On the Approximate Solution of Fractional Logistic Differential Equation Using Operational Matrices of Bernstein Polynomials

R F Al-Bar

doi:10.4236/am.2015.612184

Abstract

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.

Highlights

Fractional Logistic Equation, Riemann-Liouville Fractional Derivatives, Riemann-Liouville Fractional Integral, Operational Matrix, Bernstein Polynomials. 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 ([1] [2])
Considerable attention has been given to the efficient numerical solutions of FDEs of physical interest, because it is difficult to find exact solutions
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 [25] and the multi-term orders fractional differential equations [26] and others [27]

Summary

Introduction

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 ([1] [2]). (2015) On the Approximate Solution of Fractional Logistic Differential Equation Using Operational Matrices of Bernstein Polynomials. The fractional Logistic model can obtain by applying the fractional derivative operator on the Logistic equation. The continuous Logistic model is described by first order ordinary differential equation. The solution of Logistic equation explains the constant population growth rate which does not include the limitation on food supply or spread of diseases [15]. The solution of continuous Logistic equation is in the form of constant growth rate as in formula. Khader and Hendy [21] introduced a new approximate formula of the fractional derivative using Legendre series expansion and used it to solve numerically the fractional delay equation. Special attention is given to study the convergence analysis and estimate an upper bound of the error of the introduced formula

Preliminaries and Notations

Bernstein Polynomials and Their Properties

BPs Operational Matrix of Riemann-Liouville Fractional Integration

Implementation of Bernstein Polynomials Operational Matrix for Solving FLDE

Conclusion and Remarks