On a Boundary Value Problem for the Biharmonic Equation with Multiple Involutions

Batirkhan Turmetov,Moldir Muratbekova,Valery Karachik

doi:10.3390/math9172020

Batirkhan Turmetov, Moldir Muratbekova + Show 1 more

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https://doi.org/10.3390/math9172020

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Journal: Mathematics	Publication Date: Aug 24, 2021
Citations: 5	License type: CC BY 4.0

Affiliation: Ahmet Yesevi University

Abstract

A nonlocal analogue of the biharmonic operator with involution-type transformations was considered. For the corresponding biharmonic equation with involution, we investigated the solvability of boundary value problems with a fractional-order boundary operator having a derivative of the Hadamard-type. First, transformations of the involution type were considered. The properties of the matrices of these transformations were investigated. As applications of the considered transformations, the questions about the solvability of a boundary value problem for a nonlocal biharmonic equation were studied. Modified Hadamard derivatives were considered as the boundary operator. The considered problems covered the Dirichlet and Neumann-type boundary conditions. Theorems on the existence and uniqueness of solutions to the studied problems were proven.

Highlights

The concept of a nonlocal operator and the related concepts of a nonlocal differential equation appeared in the theory of differential equations not long ago
A special place is occupied by equations in which the deviation of the arguments has an involutive character
For classical equations, one can study nonlocal boundary value problems of the Bitsadze–Samarskii-type, in which the values of the sought function u( x ) at the boundary of the domain are related to the values of u(Sx ) [17,18,19]

Summary

Introduction

The concept of a nonlocal operator and the related concepts of a nonlocal differential equation appeared in the theory of differential equations not long ago. In [1], the authors considered equations containing fractional derivatives of the desired function and equations with deviating arguments, in other words, equations that include an unknown function and its derivatives, generally speaking, for different values of arguments Such equations are called nonlocal differential equations. Note that boundary value problems with fractional-order boundary operators for elliptic equations were studied in [22,23,24,25,26,27,28,29,30]. Consider the following boundary value problem in the domain Ω.

Auxiliary Statements

Properties of Integrodifferential Operators

The Dirichlet Problem

The Neumann Problem

The General Case of the Main Problem

Conclusions

Methods