# Picard and Adomian solutions of nonlinear fractional differential equations system containing Atangana – Baleanu derivative

## Abstract

In this paper, we apply two methods for solving nonlinear system of fractional differential equations (FDEs); these two methods are Picard and Adomian decomposition methods (ADM). The type of fractional derivative in this system will be the Atangana–Baleanu derivative. The existence and uniqueness of the solution will be proved. In addition, the convergence of ADM series solution and the maximum expected error will be discussed. Some numerical examples will be solved by using these two method and a comparison between their solutions will be done. There exist an important application to these types of systems, this application is the fractional-order rabies model and it will be solved here. From the obtained results, it is noticed that the obtained results from using these two methods are coincide with each other, and also these results are coincide with the obtained results from the classical fractional derivatives such as Caputo sense.

## Introduction

Fractional Differential equations have many applications in engineering and science; some of them are fluid flow [1, 2], electrical networks, control theory [3, 4], electromagnetic theory, viscoelasticity [5, 6], fractals theory, potential theory [2, 7], biology, chemistry [8, 9], optical and neural network systems [10,11,12]. In this paper, Picard [13,14,15] and Adomian decomposition methods [16,17,18,19,20] will be used to solve these type of systems. These two methods have many advantages; they efficiently work with different types of linear and nonlinear equations [21,22,23,24] in deterministic or stochastic [25,26,27] fields and gives an analytic solution for all these types of equations without linearization or discretization [28,29,30].

The paper will be organized as follows:

In Methods section, Picard and ADM will be introduced as the two used methods to solve the system under consideration. In Results and discussion section, Existence and uniqueness of the solution will be proved, convergence of ADM series solution and error analysis will be discussed. Finally, an important application to these types of systems will be solved which is fractional-order rabies model and other numerical examples will be solved by using these two methods and a comparison between their results will be illustrated.

## Methods

In this research, two methods will be used to solve a nonlinear system of fractional differential equations containing Atangana–Baleanu derivative. The first method is ADM and the second method is Picard method.

### Formulation of the problem

Consider a system of nonlinear FDEs of the form,

$${}^{AB}{\mathcal{D}}_{t}^{\alpha }{y}_{i}\left(t\right)+{g}_{i}\left(t\right){f}_{i}\left(\overline{y}\left(t\right)\right)={x}_{i}\left(t\right),\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\alpha \in \left(n-1,n\right),\hspace{0.33em}i=\mathrm{1,2},\dots ,n.$$
(1)

Subject to the initial conditions,

$${{y}_{i}}^{(j-1)}(0)={c}_{j},\hspace{1em} j=\mathrm{1,2},\dots ,n.$$
(2)

Where $$\overline{y}=\left({y}_{1},{y}_{2},\dots ,{y}_{n}\right)$$ and $${}^{AB}{\mathcal{D}}_{t}^{\alpha }(.)$$ is fractional derivative of Atangana–Baleanu sense that defined as:

$${}^{AB}{\mathcal{D}}_{t}^{\alpha }f\left(t\right)=\frac{B\left(\alpha \right)}{1-\alpha }{\int }_{0}^{t}{E}_{\alpha }{\left(\frac{-\alpha \left(t-s\right)}{1-\alpha }\right)}^{\alpha }{f}^\prime(s)ds$$

Where $$B(\alpha )>0$$, is a normalization function satisfying $$B(0)=B(1)=1$$ and $${E}_{\alpha }$$ is the Mittag–Leffler function of one variable. The corresponding fractional integral defined by see [3, 4],

$${}^{AB}{I}^{\alpha }f\left(t\right)=\frac{1-\alpha }{B\left(\alpha \right)}f\left(t\right)+\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}f(s){\left(t-s\right)}^{\alpha -1}ds,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}0<\alpha <1.$$

And

$$\left({}^{AB}{I}^{\alpha }\right)\left({}^{AB}{\mathcal{D}}_{t}^{\alpha }\right)f\left(t\right)=f\left(t\right)-f\left(a\right)$$

Now applying the integrating operator of order $$\alpha$$ to the system (1)-(2), this reduces it to the system of fractional integral equations,

$$\begin{array}{c}{y}_{i}\left(t\right)=\sum\limits_{i=1}^{n}\frac{{c}_{i}}{\Gamma (\alpha )}{t}^{\alpha -1}+\frac{1-\alpha }{B\left(\alpha \right)}{x}_{i}\left(t\right)+\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}(t-\tau {)}^{\alpha -1}{x}_{i}\left(\tau \right)d\tau \\ -\frac{1-\alpha }{B\left(\alpha \right)}{g}_{i}\left(t\right){f}_{i}\left(\overline{y}\left(t\right)\right)-\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}(t-\tau {)}^{\alpha -1}{g}_{i}\left(\tau \right){f}_{i}\left(\overline{y}\left(\tau \right)\right)d\tau \end{array}$$
(3)

Assume that $${x}_{i}(t)$$ bounded $$\forall t\in I=[0,T]$$$$T\in {R}^{+}$$, $$\left|{g}_{i}(\tau )\right|\le {M}_{i} \forall 0\le \tau \le T$$$${M}_{i}$$ are finite constants and $${f}_{i}(\overline{y})$$ satisfy Lipschitz condition with Lipschitz constants $${L}_{i}$$ such as,

$$\left|{f}_{i}(\overline{y})-{f}_{i}(\overline{z})\right|\le {L}_{i}\sum_{k=1}^{n}\left|{y}_{k}-{z}_{k}\right|$$
(4)

Applying ADM depends on replacing the nonlinear term with its corresponding Adomian polynomials as follows,

$${f}_{i}(\overline{y})=\sum_{k=0}^{\infty }{A}_{ik}({y}_{i0},{y}_{i1},\dots ,{y}_{ik})$$
(5)

Where,

$${A}_{ik}({y}_{i0},{y}_{i1},\dots ,{y}_{ik})=\frac{1}{k!}\frac{{d}^{k}}{d{\lambda }^{k}}{\left[{f}_{i}\left(\sum_{j=0}^{\infty }{\lambda }^{j}{y}_{ij}\right)\right]}_{\lambda =0}$$
(6)

Substitute from Eq. (5) into Eq. (3), we get

$$\begin{array}{c}{y}_{i}\left(t\right)=\sum\limits_{i=1}^{n}\frac{{c}_{i}}{\Gamma (\alpha )}{t}^{\alpha -1}+\frac{1-\alpha }{B\left(\alpha \right)}{x}_{i}\left(t\right)+\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}(t-\tau {)}^{\alpha -1}{x}_{i}\left(\tau \right)d\tau \\ -\frac{1-\alpha }{B\left(\alpha \right)}{g}_{i}\left(t\right)\sum\limits_{k=0}^{\infty }{A}_{ik}\hspace{0.17em}-\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}(t-\tau {)}^{\alpha -1}{g}_{i}\left(\tau \right)\sum\limits_{k=0}^{\infty }{A}_{ik}\hspace{0.17em}d\tau \end{array}$$
(7)

Let $${y}_{i}(t)=\sum_{k=0}^{\infty }{y}_{ik}(t)$$ in (7) we get,

$${y}_{i0}(t)=\sum_{i=1}^{n}\frac{{c}_{i}}{\Gamma (\alpha )}{t}^{\alpha -1}+\frac{1-\alpha }{B\left(\alpha \right)}{x}_{i}\left(t\right)+\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}(t-\tau {)}^{\alpha -1}{x}_{i}\left(\tau \right)d\tau ,$$
(8)
$$\begin{array}{l}{y}_{ik}\left(t\right)=-\frac{1-\alpha }{B\left(\alpha \right)}{g}_{i}\left(t\right){A}_{i(k-1)}\\ -\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}(t-\tau {)}^{\alpha -1}{g}_{i}\left(\tau \right){A}_{i\left(k-1\right)}\hspace{0.17em}d\tau , k\ge 1.\end{array}$$
(9)

Finally, the ADM series solution will be,

$${y}_{i}(t)=\sum_{k=0}^{\infty }{y}_{ik}(t)$$
(10)

### Existence and uniqueness theorem

Let E = ($$\left(I\right),{\mathbb{R}}^{\left(n\right)}$$) be the Banach space of continuous functions defined on the compact interval I that are valued in $${\mathbb{R}}^{\left(n\right)}$$. On $${\mathbb{R}}^{\left(n\right)}$$ is considered the norm $$\Vert {\varvec{y}}\Vert =\sum_{i=1}^{n}\left|{y}_{i}\right|$$ where y = $$({y}_{1},{y}_{2},\dots ,{y}_{n})$$$${\mathbb{R}}^{\left(n\right)}$$. If yE and $$\mathbf{y}\left({\text{t}}\right)=({y}_{1}(t),{y}_{2}(t),\dots ,{y}_{n}(t))$$ then $$\Vert {\varvec{y}}\Vert =\sum_{i=1}^{n}\underset{t\in J}{{\text{max}}}\left|{y}_{i}(t)\right|$$.

Theorem 1 The system (1) and (2) has a unique solution whenever $$0<\beta <1$$, $$\beta =\frac{LM}{B\left(\alpha \right)}[\left(1-\alpha \right)+\frac{\alpha {T}^{\alpha }}{\Gamma \left(\alpha +1\right)}]$$ where $$L=\sum_{m=1}^{n}{L}_{m}$$,$$M=\mathrm{max }\left\{{M}_{1},{M}_{2},\dots ,{M}_{n}\right\}$$.

Proof Equation (3) can written as,

$$\begin{array}{c}{\varvec{y}}\left(t\right)=\sum\limits_{i=1}^{n}\frac{{c}_{i}}{\Gamma (\alpha )}{t}^{\alpha -1}+\frac{1-\alpha }{B\left(\alpha \right)}{x}_{i}\left(t\right)+\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}(t-\tau {)}^{\alpha -1}{x}_{i}\left(\tau \right)d\tau \\ -\frac{1-\alpha }{B\left(\alpha \right)}{g}_{i}\left(t\right){f}_{i}\left({\varvec{y}}\left(t\right)\right)-\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}(t-\tau {)}^{\alpha -1}{g}_{i}\left(\tau \right){f}_{i}\left({\varvec{y}}\left(\tau \right)\right)d\tau \end{array}$$

Where,

$$\begin{array}{l}{\varvec{x}}\left(t\right)={\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)}^\prime,\\ {\varvec{g}}\left(t\right)=diag\left\{{g}_{1},{g}_{2},\dots ,{g}_{n}\right\},\\ {\varvec{f}}({\varvec{y}}\left(t\right))={\left({f}_{1}\left({\varvec{y}}\right),{f}_{2}\left({\varvec{y}}\right),\dots ,{f}_{n}\left({\varvec{y}}\right)\right)}^\prime.\end{array}$$

The mapping $$R:E\to E$$ defined as,

$$\begin{array}{c}R{\varvec{y}}\left(t\right)=\sum\limits_{i=1}^{n}\frac{{c}_{i}}{\Gamma (\alpha )}{t}^{\alpha -1}+\frac{1-\alpha }{B\left(\alpha \right)}x(t)+\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}(t-\tau {)}^{\alpha -1}{\varvec{x}}\left(\tau \right)d\tau \\ -\frac{1-\alpha }{B\left(\alpha \right)}{\varvec{g}}\left(t\right){\varvec{f}}({\varvec{y}}\left(t\right))-\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}(t-\tau {)}^{\alpha -1}{\varvec{g}}\left(\tau \right){\varvec{f}}({\varvec{y}}\left(\tau \right))d\tau \end{array}$$

Let $$Y,Z\in E$$:

$$\begin{array}{lllllllllll}\Vert RY\left(t\right)-RZ\left(t\right)\Vert =\Vert -\frac{1-\alpha }{B\left(\alpha \right)}{\varvec{g}}\left(t\right)\left({\varvec{f}}\left(\overline{y}\right)-{\varvec{f}}\left(\overline{z}\right)\right)\\ -\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}(t-\tau {)}^{\alpha -1}{\varvec{g}}\left(\tau \right){\varvec{f}}({\varvec{y}}\left(\tau \right))d\tau \Vert\\ \le \frac{1-\alpha }{B\left(\alpha \right)}\Vert {\varvec{g}}\left(\tau \right)\Vert \Vert {\varvec{f}}\left({\varvec{y}}\right)-{\varvec{f}}\left({\varvec{z}}\right)\Vert\\ +\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}(t-\tau {)}^{\alpha -1}\Vert {\varvec{g}}\left(\tau \right)\Vert \Vert {\varvec{f}}\left({\varvec{y}}\right)-{\varvec{f}}\left({\varvec{z}}\right)\Vert d\tau\\ \le \frac{\left(1-\alpha \right)M}{B\left(\alpha \right)}\sum_{m=1}^{n}{L}_{m}\left(\sum_{m=1}^{n}\underset{t\in J}{{\text{max}}}\left|{f}_{m}\left({\varvec{y}}\right)-{f}_{m}\left({\varvec{z}}\right)\right|\right)\\ +\frac{\alpha M}{B\left(\alpha \right)\Gamma \left(\alpha \right)}\sum_{m=1}^{n}{L}_{m}\left(\sum_{m=1}^{n}\underset{t\in J}{{\text{max}}}\left|{f}_{m}\left({\varvec{y}}\right)-{f}_{m}\left({\varvec{z}}\right)\right|\right){\int }_{0}^{t}(t-\tau {)}^{\alpha -1}d\tau\\ \le \frac{\left(1-\alpha \right)M}{B\left(\alpha \right)}\sum_{m=1}^{n}{L}_{m}\left(\sum_{m=1}^{n}\underset{t\in J}{{\text{max}}}\left|{y}_{k}-{z}_{k}\right|\right)\\ +\frac{\alpha M{T}^{\alpha }}{B\left(\alpha \right)\Gamma \left(\alpha +1\right)}\sum_{m=1}^{n}{L}_{m}\left(\sum_{m=1}^{n}\underset{t\in J}{{\text{max}}}\left|{y}_{k}-{z}_{k}\right|\right)\\ \le \left[\frac{\left(1-\alpha \right)ML}{B\left(\alpha \right)}+\frac{\alpha M{T}^{\alpha }L}{B\left(\alpha \right)\Gamma \left(\alpha +1\right)}\right]\Vert {\varvec{y}}-{\varvec{z}}\Vert\\ \le \frac{LM}{B\left(\alpha \right)}\left[\left(1-\alpha \right)+\frac{\alpha {T}^{\alpha }}{\Gamma \left(\alpha +1\right)}\right]\Vert {\varvec{y}}-{\varvec{z}}\Vert\\\le \beta \Vert Y-Z\Vert\end{array}$$

Under the condition, $$0<\beta <1,$$ the mapping $$R$$ is contraction and there exist a unique solution of the system (1)-(2).

### Proof of convergence

Theorem 2 The series solution (10) of the system (1)-(2) using ADM converges if $$\left|{y}_{i1}\right|<\infty$$ and $$0<\beta <1,\hspace{0.17em}\beta =\frac{LM}{B\left(\alpha \right)}[\left(1-\alpha \right)+\frac{\alpha {T}^{\alpha }}{\Gamma \left(\alpha +1\right)}]$$, where $$L=\sum_{k=1}^{n}{L}_{k}$$$$M=\mathrm{max }\left\{{M}_{1},{M}_{2},\dots ,{M}_{n}\right\}$$.

Proof Define a sequence $$\left\{{S}_{ip}\right\}$$ as, $${S}_{ip}=\sum_{k=0}^{p}{y}_{ik}(t)$$ is the sequence of partial sums from the series solution $$\sum_{k=0}^{\infty }{y}_{ik}\left(t\right),$$ we have,

$$f({S}_{ip})=\sum_{k=0}^{p}{A}_{ik}({y}_{i0},{y}_{i1},\dots ,{y}_{ip})$$

Let $${S}_{ip}$$ and $${S}_{iq}$$ be two arbitrary partial sums such that $$p>q$$. Now, we are going to prove that $$\left\{{S}_{ip}\right\}$$ is a Cauchy sequence in this Banach space.

$$\begin{array}{l}\Vert {S}_{ip}-{S}_{iq}\Vert =\sum\limits_{k=1}^{n}\underset{t\in J}{{\text{max}}}\left|{S}_{kp}-{S}_{kq}\right|\\ =\sum\limits_{k=1}^{n}\underset{t\in J}{{\text{max}}}\left|\sum\limits_{j=q+1}^{p}{y}_{kj}\left(t\right)\right|\\ \begin{array}{l}=\sum\limits_{k=1}^{n}\underset{t\in J}{{\text{max}}} \left|\sum\limits_{j=q+1}^{p}\left[\frac{1-\alpha }{B\left(\alpha \right)}{g}_{k}\left(t\right){A}_{i\left(k-1\right)}+\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}(t-\tau {)}^{\alpha -1}{g}_{k}\left(\tau \right){A}_{k\left(j-1\right)}\hspace{0.17em}d\tau \right]\right|\\ =\sum\limits_{k=1}^{n}\underset{t\in J}{{\text{max}}} \left|\sum_{j=q+1}^{p}\frac{1-\alpha }{B\left(\alpha \right)}{g}_{k}\left(t\right)\sum\limits_{j=q+1}^{p}{A}_{k\left(j-1\right)}+\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}{g}_{k}(\tau )(t-\tau {)}^{\alpha -1}\sum_{j=q+1}^{p}{A}_{k(j-1)}d\tau \right|\\ \begin{array}{l}=\sum\limits_{k=1}^{n}\underset{t\in J}{{\text{max}}} \left|\sum\limits_{j=q+1}^{p}\frac{1-\alpha }{B\left(\alpha \right)}{g}_{k}\left(t\right)\sum\limits_{j=q}^{p-1}{A}_{kj}+\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}{g}_{k}(\tau )(t-\tau {)}^{\alpha -1}\sum\limits_{j=q}^{p-1}{A}_{kj}d\tau \right|\\ =\sum\limits_{k=1}^{n}\underset{t\in J}{{\text{max}}}\left|\sum\limits_{j=q+1}^{p}\frac{1-\alpha }{B\left(\alpha \right)}{g}_{k}\left(t\right)\left[f\left({S}_{k\left(p-1\right)}\right)-f\left({S}_{k\left(q-1\right)}\right)\right]\right.\\ \begin{array}{l}\left.+\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}{g}_{k}(\tau )(t-\tau {)}^{\alpha -1}[f({S}_{k(p-1)})-f({S}_{k(q-1)})]d\tau \right|\\ \le \frac{1-\alpha }{B\left(\alpha \right)}\sum\limits_{k=1}^{n}\underset{t\in J}{{\text{max}}}[\left|{g}_{k}\left(t\right)\right|\left|f\left({S}_{k\left(p-1\right)}\right)-f\left({S}_{k\left(q-1\right)}\right)\right|] \\ \begin{array}{l}+\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}\sum_{k=1}^{n}\underset{t\in J}{{\text{max}}}[{\int }_{0}^{t}\left|{g}_{k}\left(t\right)\right|\left|(t-\tau {)}^{\alpha -1}\right|\left|f\left({S}_{k\left(p-1\right)}\right)-f\left({S}_{k\left(q-1\right)}\right)\right|d\tau ]\\ \le \frac{\left(1-\alpha \right)ML}{B\left(\alpha \right)}\underset{t\in J}{{\text{max}}}\sum\limits_{j=1}^{n}\left|{S}_{j(p-1)}-{S}_{j(q-1)}\right| \\ \begin{array}{l}+\frac{\alpha ML}{B\left(\alpha \right)\Gamma \left(\alpha \right)}\underset{t\in J}{{\text{max}}}\sum_{j=1}^{n}\left|{S}_{j(p-1)}-{S}_{j(q-1)}\right|{\int }_{0}^{t}(t-\tau {)}^{\alpha -1}d\tau \\ \le \frac{LM}{B\left(\alpha \right)}\left[\left(1-\alpha \right)+\frac{\alpha {T}^{\alpha }}{\Gamma \left(\alpha +1\right)}\right]\Vert {S}_{i\left(p-1\right)}-{S}_{i\left(q-1\right)}\Vert \\ \le \beta \Vert {S}_{i(p-1)}-{S}_{i(q-1)}\Vert \end{array}\end{array}\end{array}\end{array}\end{array}\end{array}$$

Let $$p=q+1$$ then,

$$\Vert {S}_{i(q+1)}-{S}_{iq}\Vert \le \beta \Vert {S}_{iq}-{S}_{i(q-1)}\Vert \le {\beta }^{2}\Vert {S}_{i(q-1)}-{S}_{i(q-2)}\Vert \le \cdots \le {\beta }^{q}\Vert {S}_{i1}-{S}_{i0}\Vert$$

Using the triangle inequality,

$$\begin{array}{c}\Vert {S}_{ip}-{S}_{iq}\Vert \le \Vert {S}_{i\left(q+1\right)}-{S}_{iq}\Vert +\Vert {S}_{i\left(q+2\right)}-{S}_{i\left(q+1\right)}\Vert +\cdots +\Vert {S}_{ip}-{S}_{i\left(p-1\right)}\Vert \\ \le \left[{\beta }^{q}+{\beta }^{q+1}+\cdots +{\beta }^{p-1}\right]\Vert {S}_{i1}-{S}_{i0}\Vert \\ \begin{array}{l}\le {\beta }^{q}\left[1+\beta +\cdots +{\beta }^{p-q-1}\right]\Vert {S}_{i1}-{S}_{i0}\Vert \\ \le {\beta }^{q}\left[\frac{1-{\beta }^{p-q}}{1-\beta }\right]\Vert {y}_{i1}(t)\Vert \end{array}\end{array}$$

Since, $$0<\beta <1$$ and $$p>q$$ then, $$(1-{\beta }^{p-q})\le 1$$. Consequently,

$$\begin{array}{c}\Vert {S}_{ip}-{S}_{iq}\Vert \le \frac{{\beta }^{q}}{1-\beta }\Vert {y}_{i1}\left(t\right)\Vert \\ \le \frac{{\beta }^{q}}{1-\beta }\underset{t\in J}{{\text{max}}} \left|{y}_{i1}(t)\right|\end{array}$$

If $$\left|{y}_{i1}(t)\right|<\infty$$ and as $$q\to \infty$$ then, $$\Vert {S}_{ip}-{S}_{iq}\Vert \to 0$$ and hence, $$\left\{{S}_{ip}\right\}$$ is a Cauchy sequence in this Banach space so, the series $$\sum_{k=0}^{\infty }{y}_{ik}(t)$$ converges.

### Error analysis

Theorem 3 The maximum absolute truncation error of the series solution (10) to the system (1)-(2) estimated to be,

$$\underset{t\in J}{{\text{max}}} \left|{y}_{i}(t)-\sum_{k=0}^{q}{y}_{ik}(t)\right|\le \frac{{\beta }^{q}}{1-\beta }\underset{t\in J}{{\text{max}}} \left|{y}_{i1}(t)\right|$$

Proof From Theorem 2 we get that

$$\Vert {S}_{ip}-{S}_{iq}\Vert \le \frac{{\beta }^{q}}{1-\beta }\underset{t\in J}{{\text{max}}} \left|{y}_{i1}(t)\right|$$

If $${S}_{ip}=$$$$\sum_{k=0}^{p}{y}_{ik}(t)$$ as $$p\to \infty$$ then,$${S}_{ip}\to {y}_{i}(t)$$ so,

$$\Vert {y}_{i}(t)-{S}_{iq}\Vert \le \frac{{\beta }^{q}}{1-\beta }\underset{t\in J}{{\text{max}}} \left|{y}_{i1}(t)\right|.$$

Hence the maximum absolute truncation error in the interval $$J$$ is,

$$\underset{t\in J}{{\text{max}}} \left|{y}_{i}(t)-\sum_{k=0}^{q}{y}_{ik}(t)\right|\le \frac{{\beta }^{q}}{1-\beta }\underset{t\in J}{{\text{max}}} \left|{y}_{i1}(t)\right|$$

#### The second method: Picard method

Applying Picard method to the system (3), the solution is constructed by the sequence,

$${y}_{i0}(t)=\sum_{i=1}^{n}\frac{{c}_{i}}{\Gamma (\alpha )}{t}^{\alpha -1}+\frac{1-\alpha }{B\left(\alpha \right)}{x}_{i}\left(t\right)+\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}(t-\tau {)}^{\alpha -1}{x}_{i}\left(\tau \right)d\tau ,$$
(11)
$$\begin{array}{l}{y}_{ik}\left(t\right)={y}_{i0}(t)-\frac{1-\alpha }{B\left(\alpha \right)}{g}_{i}\left(t\right){f}_{i}\left({y}_{i\left(k-1\right)}\left(\tau \right)\right)\\ -\frac{\alpha }{B\left(\alpha \right)\Gamma \left(\alpha \right)}{\int }_{0}^{t}(t-\tau {)}^{\alpha -1}{g}_{i}\left(\tau \right){f}_{i}\left({y}_{i\left(k-1\right)}\left(\tau \right)\right)\hspace{0.17em}d\tau , k\ge 1.\end{array}$$
(12)

Finally, the Picard solution will be,

$${y}_{i}(t)=\underset{k\to \infty }{{\text{lim}}}{y}_{ik}\left(t\right)$$
(13)

## Results and discussion

Example 1. Fractional-order rabies model

The fractional-order rabies model,

$$\begin{array}{c}{}^{AB}{\mathcal{D}}_{t}^{\alpha }{y}_{1}={-by}_{1}{y}_{2},\\ {}^{AB}{\mathcal{D}}_{t}^{\alpha }{y}_{2}={by}_{1}{y}_{2}-d{y}_{2},\end{array}$$
(14)

Subject to the initial conditions,

$${y}_{1}\left(0\right)=1,\hspace{0.33em}{y}_{2}\left(0\right)=2,$$

Was discussed before in [22], it solved by using Adams-type predictor–corrector method. Now, we will solve it by using ADM.

1. 1-

$$\begin{array}{l}{y}_{\mathrm{1,0}}=1,\hspace{1em}{y}_{1,j+1}=-b{}^{AB}{I}^{\alpha }({A}_{1,j}),\\ {y}_{\mathrm{2,0}}=2,\hspace{1em}{y}_{2,j+1}={}^{AB}{I}^{\alpha }\left({bA}_{1,j}-d{y}_{2,j}\right),\end{array}$$
(15)

Where $${A}_{1,j}$$ represent the Adomian polynomials of the nonlinear term $${y}_{1}{y}_{2}$$.

Moreover, the final solution will be,

$${y}_{1}=\sum_{i=0}^{n}{y}_{1,i},{y}_{2}=\sum_{i=0}^{n}{y}_{2,i}.$$
1. 2-

Picard Solution:

Using Picard method to the system (14), the solution algorithm will be,

$$\begin{array}{l}{y}_{\mathrm{1,0}}=1,\hspace{1em}{y}_{1,j+1}={y}_{\mathrm{1,0}}-b{}^{AB}{I}^{\alpha }[{y}_{1,j}{y}_{2,j}],\\ {y}_{\mathrm{2,0}}=2,\hspace{1em}{y}_{2,j+1}={y}_{\mathrm{2,0}}+{}^{AB}{I}^{\alpha }\left[b{y}_{1,j}{y}_{2,j}-d{y}_{2,j}\right].\end{array}$$
(16)

Moreover, the final solution will be,

$${y}_{1}=\underset{n\to \infty }{{\text{lim}}}{y}_{1,n},{y}_{2}=\underset{n\to \infty }{{\text{lim}}}{y}_{2,n}.$$

Figure 1a and b show ADM and Picard solutions of $${y}_{1}$$ and $${y}_{2}$$ where (n = 5,$$b=d=1$$) at ($$\alpha =\mathrm{0.8,0.9})$$.

From these two figures, we see that ADM solutions of $$({y}_{1} \,and\, {y}_{2})$$ are coincide with Picard solutions at the same values of $$\alpha .$$

Example 2. Consider the following nonlinear system of FDEs,

$$\begin{array}{l}{}^{AB}{\mathcal{D}}_{t}^{0.5}({}^{AB}{\mathcal{D}}_{t}^{0.5}{y}_{1})=1+{y}_{2}^{3}-{t}^{6},\\ {}^{AB}{\mathcal{D}}_{t}^{0.5}({}^{AB}{\mathcal{D}}_{t}^{0.5}{y}_{2})={y}_{1}+t,\\ {}^{AB}{\mathcal{D}}_{t}^{0.5}({}^{AB}{\mathcal{D}}_{t}^{0.5}{y}_{3})={3y}_{1}^{2},\end{array}$$
(17)

Subject to the initial conditions,

$${y}_{k}\left(0\right)=0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}k=\mathrm{1,2},3.$$

Which has the exact solution $${y}_{1}\left(t\right)=t,{y}_{2}\left(t\right)={t}^{2} \,and\, {y}_{3}\left(t\right)={t}^{3}.$$

1. 1-

Apply $${}^{AB}{I}^{\alpha }$$ to the system (17), then using ADM and replace each nonlinear term by its corresponding Adomian polynomials we obtain,

$$\begin{array}{l}{y}_{\mathrm{1,0}}=t-\frac{{t}^{7}}{7},\hspace{1em}{y}_{1,j+1}={}^{AB}{I}^{1}[{A}_{1,j}],\\ {y}_{\mathrm{2,0}}=\frac{{t}^{2}}{2},\hspace{1em}{y}_{2,j+1}={}^{AB}{I}^{1}[{y}_{1,j}],\\ {y}_{\mathrm{3,0}}=0,\hspace{1em}{y}_{3,j+1}={}^{AB}{I}^{1}[{3A}_{2,j}].\end{array}$$
(18)

Moreover, the final solution will be,

$${y}_{1}=\sum_{i=0}^{\infty }{y}_{1,n},{y}_{2}=\sum_{i=0}^{\infty }{y}_{2,n},{y}_{3}=\sum_{i=0}^{\infty }{y}_{3,n}.$$
1. 2-

Picard Solution:

Using Picard method to the system (17), the solution algorithm will be,

$$\begin{array}{l}{y}_{\mathrm{1,0}}=t-\frac{{t}^{7}}{7},\hspace{1em}{y}_{1,j+1}={y}_{\mathrm{1,0}}+{}^{AB}{I}^{1}[{({y}_{2,j})}^{3}],\\ {y}_{\mathrm{2,0}}=\frac{{t}^{2}}{2},\hspace{1em}{y}_{2,j+1}={y}_{\mathrm{2,0}}+{}^{AB}{I}^{1}\left[{y}_{1,j}\right],\\ {y}_{\mathrm{3,0}}=0,\hspace{1em}{y}_{3,j+1}={y}_{\mathrm{3,0}}+{}^{AB}{I}^{1}[3{({y}_{1,j})}^{2}].\end{array}$$
(19)

Moreover, the final solution will be,

$${y}_{1}=\underset{n\to \infty }{{\text{lim}}}{y}_{1,n},{y}_{2}=\underset{n\to \infty }{{\text{lim}}}{y}_{2,n},{y}_{3}=\underset{n\to \infty }{{\text{lim}}}{y}_{3,n}.$$

Figure 2a-c show Picard and exact solutions of $${y}_{1},{y}_{2}$$ and $${y}_{3}$$ (n = 5). While, Fig. 2d-f show ADM and exact solutions of $${y}_{1},{y}_{2}$$ and $${y}_{3}$$ (n = 5).

Tables 1, 2 and 3 show the relative absolute error between exact solutions, Picard and ADM solutions of $${y}_{1},{y}_{2}$$ and $${y}_{3}$$. A comparison between Picard with exact solutions and ADM with exact solutions are shown from these results that Picard method give more accurate results than ADM but ADM take less time than Picard (ADM time = 0.235, Picard time = 0.455).

Example 3. Consider the following nonlinear system of FDEs,

$$\begin{array}{l}{}^{AB}{\mathcal{D}}_{t}^{\alpha }{y}_{1}=1-{y}_{1},\\ {}^{AB}{\mathcal{D}}_{t}^{\alpha }{y}_{2}={y}_{1}-{y}_{2}^{2},\\ {}^{AB}{\mathcal{D}}_{t}^{\alpha }{y}_{3}={y}_{2}^{2},\end{array}$$
(20)

Subject to the initial conditions,

$${y}_{k}\left(0\right)=0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}k=\mathrm{1,2},3.$$

Where $$0<\alpha <1.$$

1. 1-

Apply $${}^{AB}{I}^{\alpha }$$ to the systems (20), then using ADM and replace each nonlinear term by its corresponding Adomian polynomials we obtain,

$$\begin{array}{l}{y}_{\mathrm{1,0}}=\frac{{t}^{\alpha }}{\Gamma \left(1+\alpha \right)},\hspace{1em}{y}_{1,j+1}=-{}^{AB}{I}^{\alpha }[{y}_{1,j}],\\ {y}_{\mathrm{2,0}}=0,\hspace{1em}{y}_{2,j+1}={}^{AB}{I}^{\alpha }[{y}_{1,j}-{A}_{j}],\\ {y}_{\mathrm{3,0}}=0,\hspace{1em}{y}_{3,j+1}={}^{AB}{I}^{\alpha }[{A}_{j}].\end{array}$$
(21)

Moreover, the final solution will be,

$${y}_{1}=\sum_{i=0}^{\infty }{y}_{1,n},{y}_{2}=\sum_{i=0}^{\infty }{y}_{2,n},{y}_{3}=\sum_{i=0}^{\infty }{y}_{3,n}.$$
1. 2-

Picard Solution:

Using Picard method to the systems (20), the solution algorithm will be,

$$\begin{array}{l}{y}_{\mathrm{1,0}}=\frac{{t}^{\alpha }}{\Gamma \left(1+\alpha \right)},\hspace{1em}{y}_{1,j+1}={y}_{\mathrm{1,0}}-{}^{AB}{I}^{\alpha }[{y}_{1,j}],\\ {y}_{\mathrm{2,0}}=0,\hspace{1em}{y}_{2,j+1}={y}_{\mathrm{2,0}}+{}^{AB}{I}^{\alpha }\left[{y}_{1,j}-{({y}_{2,j})}^{2}\right],\\ {y}_{\mathrm{3,0}}=0,\hspace{1em}{y}_{3,j+1}={y}_{\mathrm{3,0}}+{}^{AB}{I}^{\alpha }[{({y}_{2,j})}^{2}].\end{array}$$
(22)

Moreover, the final solution will be,

$${y}_{1}=\underset{n\to \infty }{{\text{lim}}}{y}_{1,n},{y}_{2}=\underset{n\to \infty }{{\text{lim}}}{y}_{2,n},{y}_{3}=\underset{n\to \infty }{{\text{lim}}}{y}_{3,n}.$$

Figure 3a-c show ADM solutions of $${y}_{1},{y}_{2}$$ and $${y}_{3}$$ at different values of $$\alpha$$ ($$\alpha =\mathrm{1,0.95,0.9,0.85}$$).

While, Fig. 3d-f show Picard solutions of $${y}_{1},{y}_{2}$$ and $${y}_{3}$$ at the same values of $$\alpha .$$

Comparing between Fig. 3a-c and d-f, we see that ADM solutions of $${y}_{1},{y}_{2}$$ and $${y}_{3}$$ coincide with Picard solutions at the same values of $$\alpha$$.

Example 4. Consider the following nonlinear system of FDEs,

$$\begin{array}{c}{}^{AB}{\mathcal{D}}_{t}^{\alpha }{y}_{1}={y}_{1}^{2}+{y}_{2},\\ {}^{AB}{\mathcal{D}}_{t}^{\alpha }{y}_{2}=1+{y}_{2}\mathrm{cos }{y}_{1},\end{array}$$
(23)

Subject to the initial conditions,

$${y}_{k}\left(0\right)=0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}k=\mathrm{1,2}.$$

Where $$\alpha \in \left(\mathrm{0,1}\right).$$

1. 1-

$$\begin{array}{c}{y}_{\mathrm{1,0}}=0,\hspace{1em}{y}_{1,j+1}={}^{AB}{I}^{\alpha }({A}_{1,j})+{}^{AB}{I}^{\alpha }({y}_{2,j}),\\ {y}_{\mathrm{2,0}}=\frac{{t}^{\alpha }}{\Gamma \left(1+\alpha \right)},\hspace{1em}{y}_{2,j+1}={}^{AB}{I}^{\alpha }\left({A}_{2,j}\right),\end{array}$$
(24)

Where $${A}_{1,j}$$ and $${A}_{2,j}$$ represent the Adomian polynomials of the nonlinear terms $${y}_{1}^{2}$$ and $${y}_{2}\,\mathrm{cos\, }{y}_{1}$$ respectively.

Moreover, the final solution will be,

$${y}_{1}=\sum_{i=0}^{\infty }{y}_{1,n},{y}_{2}=\sum_{i=0}^{\infty }{y}_{2,n}$$
1. 2-

Picard Solution:

Using Picard method to the system (23), the solution algorithm will be,

$$\begin{array}{l}{y}_{\mathrm{1,0}}=0,\hspace{1em}{y}_{1,j+1}={y}_{\mathrm{1,0}}+{}^{AB}{I}^{\alpha }[{({y}_{1,j})}^{2}+{y}_{2,j}],\\ {y}_{\mathrm{2,0}}=\frac{{t}^{\alpha }}{\Gamma \left(1+\alpha \right)},\hspace{1em}{y}_{2,j+1}={y}_{\mathrm{2,0}}+{}^{AB}{I}^{\alpha }\left[{y}_{2,j}cos({y}_{1,j})\right],\end{array}$$
(25)

Moreover, the final solution will be,

$${y}_{1}=\underset{n\to \infty }{{\text{lim}}}{y}_{1,n},{y}_{2}=\underset{n\to \infty }{{\text{lim}}}{y}_{2,n}.$$

Figure 4a and b show ADM solutions of $${y}_{1}$$ and $${y}_{2}$$ at different values of $$\alpha$$ ($$\alpha =\mathrm{1,0.95,0.9,0.85,0.8}$$).

While, Fig. 4c and d show Picard solutions of $${y}_{1}$$ and $${y}_{2}$$ at the same values of $$\alpha .$$

Figure 4e and f show ADM solution of $${y}_{1}$$ and $${y}_{2}$$ at another different values of $$\alpha$$ ($$\alpha =0.25, \alpha =0.5, \alpha =0.75, \alpha =1)$$. While, Fig. 4g and h show Picard solutions of $${y}_{1}$$ and $${y}_{2}$$ at the same values of $$\alpha .$$

Example 5. Consider the following nonlinear system of FDEs,

$$\begin{array}{l}{}^{AB}{\mathcal{D}}_{t}^{\alpha }{y}_{1}=2{y}_{2}^{2},\\ {}^{AB}{\mathcal{D}}_{t}^{\alpha }{y}_{2}=1+t{y}_{1},\\ {}^{AB}{\mathcal{D}}_{t}^{\alpha }{y}_{3}=1+{y}_{2}{y}_{3},\end{array}$$
(26)

Subject to the initial conditions,

$${y}_{k}\left(0\right)=0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}k=\mathrm{1,2},3.$$

Where $$\alpha \in \left(\mathrm{0,1}\right).$$

1. 1-

$${y}_{\mathrm{1,0}}=0,\hspace{1em}{y}_{1,j+1}={}^{AB}{I}^{\alpha }(2{A}_{1,j}),$$
(27)
$${y}_{\mathrm{2,0}}=\frac{{t}^{\alpha }}{\Gamma \left(1+\alpha \right)},\hspace{1em}{y}_{2,j+1}={}^{AB}{I}^{\alpha }\left(t{y}_{1,j}\right),$$
(28)
$${y}_{\mathrm{3,0}}=\frac{{t}^{\alpha }}{\Gamma \left(1+\alpha \right)},\hspace{1em}{y}_{3,j+1}={}^{AB}{I}^{\alpha }\left({A}_{2,j}\right),$$
(29)

Where $${A}_{1,j}$$ and $${A}_{2,j}$$ represent the Adomian polynomials of the nonlinear terms $${y}_{2}^{2}$$ and $${y}_{2}{y}_{3}$$ respectively.

Moreover, the final solution will be,

$${y}_{1}=\sum_{i=0}^{\infty }{y}_{1,n},{y}_{2}=\sum_{i=0}^{\infty }{y}_{2,n},{y}_{3}=\sum_{i=0}^{\infty }{y}_{3,n}.$$
1. 2-

Picard Solution:

Using Picard method to the system (26), the solution algorithm will be,

$$\begin{array}{c}{y}_{\mathrm{1,0}}=0,\hspace{1em}{y}_{1,j+1}={y}_{\mathrm{1,0}}+{}^{AB}{I}^{\alpha }[{({y}_{2,j})}^{2}],\\ \begin{array}{l}{y}_{\mathrm{2,0}}=\frac{{t}^{\alpha }}{\Gamma \left(1+\alpha \right)},\hspace{1em}{y}_{2,j+1}={y}_{\mathrm{2,0}}+{}^{AB}{I}^{\alpha }\left[t{y}_{1,j}\right],\\ {y}_{\mathrm{3,0}}=\frac{{t}^{\alpha }}{\Gamma \left(1+\alpha \right)},\hspace{1em}{y}_{3,j+1}={y}_{\mathrm{3,0}}+{}^{AB}{I}^{\alpha }\left[{y}_{1,j}{y}_{2,j}\right],\end{array}\end{array}$$
(30)

Moreover, the final solution will be,

$${y}_{1}=\underset{n\to \infty }{{\text{lim}}}{y}_{1,n},{y}_{2}=\underset{n\to \infty }{{\text{lim}}}{y}_{2,n},{y}_{3}=\underset{n\to \infty }{{\text{lim}}}{y}_{3,n}.$$

Figure 4a-c show ADM solutions of $${y}_{1}$$, y2 and $${y}_{3}$$ at different values of $$\alpha$$ ($$\alpha =\mathrm{0.85,0.9,0.95,1}$$).

While, Fig. 4d-f show Picard solutions of $${y}_{1}$$, y2 and $${y}_{3}$$ at the same values of $$\alpha .$$

We see from the above figures that ADM solutions of $$({y}_{1},{y}_{2} \,and\, {y}_{3})$$ are coincide with Picard solutions at the same values of $$\alpha .$$

## Conclusions

In this research, we use two interesting methods (ADM and Picard methods) to solve a system of nonlinear fractional differential equations of Atangana–Baleanu sense; these two methods give analytical solutions, which coincide with each other (see Figs. 1, 2, 3, 4 and 5). In addition, these two methods give good approximate analytical solutions as we compared them with the exact solution (see Example 2) and from these results, we see that Picard method give more accurate results than ADM but ADM take less time than Picard (see Tables 1, 2 and 3).

## Availability of data and materials

Data can be shared.

## Abbreviations

FDEs:

Fractional differential equations

## References

1. Hammad HA, De la Sen M (2021) Tripled fixed point techniques for solving system of tripled-fractional differential equations. AIMS Mathem 6(3):2330–2343

2. Daraghmeh A, Qatanani N, Saadeh A (2020) Numerical solution of fractional differential equations. Appl Math 11:1100–1115. https://doi.org/10.4236/am.2020.1111074

3. Syam MI, Al-Refai M (2019) Fractional differential equations with Atangana–Baleanu fractional derivative: analysis and applications. Chaos Solit Fractals X (2):1–5

4. Fernandez AA (2021) Complex analysis approach to Atangana-Baleanu fractional calculus. Math Meth Appl Sci 44:8070–8087. https://doi.org/10.1002/mma.5754FERNANDEZ8087

5. Khan NA, Razzaq OA, Ara A, Riaz F (2016) Numerical solution of system of fractional differential equations in imprecise environment. https://doi.org/10.5772/64150

6. Atangana A, Alabaraoye E (2013) Solving a system of fractional partial differential equations arising in the model of HIV infection of CD4+ cells and attractor one-dimensional Keller-Segel equations. Adv Difference Equ 94:1–14

7. Rida SZ, Arafa AAM (2011) New method for solving linear fractional differential equations. Int J Differ Equ 1–8. https://doi.org/10.1155/2011/814132

8. Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley-Interscience, New York

9. Podlubny I (1999) Fractional differential equations. Academic, New York

10. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, New York

11. Abd El-Salam ShA, El-Sayed AMA (2007) On the stability of some fractional-order non-autonomous systems. Electron J Qual Theory Differ Equ 6:1–14

12. El-Sayed AMA, Abd El-Salam ShA (2008) On the stability of a fractional-order differential equation with nonlocal initial condition. Electron J Qual Theory Differ Equ 29:1–8

13. El-Sayed AMA, Hashem HHG, Ziada EAA (2012) Picard and Adomian Methods for coupled systems of quadratic integral equations of fractional order. J Nonlinear Anal Optim Theory Appl 3(2):171–183

14. El-Sayed AMA, Hashem HHG, Ziada EAA (2014) Picard and Adomian decomposition methods for a quadratic integral equation of fractional order. Comput Appl Math 33:95–109

15. El-Sayed AMA, Hashem HHG, Ziada EAA (2010) Picard and Adomian methods for quadratic integral equation. Comput Appl Math 29:447–463

16. Evans DJ, Raslan KR (2005) The Adomian decomposition method for solving delay differential equation. Int J Comput Math (UK) 82:49–54

17. Zwillinger D (1997) Handbook of differential equations. Academic, USA

18. El-Sayed AMA, El-Kalla IL, Ziada EAA (2010) Analytical and numerical solutions of multi-term nonlinear fractional orders differential equations. Appl Numer Math 60(8):788–797

19. Hefferan JM, Corless RM (2006) Solving some delay differential equations with computer algebra. Mathematical Scientist 31(1):1–22

20. Ziada E (2013) Numerical solution for nonlinear quadratic integral equations. J Fract Calc Appl 7(7):1–12

21. El-Mesiry EM, El-Sayed AMA, El-Saka HAA (2005) Numerical methods for multi-term fractional (arbitrary) orders differential equations. Appl Math Comput 160(3):683–699

22. Ahmed E, El-Sayed AMA, El-Saka HAA (2007) Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. J Math Anal Appl 325(1):542–553

23. El-Sayed AMA (1993) Linear differential equations of fractional orders. J Appl Math Comput 55(1):1–12

24. Adomian G (1994) Solving frontier problems of physics: the decomposition method. Kluwer Academic Publishers, Boston

27. Adomian G (1989) Nonlinear stochastic systems: theory and applications to physics. Kluwer Academic Publishers, Dordrecht

28. Abbaoui K, Cherruault Y (1994) Convergence of Adomian’s method applied to differential equations. Comput Math Appl 28:103–109

29. Cherruault Y, Adomian G, Abbaoui K, Rach R (1995) Further remarks on convergence of decomposition method. Int J Bio-Med Comput 38:89–93

30. Shawaghfeh NT (2002) Analytical approximate solution for nonlinear fractional differential equations. J Appl Math Comput 131:517–529

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Ziada, E.A.A. Picard and Adomian solutions of nonlinear fractional differential equations system containing Atangana – Baleanu derivative. J. Eng. Appl. Sci. 71, 31 (2024). https://doi.org/10.1186/s44147-024-00361-6