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In this paper, we provide a new approach to solve approximately a system of fractional differential equations (FDEs). We extend this approach for approximately solving a fractional-order differential equation model of HIV infection of CD4<sup>+</sup>T cells with therapy effect. The fractional derivative in our approach is in the sense of Riemann-Liouville. To solve the problem, we reduce the system of FDE to a discrete optimization problem. By obtaining the optimal solutions of new problem by minimization the total errors, we obtain the approximate solution of the original problem. The numerical solutions obtained from the proposed approach indicate that our approximation is easy to implement and accurate when it is applied to a systems of FDEs.

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 [

In this paper, at first, we approximate the fractional derivative by a finite difference method and then use the AVK approach [^{+}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.

Consider the following fractional-order differential equation model of HIV infection of CD4^{+}T cells [

with the initial conditions

Following Theorem 1 of [

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

Parameter | Value/unit |
---|---|

^{+}T) | |

^{+}T) | |

^{+}T become infected with virus) | |

r (Rowth rate of CD4^{+}T population) | |

N (Number of virions produced by infected CD4^{+}T) | Varies |

^{+}T) | |

s (Source term for uninfected CD4^{+}T) | |

^{+}T population for HIV-negative persons) |

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.

Consider a general system of FDEs as follows:

where

Definition 1. For problem (3) we define the following functional

where

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 [

Theorem 1. Let h be a nonnegative continuous function on

Corollary 1. Necessary and sufficient condition for the trajectory

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

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

Define

In order to better illustrate the numerical approach, we also introduce the following difference operator:

Then,

Hence

suppose

Thus, we simply get problem (8) in the following form:

in which,

We solved this optimization problem by linear programming (LP) formulation which is done in what follows.

Lemma 1. Let pairs

where I is a compact set. Then

Proof. Since,

So

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

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

derivatives are derived from

Now, assume that

In this example, the maximum absolute errors computed by Equation (16) for

Example 2. Consider the following initial value problem:

with initial condition

We know that

K | |
---|---|

5 | |

10 | |

20 | |

40 | |

50 | |

100 |

In the case of

From numerical results we can indicate that the solution of FDE approaches to the solution of integer order differential equation, whenever

Example 3. Consider the following FDE:

where

The exact solution of this equation is

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

where

For numerical simulations we assumed 350 days for treatment period. With the change of variables

m | |||
---|---|---|---|

10 | 0.26227 | ||

20 | 0.11574 | ||

40 | 0.08285 | ||

50 | 0.02621 | ||

100 | 0.00748 |

t | ||||
---|---|---|---|---|

0.0 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

0.1 | −0.089978 | −0.090000 | −0.089586 | −0.090000 |

0.2 | −0.159889 | −0.160000 | −0.159688 | −0.160000 |

0.3 | −0.209891 | −0.210000 | −0.209707 | −0.210000 |

0.4 | −0.239974 | −0.240000 | −0.239787 | −0.240000 |

0.5 | −0.249896 | −0.250000 | −0.249738 | −0.250000 |

0.6 | −0.239998 | −0.240000 | −0.239795 | −0.240000 |

0.7 | −0.199879 | −0.210000 | −0.209830 | −0.210000 |

0.8 | −0.160109 | −0.160000 | −0.159897 | −0.160000 |

0.9 | −0.096390 | −0.090000 | −0.100098 | −0.090000 |

we converted period

P | |||||
---|---|---|---|---|---|

T | |||||

I | |||||

V |

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

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.

Samaneh Soradi Zeid,Mostafa Yousefi,Ali Vahidian Kamyad, (2016) Approximate Solutions for a Class of Fractional-Order Model of HIV Infection via Linear Programming Problem. American Journal of Computational Mathematics,06,141-152. doi: 10.4236/ajcm.2016.62015