Boundary Value Problem of the Operator ⊕k Related to the Biharmonic Operator and the Diamond Operator

Chalermpon Bunpog

doi:10.3390/math6070115

Abstract

This paper presents an alternative methodology for finding the solution of the boundary value problem (BVP) for the linear partial differential operator. We are particularly interested in the linear operator &oplus;k, where &oplus;k=♡k♢k, ♡k is the biharmonic operator iterated k-times and ♢k is the diamond operator iterated k-times. The solution is built on the Green’s identity of the operators ♡k and &oplus;k, in which their derivations are also provided. To illustrate our findings, the example with prescribed boundary conditions is exhibited.

Highlights

Boundary value problems (BVPs) for ordinary and partial differential equations have appeared in widespread applications ranging from cognitive science to engineering
These types of problems inevitably associate with the partial differential operators—for example, the Laplace operator [5,6], the ultrahyperbolic operator [7,8], and the biharmonic operator [9,10]
In the case that the operator ⊕k reduces to the Laplace operator iterated k-times 4k, 4k u(ξ ) = f (ξ ), ξ ∈ Ω, (42)

Summary

Introduction

Boundary value problems (BVPs) for ordinary and partial differential equations have appeared in widespread applications ranging from cognitive science to engineering. The heat flow in a nonuniform rod without sources accompanied with initial—boundary conditions [4] These types of problems inevitably associate with the partial differential operators—for example, the Laplace operator [5,6], the ultrahyperbolic operator [7,8], and the biharmonic operator [9,10]. The solution’s existence under some suitable boundary conditions of the operator ⊕k is manifested by using Green’s identity of the operators ♥ and ⊕k , as well as the BVP solution of the diamond operator ♦. Applications connected to the BVP of the linear partial differential operators are shown

Preliminaries

Results

Example 1

Remark

Conclusions