Integral Equations of Non-Integer Orders and Discrete Maps with Memory

Vasily E Tarasov

doi:10.3390/math9111177

Abstract

In this paper, we use integral equations of non-integer orders to derive discrete maps with memory. Note that discrete maps with memory were not previously derived from fractional integral equations of non-integer orders. Such a derivation of discrete maps with memory is proposed for the first time in this work. In this paper, we derived discrete maps with nonlocality in time and memory from exact solutions of fractional integral equations with the Riemann–Liouville and Hadamard type fractional integrals of non-integer orders and periodic sequence of kicks that are described by Dirac delta-functions. The suggested discrete maps with nonlocality in time are derived from these fractional integral equations without any approximation and can be considered as exact discrete analogs of these equations. The discrete maps with memory, which are derived from integral equations with the Hadamard type fractional integrals, do not depend on the period of kicks.

Highlights

The first mathematical model of processes with memory has been proposed by LudwigBoltzmann in 1874 and 1876 [1,2,3] for isotropic viscoelastic media
The processes with memory were described in a book by Vito Volterra in 1930 [4,5], where the integral equations were used to take into account fading memory
We derive exact solutions of fractional integral equations with periodic kicks. These solutions are obtained for arbitrary positive order of integral equations. These maps with nonlocality in time and memory are obtained from exact solutions to these fractional integral equations for discrete time points

Summary

Introduction

The first mathematical model of processes with memory has been proposed by Ludwig. Boltzmann in 1874 and 1876 [1,2,3] for isotropic viscoelastic media. The presence of memory in a process means that this process depends on the history of changes of the process in the past during a finite time interval Such processes cannot be described by differential equations containing only derivatives of integer order with respect to time. Note that discrete maps with memory were not previously derived from fractional integral equations of non-integer orders Such a derivation of discrete maps with memory is proposed for the first time in this work. We derive exact solutions of fractional integral equations with periodic kicks These solutions are obtained for arbitrary positive order of integral equations. These maps with nonlocality in time and memory are obtained from exact solutions to these fractional integral equations for discrete time points. The proposed discrete maps describe discrete-time dynamics of systems with nonlocality in time, and periodic kicks

Riemann–Liouville Fractional Integral and Derivative

Discrete Maps from Fractional Integral Equations with Riemann–Liouville

Hadamard Type Fractional Integral and Derivative Operators and Its Properties

Discrete Map from Fractional Integral Equation with Hadamard Type Fractional

Conclusions