Approximate Iterative Method for Initial Value Problem of Impulsive Fractional Differential Equations with Generalized Proportional Fractional Derivatives

Ravi P Agarwal,Snezhana Hristova,Ricardo Almeida,Donal O’Regan

doi:10.3390/math9161979

Abstract

The main aim of the paper is to present an algorithm to solve approximately initial value problems for a scalar non-linear fractional differential equation with generalized proportional fractional derivative on a finite interval. The main condition is connected with the one sided Lipschitz condition of the right hand side part of the given equation. An iterative scheme, based on appropriately defined mild lower and mild upper solutions, is provided. Two monotone sequences, increasing and decreasing ones, are constructed and their convergence to mild solutions of the given problem is established. In the case of uniqueness, both limits coincide with the unique solution of the given problem. The approximate method is based on the application of the method of lower and upper solutions combined with the monotone-iterative technique.

Highlights

Fractional differential equations are effective in both theoretical and applied mathematics and arise in models of medicine, engineering, biochemistry, thermal and mechanical systems, acoustics and modeling of materials, etc
We note that initial value problems for Riemann–Liouville fractional differential equations differ from the Caputo fractional ones and requires a separate study
In the case of uniqueness, both limits coincide with the unique solution of the given problem

Summary

Introduction

Fractional differential equations are effective in both theoretical and applied mathematics and arise in models of medicine, engineering, biochemistry, thermal and mechanical systems, acoustics and modeling of materials, etc. We note that initial value problems for Riemann–Liouville fractional differential equations differ from the Caputo fractional ones and requires a separate study. A new algorithm for approximate solving an initial value problem for scalar non-linear fractional differential equations with generalized proportional fractional derivative is proposed. This method is based on the application of the method of lower and upper solutions and the monotone-iterative technique. Note that the generalized proportional fractional derivative of Riemann–Liouville fractional type leads to an appropriate definition of the impulsive conditions similar to the initial condition (see the last two equations in problem (1).

Explicit Solution of the Impulsive Linear Fractional Equation

Monotone-Iterative Technique

Conclusions