Best approximation of a nonlinear fractional Volterra integro-differential equation in matrix MB-space

Reza Chaharpashlou,Reza Saadati

doi:10.1186/s13662-021-03275-2

Abstract

In this article, we introduce a class of stochastic matrix control functions to stabilize a nonlinear fractional Volterra integro-differential equation with Ψ-Hilfer fractional derivative. Next, using the fixed-point method, we study the Ulam–Hyers and Ulam–Hyers–Rassias stability of the nonlinear fractional Volterra integro-differential equation in matrix MB-space.

Highlights

1 Introduction Fractional calculus is considered as a branch of mathematical analysis which deals with the investigation and applications of integrals and derivatives of arbitrary order
The first approach is that Riemann-Liouville, which is based on iterating the classical integral operator n times and considering the Cauchy’s formula where n! is replaced by the Gamma function, and the fractional integral of noninteger order is defined
4 Conclusions We introduced a new model of stochastic matrix control functions which helped us to stabilize a pseudo-nonlinear fractional Volterra integral equation and get better approximation for it

Summary

Introduction

Fractional calculus is considered as a branch of mathematical analysis which deals with the investigation and applications of integrals and derivatives of arbitrary order. Fractional calculus has attracted the attention of many mathematicians, and of some researchers in other areas like physics, chemistry, and engineering. As it is well known, several physical phenomena are often better described by fractional derivatives. By proposing the study of solution stability via fractional integrals and fractional derivatives, we can generalize the results and obtain the usual ones as particular cases. We will use two recent fractional operators, that is, of general differentiation and integration [12] These concepts help us study the Hyers–Ulam (in short HU) and Hyers–Ulam–Rassias (in short HUR) stability of fractional nonlinear Volterra integro-differential equation (in short VIDE),. With ς ∈ [0, T] and a continuous function (in short CF) F(ς, μ), H(ς, θ, μ) is a CF with respect to ς , θ and μ on [0, T] × R × R, σ is a fixed number, H D0ι,+κ;Ψ μ(·) is defined in (2.1) in which 0 < ι < 1, 0 ≤ κ ≤ 1, and I01+–γ (·) is the Ψ -Riemann–Liouville fractional integral in which 0 ≤ γ < 1 [12]

Preliminaries

Main results

Conclusions