EXISTENCE AND UNIQUENESS SOLUTION FOR THREE-POINT HADAMARD-TYPE FRACTIONAL VOLTERRA BVP

Faraj Yacoob Ishak

doi:10.24843/mtk.2022.v11.i01.p363

Abstract

In this paper we study the existence and uniqueness solution for a first kind fractional Volterra boundary value problem involving Hadamard type and three-point boundary conditions. Our analysis is based on Krasnoselskii-Zabreiko’s fixed point theorem and Banach contraction principle. As an application we discuss a Hadamard type boundary value problem with fractional integral boundary conditions. We emphasize that our results are new in the context of Hadamard fractional calculus and are well illustrated with the aid of examples.

Highlights

The theory of fractional differential equations and inclusions has received a lot of attention in recent years
Fractional differential equations are derived from the mathematical modelling of systems and operations encountered in a wide variety of engineering and scientific disciplines, including physics, chemistry, aerodynamics, electrodynamics of complex media, polymer rheology, economics, control theory, signal and image processing, biophysics, and blood flow phenomena, among others (Ishak, 2020; Kilbas & Trujillo, 2003; Guotao et al, 2018; Ahmad et al, 2021; Sial et al, 2021; Ntouyas et al, 2021; Jhanthanam et al, 2019)
Another type of fractional derivative that appears in the literature alongside RiemannLiouville and Caputo derivatives is the Hadamard fractional derivative introduced in 1892 (Chen et al, 2013), which is distinguished from the preceding ones by the presence of a logarithmic function of any exponent in the kernel of the integral

Summary

INTRODUCTION

The theory of fractional differential equations and inclusions has received a lot of attention in recent years. It has become an important academic issue because to its numerous applications in the fields of physics, economics, and engineering sciences. Fractional differential equations are derived from the mathematical modelling of systems and operations encountered in a wide variety of engineering and scientific disciplines, including physics, chemistry, aerodynamics, electrodynamics of complex media, polymer rheology, economics, control theory, signal and image processing, biophysics, and blood flow phenomena, among others (Ishak, 2020; Kilbas & Trujillo, 2003; Guotao et al, 2018; Ahmad et al, 2021; Sial et al, 2021; Ntouyas et al, 2021; Jhanthanam et al, 2019). The majority of study on this issue has long been recognized to be based on RiemannLiouville and Caputo-type fractional differential equations. Existence and Uniqueness Solution for Three-Point Hadamard-Type Fractional Volterra BVP

PRELIMINARIES

MAIN RESULT

APPLICATION

EXAMPLES Example 1

CONCLUSION