Received 14 May 2016; accepted 24 June 2016; published 27 June 2016
In recent years, scientists have been interested in studying the fractional calculus and the FDEs in different fields of engineering, physics, mathematics, biology, finance, biomechanics and electrochemical processes (see  -  , for more details). Also, it has been shown that modelling the behavior of many biological systems that governed by FDEs has more advantages than classical integer-order modelling  . Readers interested in FDEs are referred to  -  . Although great efforts have been made to find numerical and analytical techniques for solving FDE, for example, predictor-corrector method  , the Adomian decomposition  , the variational iteration method  , collocation using spline functions  and matrix expression given by   , but most of these FDEs do not have analytic solutions.
In this paper, at first, we approximate the fractional derivative by a finite difference method and then use the AVK approach  to obtain a new approximate solution for the FDEs. This approach substitutes the FDEs with an equivalent minimization problem in which the optimal solution of this problem is the approximate solution of the original FDE. Moreover, since the error of this approach is minimized, the approximate solutions are the best solutions for the original problem. We employ this approximation to get numerical solution of a system of FDEs which has been used for modelling HIV infection of CD4+T cells.
The discussion of paper will be as follows: in the next section, we express the fractional HIV model and introduce the notations that used in the rest of this paper. In Section 3, we design an efficient approach to approximate the fractional derivative and use it in our numerical method for solving FDEs. Some numerical examples are displayed in Section 4. Finally, conclusions are included in the last section.
2. The Problem
Consider the following fractional-order differential equation model of HIV infection of CD4+T cells  :
with the initial conditions, and, in which the parameter values reported by Table 1.
Following Theorem 1 of  , we note that (1) along with its initial conditions possesses a unique solution which is non-negative. Throughout this paper, we set () as the Riemann-Liouville derivative of order defined by  :
The aim of this paper is to extend the application of the AVK approach to solve a fractional order model for this HIV infection model of CD4+T cells. So, in the next section, at first we convert the original FDE to an
Table 1. Variables and parameters for HIV infection model.
optimization problem based on minimization of error. By discretizing the new problem and approximating the Riemann-Liouville fractional derivative by a finite difference method, we obtaine the best approximate solution of the original FDE.
3. AVK Approach for Solving Approximately FDEs
Consider a general system of FDEs as follows:
where () is the Riemann-Liouville derivative of order, g is an riemann integrable time varying function, , and A is a compact subset in. Also called the state variable. We want to obtain an approximate solution of problem (3). Therefore, we need the following definition.
Definition 1. For problem (3) we define the following functional that is called the total error functional:
where is a non-negative functional, is any norm in space, such as where is defined as follows:
Here, we convert the problem (4) to a nonlinear programming (NLP) as follow:
Now, to reach the approximating solution for the original problem (3) it is sufficient to solve the minimization problem (6). Hence, we need the following mean theorem  and corollary.
Theorem 1. Let h be a nonnegative continuous function on, the necessary and sufficient condition for is that, on.
Corollary 1. Necessary and sufficient condition for the trajectory to be a solution of system (3) is that the optimal solution of (6) has zero objective function.
To develop the numerical solution of problem (6) approximately, we defined the grid size in time by
for some positive integer m, so the grid points in the time interval is given by,. In order to illustrate the numerical approach better, we introduce the following notations:
By the above notations, problem (6) is now approximated by the following optimization problem:
By using the ending point in any subinterval for approximating integrals, problem (7) is now approximated by the following optimization problem:
Now, we approximate fractional derivative as follows:
Define. Then, Equation (9) yields to
In order to better illustrate the numerical approach, we also introduce the following difference operator:
Hence or sampling time is very important, and must be chosen small, so the number of partitions is great. This is a trade off between sampling time and speed of problem solving. Using again trapezoidal rule in any subinterval for approximating integrals, except for the last interval that we use the midpoint approximation, and
suppose, for. Therefore,
Thus, we simply get problem (8) in the following form:
in which, for.
We solved this optimization problem by linear programming (LP) formulation which is done in what follows.
Lemma 1. Let pairs, , be the optimal solutions of the following LP problem:
where I is a compact set. Then, , is the optimal solution of the following NLP problem:
Proof. Since, , , is the optimal solution of the LP problem, so they satisfy the con- straints. Thus there is and for. Hence, , , and so
. Now, let there exist, , such that. Define, for. Then and. Moreover, and hence
So, which is a contradiction. See  more details.
Now, by lemma 1, problem (14) can be converted to the following equivalent LP problem:
By obtaining the solution of this problem, we recognize the value of unknown admissible, and.
4. Numerical Examples
In this section, we give some numerical examples and apply the method presented in the last sections for solving them. Moreover, we extend this approach for approximately solving a model of HIV infection of CD4+T cells with therapy effect including a system of FDEs. These test problems demonstrate the validity and efficiency of this approximation.
Example 1. As first example, we compute, with, for. The exact formulas of the
derivatives are derived from
Figure 1 shows the results by using approximation (10)-(13) for and various choices of m.
Now, assume that, , and are the approximated and exact solutions of system (3), respectively. We defined the absolute error of approximation as follow:
In this example, the maximum absolute errors computed by Equation (16) for and various choices of m, has been shown in Table 2.
Example 2. Consider the following initial value problem:
with initial condition.
We know that. Therefore, the analytic solution for system (17) is. Now weexpand the fractional derivative up to the problem (15). The solution is drawn in Figures 2-4 for m = 20, 50, 100 and.
Figure 1. Analytic solution and numerical approximation (10), with various choices of m and, for Example 1.
Table 2. Maximum absolute error for Example 1.
In the case of, the maximum absolute errors (16) with various choices of m is shown in Table 3.
From numerical results we can indicate that the solution of FDE approaches to the solution of integer order differential equation, whenever approaches to its integer value.
Example 3. Consider the following FDE:
Table 4 shows the exact solution and the approximate solution for equation (18) by solving problem (15) for and. The results compare well with those obtained in  .
Example 4. Now we want to solve the fractional-order differential equation model of HIV infection of CD4+T cells (1) For the parameter values given in Table 1. The system (1) can be expressed in a vector form as follows:
Figure 2. Exact and approximation solutions for problem in Example 2 with and different values of m.
Figure 3. Exact and approximation solutions for problem in Example 2 with and different values of m.
where is the state vector and
For numerical simulations we assumed 350 days for treatment period. With the change of variables,
Figure 4. Exact and approximation solutions for problem in Example 2 with and different values of m.
Table 3. Maximum absolute error for different values of for Example 2.
Table 4. Numerical values with and for Example 3.
Figure 5. Analytic solution and numerical approximation (15) for Example 3 for.
Figure 6. Analytic solution and numerical approximation (15) for Example 3 for.
we converted period to. Based on concepts was said in the previous section, the key to the derivation of the approach is to replace the system (19) by the following equivalent optimization problem:
Table 5. Maximum absolute error for and different values of for Example 4.
with the initial condition (20). To solve this optimization problem, by approximating integrals as before, we transformed (21) to a discretized problem in the following form:
In problem (21) and (22), the factor 350 is omitted because of having no effect on the solution of it. Then, the minimum problem (22) converted to a linear programming problem with the following change of variables:
Now, we approximate fractional derivatives from (10)-(13). Our approach introduces an approximate solution for the fractional HIV model based on minimization the total error. The maximum absolute errors (16) with m = 100 and different values of that shown in Table 5, confirmed the efficacy of our approach in comparison with the result obtained by  .
In this paper, the finite difference method discrete time AVK approach has been successfully used for finding the solutions of a system of FDEs such as a model for HIV infection of CD4+T cells. Our approach introduces an approximate solution for the FDEs based on the minimization of the total error. In the suggested method, the original problem reduces to an optimization problem. By discretizing the new problem and solving it, we obtain the best approximate solution of the original problem. Results represent a unifying approach for numerical approximation of differential equations of fractional order. Since this method is not based on point to point error, but according to its results, it is clear that there is no difference between the exact and approximate solutions in point to point case.
Three numerical examples are given and the results are compared with the exact solutions and with the other methods. It is shown that, as the order of fractional derivatives approaches to 1, the numerical solutions for the FDEs approach the clasicall solutions of the problem. Then we use this technique for finding approximate solutions of FDEs system of a model for HIV infection of CD4+T cells. The result demonstrates the validity of the approach.
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