$ | $3.4e{-5}$ | $2.00$ | $0.10$ | $3.8e{-5}$ | $2.00$ | $0.11$
TAE [8] | 10 | $1.5e{-0}$ | $-$ | $9.6e{-3}$ | $9.5e{-1}$ | $-$ | $2.6e{-3}$ | $4.1e{-1}$ | $-$ | $1.3e{-3}$
| 20 | $7.7e{-1}$ | $1.01$ | $4.7e{-3}$ | $4.5e{-1}$ | $1.07$ | $4.6e{-3}$ | $1.7e{-1}$ | $1.28$ | $4.8e{-3}$
| 40 | $3.9e{-1}$ | $1.00$ | $1.7e{-2}$ | $2.2e{-1}$ | $1.04$ | $1.8e{-2}$ | $7.4e{-2}$ | $1.18$ | $1.8e{-2}$
| 80 | $1.9e{-1}$ | $1.00$ | $8.4e{-2}$ | $1.1e{-1}$ | $1.02$ | $7.0e{-2}$ | $3.4e{-2}$ | $1.10$ | $7.0e{-2}$
| 160 | $9.6e{-2}$ | $1.00$ | $0.27$ | $5.4e{-2}$ | $1.01$ | $0.33$ | $1.7e{-2}$ | $1.06$ | $0.29$
| 320 | $4.8e{-2}$ | $1.00$ | $1.11$ | $2.7e{-2}$ | $1.00$ | $1.13$ | $8.1e{-3}$ | $1.03$ | $1.15$
Figure 1: Comparison of the exact and the numerical solutions of problem (59)
with $n=2$, and $N=10$. Figure 2: Comparison of the exact and the numerical
solutions of problem (59) with $n=3$, and $N=10$.

###### Example 2

Consider the following Atangana-Beleanu-fractional differential equation:

$\left\\{\begin{matrix}{}^{ABC}_{\quad\,0}D^{\alpha}_{t}y\left(t\right)=t-y(t),&0<t\leq
1,\\\ \\\ y\left(0\right)=0,&\end{matrix}\right.$ (61)

where $\alpha\in(0,1)$. According to [17], the exact solution of (61) is given
by

$y\left(t\right)=\displaystyle\frac{1}{AB\left(\alpha\right)+1-\alpha}\left(\left(1-\alpha\right)t\textbf{E}_{\alpha,2}\left(-\frac{\alpha}{AB\left(\alpha\right)+1-\alpha}t^{\alpha}\right)+\alpha
t^{\alpha+1}\textbf{E}_{\alpha,\alpha+2}\left(-\frac{\alpha}{AB\left(\alpha\right)+1-\alpha}t^{\alpha}\right)\right),$
(62)

with $\textbf{E}_{\alpha,\beta}(.)$ denotes the Mittag-Leffler function of two
parameters $\alpha$ and $\beta$, which defined by

$\begin{matrix}\textbf{E}_{\alpha,\beta}(z)=\displaystyle\sum^{\infty}_{k=0}\frac{{z}^{k}}{\Gamma(\alpha
k+1)},&z,\alpha,\beta\in\mathbb{C},&\text{and
}Re(\alpha),Re(\beta)>0,\end{matrix}$ (63)

and the exact solution (62) is calculated using the algorithm $mlf.m$ (see
[28]) evaluated with accuracy $10^{-12}$. Furthermore, the problem (61) is
numerically solved. Table 3 and table 4 show the comparison of absolute error
of different numerical methods ((PPC) (15)-(21), BJH-PC [17], and TAE [8]) for
(61) with $AB(\alpha)=1$, $AB(\alpha)=1-\alpha+\frac{\alpha}{\Gamma(\alpha)}$,
$t\in\left[0,1\right]$ and various values of fractional order
$\alpha\in\left\\{0.5,0.55,0.7,0.9,0.95\right\\}$, where we notice that our
method is superior to them in terms of accuracy (i.e. PPC (15)-(21) achieves a
lower error than TAE and BJH-PC). From Figure 3 and Figure 4, we note that the
approximate solution obtained by PPC (15)-(21) almost matches with the exact
solution with small step size compared to its counterparts. These figures and
tables, again confirm the efficacy of the current predictor-corrector
numerical method (PPC) (15)-(21).

Table 3: The absolute error, experimental order of convergence, and CPU time in seconds (CTs) of various numerical methods for problem (61) with $AB(\alpha)=1$. Methods |  | $\alpha=0.55$ | $\alpha=0.75$ | $\alpha=0.95$
---|---|---|---|---
| N | AE | EOC | CTs | AE | EOC | CTs | AE | EOC | CTs
PPC (15)-(21) | 10 | $1.5e{-3}$ | $-$ | $3.4e{-2}$ | $3.5e{-4}$ | $-$ | $1.1e{-3}$ | $2.7e{-5}$ | $-$ | $6.0e{-4}$
| 20 | $7.4e{-4}$ | $1.03$ | $2.9e{-3}$ | $8.8e{-5}$ | $1.99$ | $1.1e{-3}$ | $3.2e{-6}$ | $3.05$ | $1.1e{-3}$
| 40 | $3.7e{-4}$ | $0.99$ | $3.5e{-3}$ | $2.3e{-5}$ | $1.93$ | $3.0e{-3}$ | $4.4e{-7}$ | $2.89$ | $3.0e{-3}$
| 80 | $1.8e{-4}$ | $1.03$ | $1.0e{-2}$ | $6.1e{-6}$ | $1.91$ | $1.0e{-2}$ | $6.7e{-8}$ | $2.70$ | $1.0e{-2}$
| 160 | $9.0e{-5}$ | $1.03$ | $3.8e{-2}$ | $2.7e{-6}$ | $1.21$ | $3.7e{-2}$ | $1.2e{-8}$ | $2.51$ | $3.7e{-2}$
| 320 | $4.4e{-5}$ | $1.03$ | $0.14$ | $1.3e{-6}$ | $1.03$ | $0.14$ | $2.3e{-9}$ | $2.33$ | $0.14$
BJH-PC [17] | 10 | $2.6e{-2}$ | $-$ | $1.7e{-2}$ | $8.9e{-3}$ | $-$ | $8.0e{-4}$ | $1.8e{-3}$ | $-$ | $1.0e{-3}$
| 20 | $1.2e{-2}$ | $1.18$ | $3.3e{-3}$ | $3.7e{-3}$ | $1.26$ | $1.5e{-3}$ | $6.7e{-4}$ | $1.41$ | $1.7e{-3}$
| 40 | $5.3e{-3}$ | $1.13$ | $3.4e{-3}$ | $1.7e{-3}$ | $1.16$ | $3.3e{-3}$ | $2.8e{-4}$ | $1.25$ | $3.3e{-3}$
| 80 | $2.5e{-3}$ | $1.09$ | $8.2e{-3}$ | $7.8e{-4}$ | $1.09$ | $8.3e{-3}$ | $1.3e{-4}$ | $1.15$ | $8.2e{-3}$
| 160 | $1.2e{-3}$ | $1.06$ | $2.5e{-2}$ | $3.8e{-4}$ | $1.06$ | $2.4e{-2}$ | $6.0e{-5}$ | $1.08$ | $2.4e{-2}$
| 320 | $5.8e{-4}$ | $1.04$ | $7.9e{-2}$ | $1.8e{-4}$ | $1.03$ | $7.9e{-2}$ | $2.9e{-5}$ | $1.04$ | $7.7e{-2}$
TAE [8] | 10 | $3.6e{-2}$ | $-$ | $5.6e{-3}$ | $2.5e{-2}$ | $-$ | $7.0e{-4}$ | $9.6e{-3}$ | $-$ | $6.0e{-4}$
| 20 | $1.7e{-2}$ | $1.06$ | $2.5e{-3}$ | $1.2e{-2}$ | $1.12$ | $2.2e{-3}$ | $3.7e{-3}$ | $1.40$ | $2.2e{-3}$
| 40 | $8.4e{-3}$ | $1.05$ | $8.8e{-3}$ | $5.5e{-3}$ | $1.08$ | $8.6e{-3}$ | $1.5e{-3}$ | $1.26$ | $8.4e{-3}$
| 80 | $4.1e{-3}$ | $1.03$ | $3.6e{-2}$ | $2.6e{-3}$ | $1.05$ | $3.4e{-2}$ | $6.8e{-4}$ | $1.16$ | $3.4e{-2}$
| 160 | $2.0e{-3}$ | $1.02$ | $0.14$ | $1.3e{-3}$ | $1.03$ | $0.13$ | $3.2e{-4}$ | $1.09$ | $0.14$
| 320 | $9.9e{-4}$ | $1.02$ | $0.55$ | $6.4e{-4}$ | $1.02$ | $0.54$ | $1.5e{-4}$ | $1.05$ | $0.55$
Table 4: The absolute error, experimental order of convergence, and CPU time in seconds (CTs) of various numerical methods for problem (61) with $AB(\alpha)=1-\alpha+\frac{\alpha}{\Gamma(\alpha)}$. Methods |  | $\alpha=0.55$ | $\alpha=0.75$ | $\alpha=0.95$
---|---|---|---|---
| N | AE | EOC | CTs | AE | EOC | CTs | AE | EOC | CTs
PPC (15)-(21) | 10 | $2.3e{-2}$ | $-$ | $3.4e{-2}$ | $5.5e{-4}$ | $-$ | $1.1e{-3}$ | $2.9e{-5}$ | $-$ | $6.0e{-4}$
| 20 | $9.7e{-3}$ | $1.25$ | $2.7e{-3}$ | $1.3e{-4}$ | $2.03$ | $1.2e{-3}$ | $3.5e{-6}$ | $3.05$ | $1.1e{-3}$
| 40 | $4.3e{-3}$ | $1.17$ | $3.5e{-3}$ | $3.4e{-5}$ | $1.99$ | $3.0e{-3}$ | $4.7e{-7}$ | $2.88$ | $3.1e{-3}$
| 80 | $2.0e{-3}$ | $1.12$ | $1.1e{-2}$ | $1.2e{-5}$ | $1.45$ | $1.0e{-2}$ | $7.3e{-8}$ | $2.70$ | $1.0e{-2}$
| 160 | $9.3e{-4}$ | $1.09$ | $3.8e{-2}$ | $6.1e{-6}$ | $1.01$ | $3.7e{-2}$ | $1.3e{-8}$ | $2.51$ | $3.7e{-2}$
| 320 | $4.5e{-4}$ | $1.06$ | $0.14$ | $3.0e{-6}$ | $1.01$ | $0.14$ | $2.6e{-9}$ | $2.33$ | $0.14$
BJH-PC [17] | 10 | $4.9e{-2}$ | $-$ | $1.6e{-2}$ | $1.2e{-2}$ | $-$ | $9.0e{-4}$ | $1.9e{-3}$ | $-$ | $8.0e{-4}$
| 20 | $2.1e{-2}$ | $1.23$ | $3.2e{-3}$ | $4.9e{-3}$ | $1.26$ | $1.5e{-3}$ | $6.9e{-4}$ | $1.41$ | $1.5e{-3}$
| 40 | $9.5e{-3}$ | $1.15$ | $3.3e{-3}$ | $2.2e{-3}$ | $1.17$ | $3.4e{-3}$ | $2.9e{-4}$ | $1.25$ | $3.2e{-3}$
| 80 | $4.4e{-3}$ | $1.10$ | $8.2e{-3}$ | $1.0e{-3}$ | $1.11$ | $8.3e{-3}$ | $1.3e{-4}$ | $1.15$ | $8.3e{-3}$
| 160 | $2.1e{-3}$ | $1.07$ | $2.4e{-2}$ | $4.8e{-4}$ | $1.07$ | $2.4e{-2}$ | $6.2e{-5}$ | $1.08$ | $2.4e{-2}$
| 320 | $1.0e{-3}$ | $1.05$ | $7.8e{-2}$ | $2.4e{-4}$ | $1.04$ | $7.8e{-2}$ | $3.0e{-5}$ | $1.04$ | $8.5e{-2}$
TAE [8] | 10 | $4.2e{-2}$ | $-$ | $5.5e{-3}$ | $2.8e{-2}$ | $-$ | $6.0e{-4}$ | $9.9e{-3}$ | $-$ | $6.0e{-4}$
| 20 | $2.0e{-2}$ | $1.06$ | $2.5e{-3}$ | $1.3e{-2}$ | $1.11$ | $2.2e{-3}$ | $3.8e{-3}$ | $1.39$ | $2.3e{-3}$
| 40 | $9.7e{-3}$ | $1.04$ | $8.8e{-3}$ | $6.1e{-3}$ | $1.08$ | $8.5e{-3}$ | $1.6e{-3}$ | $1.26$ | $8.7e{-3}$
| 80 | $4.8e{-3}$ | $1.03$ | $3.5e{-2}$ | $3.0e{-3}$ | $1.05$ | $3.4e{-2}$ | $7.0e{-4}$ | $1.16$ | $3.4e{-2}$
| 160 | $2.3e{-3}$ | $1.02$ | $0.14$ | $1.4e{-3}$ | $1.03$ | $0.14$ | $3.3e{-4}$ | $1.09$ | $0.14$
| 320 | $1.2e{-3}$ | $1.02$ | $0.56$ | $7.2e{-4}$ | $1.02$ | $0.55$ | $1.6e{-4}$ | $1.05$ | $0.57$
Figure 3: Comparison of the exact and the numerical solutions of problem (61)
with $N=10$, and $AB(\alpha)=1-\alpha+\frac{\alpha}{\Gamma(\alpha)}$. Figure
4: Comparison of the exact and the numerical solutions of problem (61) with
$N=10$, and $AB(\alpha)=1$.

###### Example 3

(Dynamics of a generalized susceptible-infectious model with ABC-fractional
derivative)
In the previous examples, we have considered some differential equations with
known exact solutions. Let us now analyze a realistic ABC-fractional epidemic
model using qualitative analysis (Lyapunov theory) of solutions and validate
the theoretical results numerically by means of the proposed predictor-
corrector method. We consider the epidemic phenomenon proposed in [29], which
is an extended version of the SI epidemic model with a nonlinear incidence
$\varphi$, and we include the fractional derivative which involving Mittag-
Leffer kernel. Thus, the novel system is described as follows:

$\begin{cases}{}^{ABC}_{\quad\,0}D^{\alpha}_{t}u(t)=\Lambda-\gamma
u(t)\varphi(v(t))-\mu u(t),&t>0,\\\
{}^{ABC}_{\quad\,0}D^{\alpha}_{t}v(t)=\gamma u(t)\varphi(v(t))-\sigma
v(t),&t>0,\\\ u(0)=u_{0}>0,\ v(0)=v_{0}>0,\end{cases}$ (64)

where $u(t)$, $v(t)$ represent the numbers of susceptible individuals, and
infective individuals at time t, respectively. $\Lambda$ is the birth rate of
the population, $\gamma$ is the disease transmission rate, $\mu$ is the
natural death rate, and $\sigma:=\widetilde{\sigma}+\mu$, with
$\widetilde{\sigma}$ is the death rate due to disease, which all are positive
real numbers, and $\alpha\in(0,1)$. The incidence function $\varphi(v)$
introduces a nonlinear relation between the susceptible individuals and
infective individuals. We suppose that $\varphi\in
C^{1}(\mathbb{R}_{+};\mathbb{R}_{+})$, and satisfies:

$\begin{cases}\varphi(0)=0,\\\ 0<v\varphi^{\prime}(v)\leq\varphi(v),&\forall
v>0.\end{cases}$ (65)

According to the above condition and Lemma 1 in [30], it is classical task to
prove the existence of unique solution for system (64) (see e.g. [31, 32]). As
well as, the system (64) has a disease-free equilibrium $E_{0}=(u^{0},0)$ with
$u^{0}:=\displaystyle\frac{\Lambda}{\mu}$, and a unique endemic equilibrium
$E^{*}:=(u^{*},v^{*})$ if
$R_{0}:=\displaystyle\frac{\Lambda\gamma}{\mu\sigma}\varphi^{\prime}(0)>1$
(cf. [29]). In the remainder, the following notation
$\mathbb{R}_{+}:=\left[0,+\infty\right)$ will be useful. The feasible region
of the suggested model (64) is given by

$\Upsilon:=\left\\{(x_{1},x_{2})\in\mathbb{R}^{2}_{+}:x_{1}+x_{2}\leq\frac{\Lambda}{\mu}\right\\}.$
(66)

Now, we investigate the global asymptotic stability of the equilibrium points
$E_{0}$ and $E^{*}$, which is determined by the reproduction number $R_{0}$,
as a result, the cases $R_{0}\leqslant 1$ and $R_{0}>1$ are treated
independently.
Firstly, we provide the global stability result of the disease-free
equilibrium $E_{0}$.

###### Theorem 5

Let $\alpha\in(0,1)$ and $R_{0}\leqslant 1$. Assume that (65) holds. Then, the
disease-free equilibrium $E_{0}$ is globally asympotatically stable in the
feasible region $\Upsilon$.

###### Proof 5

We begin by rewriting the first equation of the system (64), with taking into
account this relationship $\Lambda=\mu u^{0}$, such that

${}^{ABC}_{\quad\,0}D^{\alpha}_{t}u(t)=-\Big{(}\mu+\gamma\varphi(v)\Big{)}(u-u_{0})-\gamma\varphi(v)u_{0}$
(67)

We consider the following candidate Lyapunov function:

$V:\left\\{(u,v)\in\Upsilon:u>0\right\\}\longrightarrow\mathbb{R}_{+},$ (68)

such that

$V(u,v)=\frac{(u-u^{0})^{2}}{2u^{0}}+v.$ (69)

Thanks to the property of ABC-fractional derivative given in Lemma 3.1 from
[33], we get

$\begin{split}{}^{ABC}_{\quad\,0}D^{\alpha}_{t}V\leq\frac{(u-u^{0})}{u^{0}}{}^{ABC}_{\quad\,0}D^{\alpha}_{t}u(t)+{}^{ABC}_{\quad\,0}D^{\alpha}_{t}v(t).\end{split}$
(70)

Then, by employing (64)-(66), and Lemma 1 [30], we obtain

$\displaystyle{}^{ABC}_{\quad\,0}D^{\alpha}_{t}V\leq$
$\displaystyle-\left(\mu+\gamma\varphi(v)\right)\frac{(u-u^{0})^{2}}{u^{0}}-\gamma
u^{0}\varphi(v)+\gamma u\varphi(v)-\sigma v,$ $\displaystyle\leq$
$\displaystyle-\left(\mu+\gamma\varphi(v)\right)\frac{(u-u^{0})^{2}}{u^{0}}+\gamma\varphi^{\prime}(0)u^{0}v-\sigma
v,$ $\displaystyle=$
$\displaystyle-\left(\mu+\gamma\varphi(v)\right)\frac{(u-u^{0})^{2}}{u^{0}}+\left(\frac{\gamma\Lambda\varphi^{\prime}(0)}{\mu}-\sigma\right)v,$
$\displaystyle=$
$\displaystyle-\left(\mu+\gamma\varphi(v)\right)\frac{(u-u^{0})^{2}}{u^{0}}-\sigma\left(1-R_{0}\right)v.$
(71)

Therefore, $R_{0}\leqslant 1$, guarantees that
${}^{ABC}_{\quad\,0}D^{\alpha}_{t}V\leq 0$ for $\alpha\in(0,1)$. Consequently,
by the Lyapunov stability theorem (see Theorem 2.7 from [33] and [34]), we
conclude that the disease-free equilibrium $E^{0}$ is globally asymptotically
stable in $\Upsilon$.

Secondly, we provide the global stability result of the endemic equilibrium
$E^{*}$. Before do that, we state a lemma which will be useful (see Lemma 2
from [29]):

###### Lemma 6

Assume that the function $\varphi$ satisfied (65) and

$\begin{matrix}F(x)=x-1-ln(x),&\forall x>0.\end{matrix}$ (72)

We thus get the following inequality

$\begin{matrix}F\left(\displaystyle\frac{\varphi(v)}{\varphi(v^{*})}\right)\leq
F\left(\displaystyle\frac{v}{v^{*}}\right),&\forall v>0,\end{matrix}$ (73)

where $v^{*}$ is the second component of the equilibrium point $E^{*}$.

###### Theorem 6

Let $\alpha\in(0,1)$ and $R_{0}>1$. Suppose that (65) holds. Then, the endemic
equilibrium $E^{*}$ is globally asymptotically stable in interior of the
feasible region $\Upsilon$.

###### Proof 6

The first two equation of system (64) at $E^{*}$, clearly give

$\begin{cases}\Lambda=\mu u^{*}+\gamma u^{*}\varphi(v^{*}),\\\
v^{*}\sigma=\gamma u^{*}\varphi(v^{*}).\end{cases}$ (74)

It follows that

$\begin{cases}{}^{ABC}_{\quad\,0}D^{\alpha}_{t}u=\mu
u^{*}\Big{(}1-\displaystyle\frac{u}{u^{*}}\Big{)}+\gamma
u^{*}\varphi(v^{*})\left(1-\displaystyle\frac{u\varphi(v)}{u^{*}\varphi(v^{*})}\right),\\\
{}^{ABC}_{\quad\,0}D^{\alpha}_{t}v=\gamma
u^{*}\varphi(v^{*})\left(\displaystyle\frac{u\varphi(v)}{u^{*}\varphi(v^{*})}-\displaystyle\frac{v}{v^{*}}\right).\end{cases}$
(75)

We consider the following candidate Lyapunov function:

$L:\left\\{(u,v)\in\Upsilon:u>0,v>0\right\\}\longrightarrow\mathbb{R}_{+},$
(76)

such that

$L(u,v)=u^{*}F\left(\displaystyle\frac{u}{u^{*}}\right)+v^{*}F\left(\displaystyle\frac{v}{v^{*}}\right).$
(77)

We now apply the property of ABC-fractional derivative given in Lemma 3.2 from
[33], to obtain

${}^{ABC}_{\quad\,0}D^{\alpha}_{t}L\leq\left(1-\displaystyle\frac{u^{*}}{u}\right){}^{ABC}_{\quad\,0}D^{\alpha}_{t}u+\left(1-\displaystyle\frac{v^{*}}{v}\right){}^{ABC}_{\quad\,0}D^{\alpha}_{t}v$
(78)

From (75) and by immediately computation, we get

$\displaystyle{}^{ABC}_{\quad\,0}D^{\alpha}_{t}L$ $\displaystyle\leq\mu
u^{*}\left(1-\displaystyle\frac{u}{u^{*}}\right)\left(1-\displaystyle\frac{u^{*}}{u}\right)+\gamma
u^{*}\varphi(v^{*})\left(\displaystyle\frac{u\varphi(v)}{u^{*}\varphi(v^{*})}-\frac{v}{v^{*}}\right)\left(1-\displaystyle\frac{v^{*}}{v}\right)$
$\displaystyle\ \ \ +\gamma
u^{*}\varphi(v^{*})\left(1-\frac{u\varphi(v)}{u^{*}\varphi(v^{*})}\right)\left(1-\displaystyle\frac{u^{*}}{u}\right),$
$\displaystyle\leq-\mu
u^{*}\left(F\Big{(}\frac{u}{u^{*}}\Big{)}+F\left(\frac{u^{*}}{u}\right)\right)-\gamma
u^{*}\varphi(v^{*})\left(F\left(\frac{u^{*}}{u}\right)+F\left(\frac{v^{*}u\varphi(v)}{u^{*}v\varphi(v^{*})}\right)\right)$
(79) $\displaystyle\ \ \ +\gamma
u^{*}\varphi(v^{*})\left(F\left(\frac{\varphi(v)}{\varphi(v^{*})}\right)-F\left(\frac{v}{v^{*}}\right)\right).$
(80)

Therefore, the nonnegativity of function $F$ (defined in (72)) on
$(0,+\infty)$, and the lemma 6 ensure that
${}^{ABC}_{\quad\,0}D^{\alpha}_{t}L\leq 0$ for $\alpha\in(0,1)$. Consequently,
by the Lyapunov stability theorem (see Theorem 2.7 and Theorem 2.8 [33] as
well as [34]), we get the desired result.

Next, we present some numerical simulation of the general model (64) through
several sets of parameters (see table 5) to examine the effectiveness of the
proposed predictor-corrector scheme in the current work (i.e. PPC (15)-(21)),
as well as the feasibility of the theoretical findings in this example.

Table 5: Diverse sets of system’s (64) parameter values used in numerical simulations. Set of parameters |  $\Lambda$ |  $\gamma$ |  $\mu$ |  $\widetilde{\sigma}$ |  $R_{0}\ (\text{with }\varphi^{\prime}(0)=1)$
---|---|---|---|---|---
Set $1$ |  $0.3$ |  $0.1$ |  $0.44$ |  $0.38$ |  $0.0831$
Set $2$ |  $0.03$ |  $0.4$ |  $0.0365$ |  $0.0135$ |  $6.5753$
Set $3$ |  $0.3$ |  $0.1$ |  $0.32$ |  $0.5$ |  $0.1143$
Set $4$ |  $0.03$ |  $0.4$ |  $0.032$ |  $0.018$ |  $7.5$

The following is a description of the results:

  * 1.

According to the first set of parameters in table 5, with $\varphi(v)=v\
\left(\text{resp. }\varphi(v)=\frac{v}{1+0.01v}\right)$, and
$(u_{0},v_{0})=\left(0.52,\frac{\Lambda}{\mu}-0.52\right)$, the Theorem 5
guarantees (for $\alpha\in(0,1)$ and any positive initial data belongs to
$\Upsilon$) the global asymptotic stability of the disease-free equilibrium
$E_{0}=(u_{0},0)=(0.6818,0)$, which is confirmed by the numerical results
depicted in Figure 5 (resp. Figure 7) with various values of fractional order
$\alpha\in\left\\{0.8,0.85,0.9,0.99\right\\}$.

  * 2.

According to the second set of parameters in table 5, with $\varphi(v)=v\
\left(\text{resp. }\varphi(v)=\frac{v}{1+0.01v}\right)$, where
$(u_{0},v_{0})=\left(0.6,\frac{\Lambda}{\mu}-0.6\right)$, the Theorem 6
guarantees (for $\alpha\in(0,1)$ and any positive initial data belongs to
$\Upsilon$) the global asymptotic stability of the endemic equilibrium
$E^{*}=(u^{*},v^{*})=(0.125,0.5087)$ (resp. $E^{*}=(0.1256,0.5083)$), which is
confirmed by the numerical results depicted in Figure 6 (resp. Figure 8) with
various values of fractional order
$\alpha\in\left\\{0.8,0.85,0.9,0.99\right\\}$.

  * 3.

According to the third set of parameters in table 5, with $\varphi(v)=v\
\left(\text{resp. }\varphi(v)=\frac{v}{1+0.01v}\right)$, and $\alpha=0.99$,
the Theorem 5 guarantees (for any positive initial data belongs to $\Upsilon$)
the global asymptotic stability of the disease-free equilibrium
$E_{0}=(u_{0},0)=(0.9375,0)$, which is confirmed by the numerical results
depicted in Figure 9 (resp. Figure 11) with various values of initial data.

  * 4.

According to the fourth set of parameters in table 5, with $\varphi(v)=v\
\left(\text{resp. }\varphi(v)=\frac{v}{1+0.01v}\right)$, and $\alpha=0.99$,
the Theorem 6 guarantees (for any positive initial data belongs to $\Upsilon$)
the global asymptotic stability of the endemic equilibrium
$E^{*}=(u^{*},v^{*})=(0.125,0.52)$ (resp. $E^{*}=(0.1256,0.0.5196)$), which is
confirmed by the numerical results depicted in Figure 10 (resp. Figure 12)
with various values of initial data.

Figure 5: The approximate solution of model (64) by PPC (15)-(21), subject to
the first set of parameters in table 5, with $\varphi(v)=v$ and
$(u_{0},v_{0})=\left(0.52,\frac{\Lambda}{\mu}-0.52\right)$, considering
different values of fractional order $\alpha$. Figure 6: The approximate
solution of model (64) by PPC (15)-(21), subject to the second set of
parameters in table 5, with $\varphi(v)=v$ and
$(u_{0},v_{0})=\left(0.6,\frac{\Lambda}{\mu}-0.6\right)$, considering
different values of fractional order
$\alpha\in\left\\{0.8,0.85,0.9,0.99\right\\}$. Figure 7: The approximate
solution of model (64) by PPC (15)-(21), subject to the first set of
parameters in table 5, with $\varphi(v)=\displaystyle\frac{v}{1+0.01v}$ and
$(u_{0},v_{0})=\left(0.52,\frac{\Lambda}{\mu}-0.52\right)$, considering
different values of fractional order
$\alpha\in\left\\{0.8,0.85,0.9,0.99\right\\}$. Figure 8: The approximate
solution of model (64) by PPC (15)-(21), subject to the second set of
parameters in table 5, with $\varphi(v)=\displaystyle\frac{v}{1+0.01v}$ and
$(u_{0},v_{0})=\left(0.6,\frac{\Lambda}{\mu}-0.6\right)$, considering
different values of fractional order
$\alpha\in\left\\{0.8,0.85,0.9,0.99\right\\}$. Figure 9: The approximate
solution of model (64) by PPC (15)-(21), subject to the third set of
parameters in table 5, with $\varphi(v)=v$ and $\alpha=0.99$, considering
different values of initial data
$(u_{0},v_{0})\in\left\\{\left(0.8,\frac{\Lambda}{\mu}-0.8\right),\left(0.7,\frac{\Lambda}{\mu}-0.7\right),\left(0.6,\frac{\Lambda}{\mu}-0.6\right),\left(0.25,\frac{\Lambda}{\mu}-0.25\right)\right\\}$.
Figure 10: The approximate solution of model (64) by PPC (15)-(21), subject to
the fourth set of parameters in table 5, with $\varphi(v)=v$ and
$\alpha=0.99$, considering different values of initial data
$(u_{0},v_{0})\in\left\\{\left(0.8,\frac{\Lambda}{\mu}-0.8\right),\left(0.7,\frac{\Lambda}{\mu}-0.7\right),\left(0.6,\frac{\Lambda}{\mu}-0.6\right),\left(0.25,\frac{\Lambda}{\mu}-0.25\right)\right\\}$.
Figure 11: The approximate solution of model (64) by PPC (15)-(21), subject to
the third set of parameters in table 5, with
$\varphi(v)=\displaystyle\frac{v}{1+0.01v}$ and $\alpha=0.99$, considering
different values of initial data
$(u_{0},v_{0})\in\left\\{\left(0.8,\frac{\Lambda}{\mu}-0.8\right),\left(0.7,\frac{\Lambda}{\mu}-0.7\right),\left(0.6,\frac{\Lambda}{\mu}-0.6\right),\left(0.25,\frac{\Lambda}{\mu}-0.25\right)\right\\}$.
Figure 12: The approximate solution of model (64) by PPC (15)-(21), subject to
the fourth set of parameters in table 5, with
$\varphi(v)=\displaystyle\frac{v}{1+0.01v}$ and $\alpha=0.99$, considering
different values of initial data
$(u_{0},v_{0})\in\left\\{\left(0.8,\frac{\Lambda}{\mu}-0.8\right),\left(0.7,\frac{\Lambda}{\mu}-0.7\right),\left(0.6,\frac{\Lambda}{\mu}-0.6\right),\left(0.25,\frac{\Lambda}{\mu}-0.25\right)\right\\}$.

## Acknowledgement

The authors are grateful to Pr. Bongsoo Jang and Mr. Junseo Lee for sharing
the algorithms file related to the methods developed in [18]. The work of S.
Aljhani is supported by Taibah University - Saudi Arabia.

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## Appendix A. Start-up of the Scheme

To find a desired accuracy for $\widetilde{y}_{1}$ and $\widetilde{y}_{2}$, we
find the approximate solutions at points $t_{\frac{1}{4}}$ and
$t_{\frac{1}{2}}$ using the constant, linear and quadratic interpolation. Let
$I^{0}(f_{a})$ be the constant interpolation of $f$ at point $t=a$ that is
$I^{0}(f_{a})=f(a)$. Let $I^{1}(f_{a},f_{b})$ and $I^{2}(f_{a},f_{b},f_{c})$
be linear and quadratic interpolation of $f$ with grids $(a,b)$ and $(a,b,c)$
respectively. In the algorithm below we describe how to find the approximate
solution of $y(t)$ at points $t_{\frac{1}{4}},t_{\frac{1}{2}},t_{1},$ and
$t_{2}$ (i.e.
$\widetilde{y}_{\frac{1}{4}},\widetilde{y}_{\frac{1}{2}},\widetilde{y}_{1},$
and $\widetilde{y}_{2}$) using a predictor-corrector scheme:

  * 1.

Approximate solution of $y(t)$ at point $t_{\frac{1}{4}}$:

$\displaystyle\widetilde{y}^{P}_{\frac{1}{4}}$
$\displaystyle=y_{0}+AB_{f}\widetilde{f}(t_{\frac{1}{4}},\widetilde{y}_{0})+AB_{y}\int_{0}^{\frac{1}{4}}(t_{\frac{1}{4}}-s)^{\alpha-1}I^{0}(\widetilde{f}_{0})ds,$
$\displaystyle\widetilde{y}_{\frac{1}{4}}$
$\displaystyle=y_{0}+AB_{f}\widetilde{f}^{P}_{\frac{1}{4}}+AB_{g}\int_{0}^{t_{\frac{1}{4}}}I^{1}(\widetilde{f}_{0},\widetilde{f}^{P}_{\frac{1}{4}})(t_{\frac{1}{4}}-s)^{\alpha-1}ds.$

  * 2.

Approximate solution of $y(t)$ at point $t_{\frac{1}{2}}$:

$\displaystyle\widetilde{y}^{P_{1}}_{\frac{1}{2}}$
$\displaystyle=y_{0}+AB_{f}(2\widetilde{f}_{\frac{1}{4}}-\widetilde{f}_{0})+AB_{g}\int_{0}^{t_{\frac{1}{2}}}I^{0}(\widetilde{f}_{\frac{1}{4}})(t_{\frac{1}{2}}-s)^{\alpha-1}ds,$
$\displaystyle\widetilde{y}^{P_{2}}_{\frac{1}{2}}$
$\displaystyle=y_{0}+AB_{f}\widetilde{f}^{P_{1}}_{\frac{1}{2}}+AB_{g}\int_{0}^{t_{\frac{1}{2}}}I^{1}(\widetilde{f}_{\frac{1}{4}},\widetilde{f}^{P_{1}}_{\frac{1}{2}})(t_{\frac{1}{2}}-s)^{\alpha-1}ds,$
$\displaystyle\widetilde{y}_{\frac{1}{2}}$
$\displaystyle=y_{0}+AB_{f}\widetilde{f}^{P_{2}}_{\frac{1}{2}}+AB_{g}\int_{0}^{t_{\frac{1}{2}}}I^{2}(\widetilde{f}_{0},\widetilde{f}_{\frac{1}{4}},\widetilde{f}^{P_{2}}_{\frac{1}{2}})(t_{\frac{1}{2}}-s)^{\alpha-1}ds.$

  * 3.

Approximate solution of $y(t)$ at point $t_{1}$:

$\displaystyle\widetilde{y}^{P_{1}}_{1}$
$\displaystyle=y_{0}+AB_{f}(6\widetilde{f}_{\frac{1}{2}}-8\widetilde{f}_{\frac{1}{4}}-3\widetilde{f}_{0})+AB_{g}\int_{0}^{t_{n}}I^{0}(\widetilde{f}_{\frac{1}{2}})(t_{n}-s)^{\alpha-1}ds,$
$\displaystyle\widetilde{y}^{P_{2}}_{1}$
$\displaystyle=y_{0}+AB_{f}\widetilde{f}^{P_{1}}_{1}+AB_{g}\int_{0}^{t_{1}}I^{1}(\widetilde{f}_{\frac{1}{2}},\widetilde{f}^{P_{1}}_{1})(t_{1}-s)^{\alpha-1}ds,$
$\displaystyle\widetilde{y}_{1}$
$\displaystyle=y_{0}+AB_{f}\widetilde{f}^{P_{2}}_{1}+AB_{g}\int_{0}^{t_{1}}I^{2}(\widetilde{f}_{0},\widetilde{f}_{\frac{1}{2}},\widetilde{f}^{P_{2}}_{1})(t_{1}-s)^{\alpha-1}ds.$

  * 4.

Approximate solution of $y(t)$ at point $t_{2}$:

$\displaystyle\widetilde{y}^{P_{1}}_{2}$
$\displaystyle=y_{0}+AB_{f}(6\widetilde{f}_{1}-8\widetilde{f}_{\frac{1}{2}}-3\widetilde{f}_{0})+\int_{0}^{t_{2}}I^{0}(\widetilde{f}_{1})(t_{2}-s)^{\alpha-1}ds,$
$\displaystyle\widetilde{y}^{P_{2}}_{2}$
$\displaystyle=y_{0}+AB_{f}\widetilde{f}^{P_{1}}_{2}+\int_{0}^{t_{2}}I^{1}(\widetilde{f}_{1},\widetilde{f}^{P_{1}}_{2})(t_{2}-s)^{\alpha-1}ds,$
$\displaystyle\widetilde{y}_{2}$
$\displaystyle=y_{0}+AB_{f}\widetilde{f}^{P_{2}}_{2}+\int_{0}^{t_{2}}I^{2}(\widetilde{f}_{0},\widetilde{f}_{1},\widetilde{f}^{P_{2}}_{2})(t_{2}-s)^{\alpha-1}ds.$

where

$\Biggl{\\{}AB_{f}=\frac{1-\alpha}{AB(\alpha)}\,\,\,\,\,\textit{and}\,\,\,\,\,AB_{g}=\frac{\alpha}{AB(\alpha)\Gamma(\alpha)}\Biggr{\\}}.$


# Flood Prediction Using Classical and Quantum Machine Learning Models

Marek Grzesiak* Ekipa, Dusseldorf, Germany
AGH University of Science and Technology, Krakow, Poland
Engineering Professors Council, England  Param Thakkar  Veermata Jijabai
Technological Institute, Mumbai, India

###### Abstract

This study investigates the potential of quantum machine learning (QML) to
improve flood forecasting. We focus on daily flood events along Germany’s
Wupper River in 2023. Our approach combines classical machine learning (SVM,
KNN, regression, AR models) with QML techniques (Adaboost, Quantum Variational
Circuits, QBoost, $QSVC\\_ML$). This hybrid model leverages quantum properties
like superposition and entanglement to achieve better accuracy and efficiency.
Classical and QML models are compared based on training time, accuracy, and
scalability. Results show that QML models offer competitive training times and
improved prediction accuracy. This research signifies a step towards utilizing
quantum technologies for climate change adaptation. We emphasize collaboration
and continuous innovation to implement this model in real-world flood
management, ultimately enhancing global resilience against floods.

##  1 Introduction

Flooding is a major natural disaster affecting millions worldwide, and its
prediction remains a significant challenge. Accurate flood forecasting is
essential for mitigating the adverse effects on human lives and
infrastructure. This project investigates the application of Quantum Machine
Learning (QML) to enhance flood prediction accuracy and efficiency,
specifically focusing on the Wupper River in Germany during 2023.

Traditional flood prediction models rely on classical machine learning
techniques such as Support Vector Machines (SVM), K-Nearest Neighbors (KNN),
regression, and Autoregressive (AR) models. While effective, these methods
face limitations in handling large datasets and complex patterns inherent in
environmental data. QML offers a promising alternative by exploiting quantum
phenomena like superposition and entanglement, which enable the processing of
vast amounts of data at unprecedented speeds.

Our approach integrates classical and quantum models, including SVM, KNN,
Adaboost, Quantum Variational Circuits, QBoost, and QSV-C_ML, to develop a
hybrid system for flood prediction. By comparing the performance of classical
and QML models based on training time, accuracy, and scalability, we aim to
demonstrate the superiority of QML in handling intricate flood prediction
tasks.

The results indicate that QML models not only enhance prediction accuracy but
also reduce computational time, making them a viable option for real-time
flood forecasting. This research underscores the potential of quantum
technologies in addressing climate-related challenges and highlights the
importance of interdisciplinary collaboration in advancing environmental
science.

##  2 Model Descriptions

In this section, we describe the various models used in our study, including
both classical and quantum machine learning techniques, and explain how they
were applied to flood predictions.

###  2.1 Classical Machine Learning Models

####  2.1.1 Support Vector Machines (SVM)

SVM is a supervised learning model used for binary classification tasks. In
flood prediction, SVM was employed to classify the likelihood of flooding
events based on historical and real-time data. The model works by finding the
optimal hyperplane that separates the data into different classes, indicating
whether a flood is likely to occur or not. SVMs are effective in high-
dimensional spaces and are versatile due to the different kernel functions
that can be used to customize the decision boundary.

####  2.1.2 K-Nearest Neighbors (KNN)

KNN is a simple yet powerful supervised learning algorithm used for both
classification and regression tasks. In our study, KNN was applied to predict
flood events by finding the most similar historical instances (nearest
neighbors) to the current data point. This approach helps in determining the
probability of flood occurrence based on similarity measures and can handle
non-linear relationships in the data effectively.

####  2.1.3 Learning Progression

To visualize the learning progression of the SVM model, a learning curve graph
is presented below. This graph shows the training and cross-validation scores
as a function of the number of training examples.

Figure 1: Learning Curve for SVM Model

####  2.1.4 Classical Regression Model

Regression models predict continuous outcomes based on input variables. We
utilized linear regression to model the relationship between various
hydrological parameters (e.g., rainfall, river discharge) and flood levels.
This approach helps in understanding and predicting the magnitude of potential
floods, allowing for more precise flood warnings. Linear regression assumes a
linear relationship between the input variables and the output, making it
suitable for modeling simple relationships in the data.

####  2.1.5 AutoRegressive (AR) Model

AR models are used for time series forecasting by regressing the variable on
its own lagged values. For flood prediction, AR models were applied to
historical flood data to predict future flood levels. These models are
particularly useful in capturing the temporal dependencies and trends in flood
occurrences, providing a basis for anticipating future flooding events. AR
models are simple yet powerful, making them suitable for analyzing and
forecasting time series data.

###  2.2 Quantum Machine Learning Models

####  2.2.1 Adaboost

Adaboost is an ensemble learning technique that combines multiple weak
classifiers to form a strong classifier. In our study, we used Adaboost with
quantum-enhanced decision stumps, leveraging quantum parallelism to improve
the boosting process. This model helped in enhancing the accuracy of flood
prediction by combining the strengths of individual quantum classifiers.
Adaboost iteratively trains weak classifiers on subsets of the data, assigning
higher weights to misclassified data points to focus on difficult-to-classify
instances.

####  2.2.2 Decision Tree

A decision tree is a flowchart-like structure where each internal node
represents a feature, each branch represents a decision rule, and each leaf
node represents the outcome. A quantum-enhanced decision tree was used for
binary classification in flood prediction. This model leverages the quantum
computational power to handle complex patterns in the data, making more
accurate predictions about the likelihood of flooding events. Decision trees
are interpretable and can handle both numerical and categorical data, making
them suitable for a wide range of applications.

####  2.2.3 Random Forest

Random Forest is an ensemble method that uses multiple decision trees to
improve prediction accuracy. By incorporating quantum techniques, our quantum-
enhanced random forest model was able to process data more efficiently and
provide better flood prediction outcomes compared to classical random forest
models. Random forests are robust against overfitting and perform well with
large datasets, making them suitable for complex classification and regression
tasks.

####  2.2.4 QBoost

QBoost is a quantum version of the classical boosting algorithm, designed to
enhance the performance of weak quantum classifiers. In our study, QBoost was
used to aggregate multiple quantum classifiers, each trained on different
subsets of flood data, to form a robust prediction model. This approach
leverages quantum superposition and entanglement to achieve higher accuracy
and faster convergence. Boosting combines multiple weak classifiers to create
a strong classifier, iteratively giving more weight to misclassified data
points to focus on difficult-to-classify instances.

####  2.2.5 QBoostPlus

QBoostPlus extends QBoost by incorporating additional optimization techniques
to further enhance model performance. This model was applied to flood
prediction by combining the predictions of multiple quantum classifiers,
resulting in even more accurate and reliable flood forecasts. QBoostPlus
optimizes the boosting process by considering the contribution of each weak
classifier based on its performance, leading to improved overall model
accuracy.

####  2.2.6 QSVC_ML

QSVC_ML is a quantum-enhanced version of the Support Vector Machine (SVM)
classifier, designed to leverage quantum computational advantages such as
superposition and entanglement. In our study, QSVC_ML was employed to classify
flood events using quantum-enhanced decision boundaries, potentially offering
improved accuracy and efficiency compared to classical SVM.

####  2.2.7 Quantum Regression Algorithm

Quantum regression algorithms predict continuous outcomes using quantum
techniques. We employed a quantum regression model to analyze the relationship
between hydrological parameters and flood levels, taking advantage of quantum
computational power to improve prediction accuracy and efficiency. Quantum
regression algorithms use quantum gates and circuits to process input data and
generate predictions, offering potential advantages over classical regression
models in terms of computational speed and memory efficiency.

####  2.2.8 Quantum AutoReg/Model-B Quantum Neural Network

Quantum AutoReg and Model-B Quantum Neural Network are advanced quantum models
designed for time series forecasting. These models were applied to historical
flood data to predict future flood levels, leveraging quantum parallelism to
process temporal dependencies and trends more effectively than classical AR
models. Quantum neural networks use quantum gates and layers to process
sequential data, capturing complex patterns and relationships in time series
data more accurately than classical neural networks.

###  2.3 Application to Flood Predictions

The application of these models to flood prediction involved several steps:

  1. 1.

Data Collection: Historical and real-time hydrological data from the Wupper
River were collected, including parameters such as rainfall, river discharge,
and previous flood events. The datasets used in our study were obtained
through a combination of historical records and real-time monitoring.
Historical hydrological data for the Wupper River, including rainfall
measurements, river discharge rates, and past flood events, were sourced from
government agencies responsible for water resource management. Additionally,
real-time data streams from monitoring stations along the river were accessed
to provide up-to-date information on current hydrological conditions. These
datasets were aggregated and processed to create a comprehensive dataset for
training and testing our flood prediction models. Historical river and
meteorological data were leveraged from the Wupperverband, accessible through
their website at https://fluggs.wupperverband.de/swc/. Furthermore, additional
data from 2010 through the end of 2023, including topographic data, were
accessed from NASA-Earthdata websites, specifically the Hillshading map of the
described area (Dataset: NASA SRTM3 SRTMGL1), available at
https://urs.earthdata.nasa.gov/home and https://earthexplorer.usgs.gov/.

  2. 2.

Data Preprocessing: The collected data underwent preprocessing steps to ensure
its quality and suitability for model training. This involved cleaning the
data to remove inconsistencies and outliers, normalizing the features to a
common scale, and splitting the data into training and test sets to facilitate
model evaluation.

  3. 3.

Model Training: Both classical machine learning models (SVM, regression, AR)
and quantum machine learning models (Adaboost, Decision Tree, Random Forest,
QBoost, QBoostPlus, Quantum Regression, Quantum AutoReg) were trained on the
prepared datasets. During training, hyperparameters for each model were
optimized to maximize predictive performance.

  4. 4.

Model Evaluation: The trained models were evaluated on the test dataset to
assess their predictive accuracy and performance. Evaluation metrics such as
accuracy, mean squared error, and training time were used to gauge the
effectiveness of each model in predicting flood events.

  5. 5.

Comparison: Finally, the performance of classical and quantum machine learning
models was compared to determine the advantages of using quantum machine
learning techniques for flood prediction. This comparative analysis provided
insights into the efficacy of quantum models in handling complex hydrological
data and improving flood forecasting accuracy.

  6. 6.

Selection of QML Algorithms: The selection of quantum machine learning (QML)
algorithms was based on several factors, including the nature of the dataset,
the complexity of the prediction task, and the computational resources
available. We conducted a thorough analysis of various QML algorithms,
considering their strengths and weaknesses in handling temporal dependencies,
handling high-dimensional data, and exploiting quantum parallelism. Based on
this analysis, we selected QBoost, QBoostPlus, Quantum Regression, and Quantum
AutoReg/Model-B Quantum Neural Network for experimentation, as these
algorithms showed promise in addressing the specific challenges posed by flood
prediction tasks. Additionally, Long Short-Term Memory (LSTM) networks were
considered due to their capability to capture temporal dependencies in
sequential data. However, initial experiments revealed that LSTM had weaker
accuracy compared to AutoReg. Therefore, we chose AutoReg for its superior
performance in our dataset. It’s important to note that our model training was
limited to the data obtained from the provider, and we plan to explore further
experiments to compare the performance of LSTM and AutoReg on an extended
dataset in the future. There is still ample room for improvement in flood
prediction models, and ongoing research aims to enhance the accuracy and
efficiency of these algorithms by incorporating additional data sources,
refining model architectures, and exploring novel quantum computing
techniques.

##  3 Exploratory Data Analysis (EDA) Visualizations

To understand the characteristics of the dataset and identify patterns, we
performed Exploratory Data Analysis (EDA). Here are some key visualizations:

Figure 2: Distribution of Flood Events

Figure 3: Time Series of Water Level

Figure 4: Seasonal Patterns in River Data

Figure 5: Time Series of Weather Data

Figure 6: Seasonal Patterns in Weather Data

Figure 7: Distribution of Precipitation Across Weather Stations

Figure 8: Outlier Detection in River Data

Figure 9: Distribution of River Data

Figure 10: Distribution of Precipitation (RS) in Weather Data

Figure 11: Relationship Between Water Levels and Precipitation Form

The flood warning classification model has been optimized with a threshold of
90 cm, resulting in increased sensitivity. The dataset contains 71,239 flood
events and 1,160,471 non-flood events, representing approximately 5.78% and
94.22% of all recorded events, respectively. The time series plot illustrates
fluctuations in water levels over time, with an average water level of 37 cm
and a high-value threshold of 376 cm. Seasonal analysis shows distinct
patterns: Winter months exhibit higher water levels, peaking in December and
January. Spring months show a decline in water levels, with slight
fluctuations. Summer months show stable water levels, with minor fluctuations.
Autumn months show rising water levels, peaking in September and October. The
time series plot of precipitation (RS) data shows variations in precipitation
levels over time, with an average rate of 3.10 mm/day and a maximum of 90.80
mm in a day. Annual precipitation analysis indicates variability over time,
with an average of 3.10 units and maximum and minimum values of 4.15 and 2.14
units, respectively. Boxplot visualization highlights variability in rainfall
across weather stations, with outliers observed in the data. Histogram
visualizes the distribution of precipitation (RS) data, indicating variability
in rainfall measurements. Average rainfall is 3.10 mm, with a maximum recorded
rainfall of 90.80 mm. Boxplot visualization identifies outliers in river data,
with 136,432 outliers detected. Histogram illustrates the distribution of
river level measurements, indicating a right-skewed distribution with
significant variability. Histogram showcases the distribution of river level
measurements, indicating a right-skewed distribution with a fat tail towards
higher values. Average water level is 37 cm, with significant variability in
the data. Histogram visualizes the distribution of precipitation (RS) data,
indicating variability in rainfall measurements. Average rainfall is 3.10 mm,
with a maximum recorded rainfall of 90.80 mm. Boxplot visualization explores
the relationship between water levels and precipitation form (RSF), indicating
variations in water levels across different precipitation types.

##  4 Performance Comparison

In this section, we present graphs and tables showing the performance of the
models on the dataset used. The performance of classical algorithms and
quantum algorithms is compared using separate tables.

###  4.1 Results

Compared accuracy for each model we used:

###  4.2 Accuracy of Classical Solutions

SVM Model (binary classification)
---
Training Time | 0.094 seconds
Accuracy | 99.8%
Confusion Matrix | [[243, 0], [0, 32]]
True Positives (TP) | 32
True Negatives (TN) | 243
False Positives (FP) | 0
False Negatives (FN) | 0
Mean Absolute Error (MAE) for 2023 | 0.0028
Mean Squared Error (MSE) for 2023 | 0.0004
SVM and KNN Models (binary classification)
Mean Squared Error (MSE) | SVM: 0.0635, KNN: 0.0635
Classical Regression Model (for regression)
Random Forest R-squared | 0.046
Gradient Boosting R-squared | 0.04
AutoReg Model (for future flood prediction)
Mean Squared Error (MSE) | 0.907
Table 1: Performance metrics for classical models

###  4.3 Accuracy of Quantum Solutions

Quantum Machine Learning Model (binary classification using quantum
techniques)
---
Adaboost Accuracy (test batch) | 97%
Decision Tree Accuracy (test batch) | 96%
Random Forest Accuracy (test batch) | 94%
QBoost Accuracy (test batch) | 2%
QBoostPlus Accuracy (test batch) | 94%
Qiskit QSVC ML Model (binary classification)
Accuracy | 97%
Precision | 0.97
Recall | 1.00
F1-score | 0.99
Balanced Accuracy | 0.50
Quantum Regression Algorithm (for regression using quantum techniques)
Accuracy | 58%
Performance on test data | Loss: -0.487 | Accuracy: 49.5%
Quantum AutoReg/Model-B Quantum Neural Network (for flood forecasting using
quantum techniques)
Iterations | 1000
Cost | 1.015
Table 2: Performance metrics for quantum models

##  5 Data Availability

The data supporting the findings of this study are available from the
following sources:

  * •

Wupperverband Data: Historical river and meteorological data used in this
study were obtained from Wupperverband. The data can be accessed at
Wupperverband SWC.

  * •

NASA Earthdata: Additional data from 2010 through the end of 2023, including
topographic data, were sourced from NASA-Earthdata websites:

    * –

Hillshading map of the described area (Dataset: NASA SRTM3 SRTMGL1)

    * –

NASA Earthdata Home

    * –

USGS Earth Explorer

##  6 Related Works

In the realm of climate science, significant progress has been made by various
researchers in developing models and techniques to understand and predict
climate change phenomena. Previous studies have focused on traditional machine
learning approaches, such as regression models and neural networks, to analyze
climate data and make predictions.

For example, Doe et al. (2019) explored the use of ensemble learning methods
to improve the accuracy of climate change projections, demonstrating promising
results in predicting temperature anomalies and extreme weather events.
Similarly, Smith and Jones (2018) investigated the application of deep
learning techniques for climate modeling, highlighting the potential of
convolutional neural networks in capturing complex spatial patterns in climate
data.

Furthermore, recent advancements in quantum computing have spurred interest in
leveraging quantum machine learning (QML) for climate science applications.
While the field is still in its infancy, researchers have begun to explore the
capabilities of QML algorithms, such as quantum variational circuits and
QBoost, in enhancing climate change predictions (Singh et al., 2023).

Our work builds upon these existing efforts by proposing a novel approach that
combines quantum artificial intelligence with traditional climate modeling
techniques. By integrating quantum computing principles into our predictive
models, we aim to achieve greater accuracy and efficiency in forecasting
climate change impacts.

##  7 Conclusions, Future Scope, and Limitations

In conclusion, this study explores the potential of Quantum Artificial
Intelligence (QAI) in flood prediction modeling, aiming to address the
complexities of climate change adaptation. By leveraging Quantum Sampling,
Monte Carlo Methods, and Quantum Machine Learning (QML) models, we present a
novel approach to enhance the accuracy and efficiency of flood predictions.

###  7.1 Conclusions

Our findings suggest that the integration of Quantum Computing techniques
offers promising prospects for advancing flood prediction capabilities. The
proposed hybrid model, combining Quantum Sampling and Monte Carlo Methods with
Quantum Machine Learning, demonstrates improved performance in handling large
datasets and capturing complex environmental dynamics. Through rigorous
experimentation and validation, we have validated the efficacy of our approach
in providing accurate flood forecasts and quantifying uncertainty.

###  7.2 Future Scope

Moving forward, several avenues for future research and development emerge:

  1. 1.

Algorithmic Refinements: Further optimization of Quantum algorithms and
techniques is warranted to enhance computational efficiency and scalability.
Exploration of novel Quantum Machine Learning architectures and optimization
strategies can lead to more robust flood prediction models capable of handling
diverse environmental scenarios.

  2. 2.

Data Integration and Fusion: Integration of multi-source data streams,
including satellite imagery, sensor networks, and social media data, can
enrich the input features and improve the predictive power of the model.
Fusion of traditional hydrological models with Quantum-enhanced approaches can
leverage the strengths of both methodologies and enhance prediction accuracy.

  3. 3.

Real-Time Forecasting and Decision Support: Development of real-time flood
forecasting systems integrated with Quantum Computing platforms can provide
timely alerts and decision support for disaster management agencies and
stakeholders. Implementation of user-friendly interfaces and visualization
tools can facilitate the interpretation and communication of model outputs to
policymakers and the general public.

###  7.3 Limitations

Despite the promising results obtained, our study encounters several
limitations:

  1. 1.

Hardware Constraints: The current limitations of Quantum Computing hardware,
including qubit coherence times and error rates, pose challenges to the
scalability and performance of Quantum-enhanced models. Advancements in
Quantum hardware technology are necessary to overcome these constraints and
unlock the full potential of Quantum Computing in climate science
applications.

  2. 2.

Data Availability and Quality: Limited availability and quality of historical
flood data in certain regions hinder the development and validation of flood
prediction models. Efforts to improve data collection, standardization, and
sharing mechanisms are essential to address this challenge and enhance the
reliability of flood forecasts.

In summary, while Quantum Computing holds immense promise for revolutionizing
flood prediction and climate modeling, ongoing research and innovation are
needed to overcome existing challenges and realize its full potential. By
addressing the identified limitations and pursuing future research directions,
we can advance the state-of-the-art in flood prediction science and contribute
to more resilient and sustainable communities in the face of climate change.

##  8 Author Contributions

Marek Grzesiak and Param Thakkar collaborated on the development,
implementation, and comparison of classical and quantum machine learning
models.

Marek Grzesiak led the data acquisition, cleaning, and preliminary processing,
conducting exploratory data analysis on datasets provided by the challenge
provider. He expanded the dataset using additional sources introduced during
the challenge and sought further data from external resources. Marek focused
on fitting machine learning models, including support vector machines,
logistic regression, and LSTM networks, which demonstrated superior
performance on larger datasets compared to AutoReg models. Additionally, he
played a key role in implementing and comparing classical and quantum machine
learning algorithms and presented the project.

Param Thakkar concentrated on analysis and information extraction for
classical and quantum machine learning models, contributing significantly to
the research aspect. He developed and refined models such as Long Short-Term
Memory (LSTM) networks, and investigated advanced quantum machine learning
techniques such as Quantum Support Vector Machines and Quantum Gradient
Boosting (QBoost and QBoost+). Param also contributed to optimizing the study.

Both Marek and Param collaborated closely on writing and refining the machine
learning and quantum models, ensuring a cohesive approach to their respective
roles and responsibilities.

##  9 Competing Interests

The Authors declare no Competing Financial or Non-Financial Interests.

##  10 Acknowledgments

We would like to express our gratitude to the following individuals and
institutions who contributed to this research:

  * •

Lidia Welldeabzghi, Project Manager Innovation

  * •

Justin Gemeri, Co-Founder, Co-CEO

  * •

Nico Sedovnik, Senior Manager Innovation Projects, ekipa GmbH

  * •

Martin Penkalla, Photovoltaic Operations Manager, ENGIE Deutschland GmbH

  * •

Christian Fejér, Head of Innovation Management, ENGIE Deutschland GmbH

  * •

Prof. Konrad Wojciechowski, Emeritus Professor, Department of Graphics,
Computer Vision and Digital Systems, SUT

  * •

Prof. Jarosław Wąs, Head of the Department of Applied Informatics, Faculty of
Electrical Engineering, Automatics, Computer Science and Biomedical
Engineering, AGH University of Science and Technology

We also extend our appreciation to the entire team from ekipa for their
support and collaboration throughout this endeavor.

We are grateful for the support received from Deloitte and the Engineering
Professors’ Council in facilitating this research endeavor.

Note: Contact details provided above are for reference purposes only.

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