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\title{A new optimized method for solving variable-order fractional differential equations}

\author{H. Hassani$^1$, M.Sh. Dahaghin$^2$, M.H. Heydari$^3$\\
\footnotesize{$^{1,2}$Department of Mathematics, Shahrekord University, Shahrekord, Iran.}\\
\footnotesize{$^{3}$Department of Mathematics, Yazd University, Yazd, Iran.}\\
\footnotesize{e-mail:$^1$ hosseinhassani40@yahoo.com, $^2$ msh-dahaghin@sci.sku.ac.ir, $^3$ heydari@stu.yazd.ac.ir}}


\begin{document}
\maketitle


\begin{abstract}
	Variable-order fractional derivatives are an extension of constant-order fractional derivatives and have been introduced in several physical fields.
	 Since the equations described by the variable-order derivatives are highly complex and also difficult to handle analytically, it is advisable to consider their numerical solutions.
In this paper, a new optimized method based on polynomials is proposed for solving variable-order fractional differential equations (VOFDEs) and systems of variable-order fractional differential equations (SVOFDEs). To do this, a general polynomial of degree $k$ with unknown coefficients is considered as an approximate solution for the problem under study. By using the initial conditions some of these unknown coefficients are obtained. Finally the rest of these unknown coefficients are obtained optimally by minimizing error of 2-norm of the approximate solution in a desired interval. In order to demonstrate the efficiency of the proposed method some numerical examples are given. The obtained results show that, the proposed method is very accurate for such problems.
\end{abstract}

{\bf Keywords:} Variable-order fractional differential equations, Optimized method, Polynomial, Free coefficients, Fixed coefficients.

\section{Introduction}
FDEs are generalized form integer order ones, which are obtained by replacing integer order derivatives by fractional order ones \cite{1,2}. FDEs have been successfully applied in various fields of physics and engineering such as biophysics, bioengineering, quantum mechanics, finance, control theory, image and signal processing, viscoelasticity and material sciences \cite{3,4}. Most of FDEs do not have exact solutions, so approximate and numerical techniques \cite{5,6,7,8}, should be used. Several numerical and approximate methods such as variational iteration method \cite{5}, homotopy perturbation method \cite{9}, Adomian decomposition method \cite{10}, homotopy analysis method \cite{11} and collocation methods in  \cite{12,13} have been given in recent years to solve FDEs.\\
The variable-order fractional derivative, which is an extension of constant-order fractional derivative has been introduced in several physical fields \cite{14,15,16}. Many authors have introduced different definitions of variable-order differential operators, each of these with a specific meaning to suit desired goals. These definitions such as Riemann-Liouville, Grünwald, Caputo, Riesz \cite{17,18,19}, and some notes as Coimbra definition \cite{20,21}. For a VOFDE, it is usually difficult to obtain an analytical solution. Thus there is a need to develop numerical methods for VOFDEs. Although there exist enormous literatures on the numerical investigation for constant fractional order differential equations, the investigation of numerical methods of variable- order FDEs are quite limited. Several numerical methods have been proposed for VOFDEs in recent years, e.g. \cite{22,23,24,25,26,27,28,29,30,31}.\\
In this paper, we consider the general form of the VOFDEs as:
\begin{equation}\label{a1}
{}^c_0 D^{\alpha(t)}_tu(t)=f(t,u(t),{}^c_0 D^{\alpha_{1}(t)}_tu(t),{}^c_0 D^{\alpha_{2}(t)}_tu(t),\ldots,{}^c_0 D^{\alpha_{n}(t)}_tu(t)),~~~~t\in[0,1]
\end{equation}
subject to the initial conditions:
\begin{equation}\label{a2}
u^{(i)}(0)=u^{(i)}_{0},~~~~~~~~~~~i=0,1,...,q-1,
\end{equation}
where $q$ is the integer such that $q-1<\alpha(t)\leq q$, $0<\alpha_{1}(t)<\alpha_{2}(t)<\ldots<\alpha_{n}(t)<\alpha(t)$. Also, the real numbers $u^{(i)}_{0}$, $i=0,1,...,q-1$, are assumed to be given. Moreover, $D^{\alpha(t)}_{t}u(t)$ denotes the variable-order fractional derivative of order $\alpha(t)$ in the Caputo sense for $u(t)$, which is defined in \cite{25,26} by:
\begin{equation}\label{a3}
{}^c_0 D^{\alpha(t)}_tu(t)=\frac{1}{\Gamma(q-\alpha(t))}\int_{0}^{t}(t-s)^{q-\alpha(t)-1}\frac{d^{q}u(s)}{ds^{q}}ds,~~t>0.
\end{equation}
It should be denote that in Eq. (3) for any $q-1<\alpha(t)\leq q$, we have the following useful property \cite{26}:
\begin{equation}\label{a4}
{}^c_0 D^{\alpha(t)}_tt^{m}=\left\{
 \begin{array}{ll}
 \displaystyle \frac{\Gamma(m+1)}{\Gamma(m-\alpha(t)+1)}t^{m-\alpha(t)}, & q\leq m\in \mathbb{N}, \\ \noalign{\medskip}
 0, & o.~w.
 \end{array}
 \right.
\end{equation}
Also, we know that the Taylor series approximation for $u(t)$ at $t_0$ is:
\begin{equation}\label{a5}
u(t)=\sum_{i=0}^{i=n}a_{i}~(t-t_{0})^{i}
\end{equation}
with the coefficients:
\begin{equation}\label{a6}
a_i=\frac{1}{i!}u^{(i)}(t_0)~~,~~i=0,1,\ldots,n.
\end{equation}
The main aim of this paper is to propose an efficient and accurate optimized method based on the polynomials for solving VOFDEs. In particular the efficiency and reliability of the proposed technique will be demonstrated through extensive numerical analysis, considering several examples with known exact solution.\\
The structure of the remainder of this paper is as follows: In Section 2, the proposed method is described for solving the problem under study. In Section 3, some numerical examples are given. Finally, in Section 4 a conclusions is derived.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Description of the proposed method}\label{S2}
 In this section, we apply a new optimized method based on polynomials to find approximate solutions for VOFDEs in Eq. \ref{a1}. To do this, let approximate the unknown function $\tilde{u}(t)$ by a polynomial of degree $m$ as:
\begin{equation}\label{a7}
\tilde{u}(t,a_1,a_2,...,a_m)=\sum_{i=0}^{i=m}a_i~t^i.
\end{equation}
We use the Taylor series by a different approach to obtain the unknown coefficients $a_{i},~ i=0, 1,\ldots, m$. These coefficients are divided to fixed and free coefficients. The fixed coefficients are determined by using the initial conditions (2), and the free coefficients are obtained optimally by minimizing the error in 2-norm of the residual function in each desired interval. Note that, if we have two initial conditions, the coefficients $a_{0}$ and $a_{1}$ are the fixed coefficients which are obtained by the initial conditions and also the coefficients $a_{2}, a_{3}, \ldots, a_{m}$ are the free coefficients which are obtained by the mentioned method.
%\begin{thm}\label{TE1}
%Let $X$ be a normed space and $Y\subseteq X$. Then
%\begin{equation}\label{a7}
%\forall x_{0}\in X~~~~\exists y_{0}\in~Y~~:~~~\parallel x_{0}-y_{0}\parallel~\leq ~\parallel x_{0}-y\parallel~~~~~\forall~~y\in~Y.\nonumber
%\end{equation}
%\end{thm}
\subsection{Function Approximation}
Let $X=L^{2}[0,1]$, and assume that $P_{m}(t)=[1,t,t^{2},\ldots,t^{m}]\subset X$, $Y=span \{1,t,t^{2},\ldots,t^{m}\}$ and $x$ be an arbitrary element in $X$. Since $Y$ is a finite dimensional vector subspace of $X$, $x$ has a unique best approximation out of $Y$ such as $y_{0}\in Y$, that is
\begin{equation}\label{a71}
\forall y\in Y,~~~\parallel x-y_{0}\parallel~\leq ~\parallel x-y\parallel.\nonumber
\end{equation}
Since $y_{0}\in Y$, there exist the unique coefficients $a_{0},~a_{1},\ldots,~a_{m}$, such that
\begin{equation}\label{a711}
x\simeq y_{0}=\sum_{i=0}^{i=m}~a_{i}~t^{i}=~A^{T}~P_{m}(t).\nonumber
\end{equation}
and
\begin{equation}\label{a7111}
A^{T}=[a_{0},a_{1},\ldots,a_{m}].\nonumber
\end{equation}
Let the function $\tilde{u}(t)$ defined in Eq. (\ref{a7}) be an approximate solution for Eq. (\ref{a1}). By using the initial conditions, we have
\begin{equation}\label{ee0}
a_{i}=\frac{1}{i!}\tilde{u}^{(i)}(0),~~~~~~~i=0,1,\ldots,n-1.
\end{equation}
The coefficients $a_{0}, a_{1},\ldots, a_{n-1}$ are chosen as the fixed coefficients. Now by replacing $\tilde{u}(t)$ and corresponding derivatives in Eq. (\ref{a1}), we define the residual function:
\begin{equation} \label{a1111}
g(t,a_{n},a_{n+1},\ldots,a_{m})={}^c_0 D^{\alpha(t)}_t\tilde{u}(t)-f(t,\tilde{u}(t),{}^c_0 D^{\alpha_{1}(t)}_t\tilde{u}(t),{}^c_0 D^{\alpha_{2}(t)}_t\tilde{u}(t),\ldots,{}^c_0 D^{\alpha_{n}(t)}_t\tilde{u}(t)),
\end{equation}
and then the error function:
\begin{equation}\label{a9}
\bold{e}(a_{n},a_{n+1},\ldots,a_{m})=\int_0^1g^2(t,a_{n},a_{n+1},\ldots,a_{m})dt.
\end{equation}
Now to obtain an approximate solution for Eq. (\ref{a1}), we chose the free coefficients, $a_{n},a_{n+1},\ldots,a_{m}$ optimally. to do this, we solve the following system of algebraic equations:
\begin{equation}\label{a10}
\displaystyle \begin{array}{lll}
\frac{\partial \bold{e}(a_{n},a_{n+1},\ldots,a_{m})}{\partial a_{i}}=0~,~~i=n,n+1,\ldots,m
\end{array}.\nonumber
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Numerical results}\label{S3}
The purpose of this section is to show that the proposed method designed in this paper provides good approximations for VOFDEs. It is worth mentioning that all numeric computation is performed by MAPLE software with enough decimal digits.
\begin{example}\label{ex1}
Consider the following VOFDE:
\begin{equation}\label{a100}
{}^c_0 D^{\alpha(t)}_tu(t)+u(t)=\frac{2t^{2-\alpha(t)}}{\Gamma(3-\alpha(t))}-\frac{t^{1-\alpha(t)}}{\Gamma(2-\alpha(t))}+t^2-t
\end{equation}
where $0<\alpha(t)\leq 1$ and the initial condition is $u(0)=0$. It can be verified that the exact solution for this problem is $u(t)=t^2-t$. This problem is also solved by the proposed method for $\alpha(t)=1-0.5e^{-t}$. We estimate $u(t)$ by truncation Eq. (7) after the five terms, then we have:
\begin{equation}
\tilde{u}(t,a_0,a_1,a_2,a_3,a_4)=a_{0}+a_{1}t+a_{2}t^{2}+a_{3}t^{3}+a_{4}t^{4}.
\end{equation}
Coefficient $a_0$ is chosen as the fixed coefficient. By the initial condition we have, $a_0=0$. Coefficients $a_1$, $a_2$, $a_3$ and $a_4$ are chosen as the free coefficients. Therefore the approximate solution for Eq. (\ref{a100}) is given as:
\begin{equation}\label{a101}
\tilde{u}(t,a_1,a_2,a_3,a_4)=a_{1}t+a_{2}t^{2}+a_{3}t^{3}+a_{4}t^{4}.
\end{equation}
Substitute Eq. (\ref{a101}) into Eq. (\ref{a100}) and define the residual function:
\begin{equation}
g(t,a_1,a_2,a_3,a_4)={}^c_0 D^{\alpha(t)}_t\tilde{u}(t,a_1,a_2,a_3,a_4)+\tilde{u}(t,a_1,a_2,a_3,a_4)
-\frac{2t^{2-\alpha(t)}}{\Gamma(3-\alpha(t))}+\frac{t^{1-\alpha(t)}}{\Gamma(2-\alpha(t))}-t^2+t.
\end{equation}
and the error function:
\begin{equation}
\bold{e}(a_1,a_2,a_3,a_4)=\int_0^1g^2(t,a_1,a_2,a_3,a_4)dt.
\end{equation}
The values for the free coefficients are obtained by minimizing $\bold{e}(a_1,a_2,a_3,a_4)$ as:
\begin{equation}\label{a12}
\displaystyle \begin{array}{lll}
\frac{\partial \bold{e}(a_1,a_2,a_3,a_4)}{\partial a_i}=0~,~~i=1,2,3,4.
\end{array}\nonumber
\end{equation}
By solving the above system of algebraic equations the free coefficients are obtained as:
\begin{align*}
a_1=-1,~~~~~~a_2=1,~~~~~~a_3=0,~~~~~~a_4=0,
\end{align*}
and therefore we gain the exact solution.
\end{example}


\begin{example}\label{ex2}
Consider the following VOFDE:
\begin{equation} \label{b1}
{}^c_0 D^{\alpha(t)}_tu(t)+\sin t~{}^c_0 D^{\beta(t)}_tu(t)+\cos t~u(t)=\frac{6t^{3-\alpha(t)}}{\Gamma(4-\alpha(t))}+\frac{6\sin t~t^{3-\beta(t)}}{\Gamma(4-\beta(t))}+t^3\cos t
\end{equation}
where the initial conditions are $u(0)=u^{'}(0)=0$, and $1<\alpha(t)\leq 2,~0<\beta(t) \leq 1$. The exact solution is $u(t)=t^{3}$. This problem is also solved by the proposed method for $\alpha(t)=2-\sin^{2}(t)$ and $\beta(t)=1-\frac{e^{-t^{3}}}{6}$. Consider the truncation of Eq. (7) with first five terms as:
\begin{equation}
\tilde{u}(t)=a_{0}+a_{1}t+a_{2}t^{2}+a_{3}t^{3}+a_{4}t^{4}.
\end{equation}
By applying the initial conditions, we obtain the fixed coefficients as:
\begin{equation*}
a_{0}=\tilde{u}(0)=0~~~~,~~~~a_{1}=\tilde{u}^{'}(0)=0.
\end{equation*}
Coefficients $a_{2}$, $a_{3}$ and $a_{4}$ are chosen as the free coefficients. By applying the same process in Example 1, the free coefficients are obtained as:
\begin{equation*}
a_{2}=0,~~~~~~a_{3}=1,~~~~~~a_{4}=0,
\end{equation*}
and therefore we gain the exact solution.
\end{example}
\begin{example}\label{ex3}
Consider the problem:
\begin{equation} \label{a13}
{}^c_0 D^{\alpha(t)}_tu(t)+e^{t}~{}^c_0 D^{\beta(t)}_tu(t)+\frac{2}{2t-1}~{}^c_0 D^{\gamma(t)}_tu(t)+\sqrt
t~u(t)=\frac{2e^{t}t^{2-\beta(t)}}{\Gamma(3-\beta(t))}+\frac{4t^{2-\gamma(t)}}{(2t-1)\Gamma(3-\gamma(t))}+t^{\frac{5}{2}}
\end{equation}
where $2<\alpha(t)\leq 3$, $1<\beta(t) \leq 2$ and $0<\gamma(t) \leq 1$, with the initial conditions $u(0)=u'(0)=0$, $u''(0)=2$. The exact solution is $u(t)=t^2$. We also solve this problem by the proposed method for $\alpha(t)=3-\frac{1}{3}e^{-t}$, $\beta(t)=2-\cos^2t$ and $\gamma(t)=1-\frac{1}{2}\cos t$. Consider the truncation of Eq. (7) with first four terms as:
\begin{equation}
\tilde{u}(t)=a_{0}+a_{1}t+a_{2}t^{2}+a_{3}t^{3}.
\end{equation}
By applying the initial conditions, we obtain the fixed coefficients as:
$$a_{0}=\tilde{u}(0)=0~~~,~~~a_{1}=\tilde{u}^{'}(0)=0~~~,~~~a_{2}=\frac{\tilde{u}^{''}(0)}{2}=1.$$
Coefficient $a_{3}$ is chosen as the free coefficient. By applying the same process in previous examples, the free coefficient is obtained as:
$$a_{3}=0.$$
Thus, we get $\tilde{u}(t)=t^2$, which is the exact solution.
\end{example}
\begin{example}\label{ex4}
Consider the following system of VOFDEs:
\begin{equation}
\left\{
\begin{array}{ll}
 {}^c_0 D^{\alpha(t)}_tu(t)+v(t)= \frac{2 t^{2-\alpha(t)}}{\Gamma(3-\alpha(t))}+t^{3} \\ \noalign{\medskip}
{}^c_0 D^{\beta(t)}_tv(t)- t^{2}u(t)=
\frac{6t^{3-\beta(t)}}{\Gamma(4-\alpha(t))}-t^{4}
\end{array}
\right.
\end{equation}
where $0<\alpha(t), \beta(t)\leq 1$, with the initial conditions $u(0)=v(0)=0$. The exact solutions are $u(t)=t^2$ and $v(t)=t^3$. We also have solved this system by the proposed method for $\alpha(t)=1-\cos^2(t)$ and $\beta(t)=1-\frac{1}{5}e^{-t}$. Consider the truncation of Eq. (7) with first four terms as:
\begin{equation}
\left\{\begin{array}{ll}
\tilde{u}(t)=a_{0}+a_{1}t+a_{2}t^{2}+a_{3}t^{3}, \\
\noalign{\medskip} \tilde{v}(t)=b_{0}+b_{1}t+b_{2}t^{2}+b_{3}t^{3}.
\end{array}\right.
\end{equation}
By applying the initial conditions, we obtain the fixed coefficients as:
$$a_0=\tilde{u}(0)=0~~~,~~~b_0=\tilde{v}(0)=0.$$
Coefficients $a_1$, $a_2$, $a_3$, $b_1$, $b_2$, $b_3$ are chosen as the free coefficients. Define the error function:
\begin{equation} \label{a14}
\bold{e}(a_1,a_2,a_3,b_1,b_2,b_3)=\int_0^1\bigg(g_1^2(t,a_1,a_2,a_3,b_1,b_2,b_3)+g_2^2(t,a_1,a_2,a_3,b_1,b_2,b_3)\bigg)dt
\end{equation}
where
\begin{equation}
\left\{
\begin{array}{ll}
g_{1}(t,a_{1},a_{2},a_{3},b_{1},b_{2},b_{3})={}^c_0 D^{\alpha(t)}_t\tilde{u}(t)+\tilde{v}(t)-
\frac{2t^{2-\alpha(t)}}{\Gamma(3-\alpha(t))}-t^{3} \\ \noalign{\medskip}
g_{2}(t,a_{1},a_{2},a_{3},b_{1},b_{2},b_{3})={}^c_0 D^{\beta(t)}_t\tilde{v}(t)-
t^{2}\tilde{u}(t)-\frac{6t^{3-\beta(t)}}{\Gamma(4-\alpha(t))}+t^{4}.
\end{array} \right.
\end{equation}
The values for the free coefficients are obtained by minimizing $\bold{e}(a_{1},a_{2},a_{3},b_{1},b_{2},b_{3})$ as following:
$$a_{1}=0~~~,~~~~b_{1}=0~~,~~a_{2}=1~~~~,~~~~b_{2}=0~~,~~a_{3}=0~~~~,~~~~b_{3}=1.$$
Thus, we using Eq. (21) we get $\tilde{u}(t)=t^2$ and $\tilde{v}(t)=t^3$, which are the exact solutions.
\end{example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}\label{S4}
In this paper, an efficient and accurate computational method based on polynomials was proposed to obtain approximate solutions for the VOFDEs. The proposed method is very convenient for solving the problem under study. Several examples are given to demonstrate the powerfulness of the proposed method. Also this method has been successfully applied to calculate the approximate solutions for systems of VOFDEs.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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