A non-local problem with discontinuous matching condition for loaded mixed type equation involving the Caputo fractional derivative

Kishin Sadarangani,Obidjon Khayrullayevich Abdullaev

doi:10.1186/s13662-016-0969-1

Abstract

This research work is devoted to investigations of the existence and uniqueness of the solution of a non-local boundary value problem with discontinuous matching condition for the loaded equation. Considering parabolic-hyperbolic type equations involves the Caputo fractional derivative and loaded part joins in Riemann-Liouville integrals. The uniqueness of a solution is proved by the method of integral energy and the existence is proved by the method of integral equations.

Highlights

1 Introduction and formulation of a problem It is well known that fractional derivatives have been successfully applied to problems in system biology [ ], physics [ – ] and hydrology [, ]
Physical models fractional differential operators have recently renewed attention from scientist which is mainly due to applications as models for physical phenomena exhibiting anomalous diffusion
In a series of papers the authors considered some classes of initial value problems for functional differential equations involving Riemann-Liouville and Caputo fractional derivatives of order < α ≤

Summary

Introduction

Introduction and formulation of a problemIt is well known that fractional derivatives have been successfully applied to problems in system biology [ ], physics [ – ] and hydrology [ , ]. Very recently some basic theory for the initial boundary value problem (BVP)s of fractional differential equations involving a Riemann-Liouville differential operator of order < α ≤ has been discussed by Lakshmikantham and Vatsala [ , ]. In a series of papers (see [ , ]) the authors considered some classes of initial value problems for functional differential equations involving Riemann-Liouville and Caputo fractional derivatives of order < α ≤ .

Results

Conclusion