_{1}

In this paper, operational matrices of Bernstein polynomials (BPs) are presented for solving the non-linear fractional Logistic differential equation (FLDE). The fractional derivative is described in the Riemann-Liouville sense. The operational matrices for the fractional integration in the Riemann-Liouville sense and the product are used to reduce FLDE to the solution of non-linear system of algebraic equations using Newton iteration method. Numerical results are introduced to satisfy the accuracy and the applicability of the proposed method.

It is well-known that the fractional differential equations (FDEs) have been the focus of many studies due to their frequent appearance in various applications, such as in fluid mechanics, viscoelasticity, biology, physics and engineering applications, for more details see for example ([

The fractional Logistic model can obtain by applying the fractional derivative operator on the Logistic equation. The model is initially published by Pierre Verhulst in 1838 ( [

The solution of Logistic equation explains the constant population growth rate which does not include the limitation on food supply or spread of diseases [

the multiplication factor up to saturation limit which is maximum carrying capacity [

where N is the population with respect to time,

In this article, we consider FLDE of the form

the parameter

We also assume an initial condition

The exact solution to this problem at

The existence and the uniqueness of the proposed problem (1) are introduced in details in ( [

Khader and Hendy [

In this section, we present some necessary definitions and mathematical preliminaries of the fractional calculus theory and the Bernstein polynomials that will be required in the present paper.

The Fractional Integral and Derivative OperatorsWe present some necessary definitions and mathematical preliminaries of the fractional calculus theory that will be required in the present paper.

Definition 1.

The Riemann-Liouville fractional integral operator

Definition 2.

The Riemann-Liouville fractional derivative operator

Definition 3.

The Caputo fractional derivative operator

Similar to integer-order differentiation, Caputo fractional derivative operator is a linear operation

where

We use the ceiling function

For more details on fractional derivatives definitions and its properties see ( [

Lemma 1.

If

Definition 4.

The

where

The Bernstein polynomials have the following properties

1)

2)

3)

4)

5) They satisfy symmetry

6)

7)

Since the set

We can write

Bernstein polynomials [

Theorem 1. [

The Bernstein polynomials operational matrix

Definition 5.

We can define the dual matrix

where

Lemma 2. [

Let

we can find the unique vector

from space

Definition 6.

Let

It can be shown that is uniformly convergent on the interval

Theorem 2.

Given a function

The Bernstein polynomials operational matrix are used for solving many class of fractional differential equations, they used to solve numerically the fractional heat-and wave-like equations [

In this section, we introduce a numerical algorithm using Bernstein polynomials operational matrix method for solving the fractional Logistic differential equation of the form (1).

The proposed technique will apply as in the following steps:

1) We use the initial condition (2) to reduce the given problem (1) to a problem with zero initial condition. So, we define

where

2) Substituting (17) in (1) and (2), we have an initial-value problem as follows

where

3) Using (10) in Lemma 1 we can write

4) Using Lemma 3.3 in [

where P and Q are known

5) From (9), (13), (19), (20) and (21), we have

where

6) By substituting (21) and (22) into (18), we obtain

7) Then, from Lemma 3.5 in [

Therefore we can reduce (23) by (24)-(26) as

We obtain the following non-linear system of algebraic equations

8) By solving this system we can obtain the vector C. Then, we can get

The numerical results of the proposed problem (1) are given in

From these figures we can conclude that the obtained numerical solutions are in excellent agreement with the exact solution.

In this article, we used operational matrices of the Riemann-Liouville fractional integral and the product by Bernstein polynomials for solving the fractional Logistic differential equation. The properties of these operational

matrices are used to reduce FLDE to a non-linear system of algebraic equations which solved by Newton iteration method. From the obtained numerical results, we can conclude that this method gives results with an excellent agreement with the exact solution. All numerical results are obtained using Matlab program 8.

We thank the Editor and the referee for their comments.

R. F.Al-Bar, (2015) On the Approximate Solution of Fractional Logistic Differential Equation Using Operational Matrices of Bernstein Polynomials. Applied Mathematics,06,2096-2103. doi: 10.4236/am.2015.612184