On numerical techniques for solving the fractional logistic differential equation

Yves Yannick Yameni Noupoue,Muath Awadalla,Yücel Tandoğdu

doi:10.1186/s13662-019-2055-y

Abstract

This paper studied the existence and uniqueness of the solution of the fractional logistic differential equation using Hadamard derivative and integral. Previous work has shown that there is not an exact solution to this fractional model. Hence several numerical approaches, such as generalized Euler’s method (GEM), power series expansion (PSE) method, and Caputo–Fabrizio (CF) method, were used to compute the solution. The classical solution obtained from the first order non-linear differential equation was also considered to enable the comparison of error levels.

Highlights

The name fractional calculus stems from the fact that the order of derivatives and integrals are fractions rather than integers
Consider a non-linear fractional differential equation, where the fractional derivative is taken in the Grünwald–Letnikov sense, with initial condition, defined by
3.3 The Caputo–Fabrizio method (CF) Consider a non-linear fractional differential equation with initial condition, where the fractional derivative is taken in the Caputo–Fabrizio sense, defined by CF Dq0u (t) = g t, u(t), u(0) = u0

Summary

Introduction

The name fractional calculus stems from the fact that the order of derivatives and integrals are fractions rather than integers. Definition 2.7 ([32]) The Hadamard fractional derivative of order q > 0 of a continuous function g : [a, ∞) → R is defined as. Consider a non-linear fractional differential equation, where the fractional derivative is taken in the Grünwald–Letnikov sense, with initial condition, defined by. 3.3 The Caputo–Fabrizio method (CF) Consider a non-linear fractional differential equation with initial condition, where the fractional derivative is taken in the Caputo–Fabrizio sense, defined by CF Dq0u (t) = g t, u(t) , u(0) = u0. Note that the initial value condition implies c1 = Na. Let H = C([a, T], R) denote the Banach space of all continuous functions from [a, T] to R, we identify the operator E : H → H endowed with the norm N = supa≤t≤T |N(t)|, t q–1 1 t t q–1 Q(s, N (s)). The existence of the solution of the initial value problem given by Eq (4.3) holds by Krasnoselskii’s fixed point theorem

Uniqueness of solution

Conclusion

Methods