The Galerkin Method for Fourth-Order Equation of the Moore–Gibson–Thompson Type with Integral Condition

Ahlem Mesloub,Abderrahmane Zaraï,Bahri-Belkacem Cherif,Fatiha Mesloub,Mohamed Abdalla,Kamyar Hosseini

doi:10.1155/2021/5532691

Abstract

In this manuscript, we consider the fourth order of the Moore–Gibson–Thompson equation by using Galerkin’s method to prove the solvability of the given nonlocal problem.

Highlights

Research on the nonlinear propagation of sound in a situation of high amplitude waves has shown literature on physically well-founded partial differential models
The behavior of acoustic waves depends strongly on the medium property related to dispersion, dissipation, and nonlinear effects
It arises from modeling high-frequency ultrasound (HFU) waves

Summary

Introduction

Research on the nonlinear propagation of sound in a situation of high amplitude waves has shown literature on physically well-founded partial differential models (see, e.g., [1– 23]). Existence and uniqueness of the generalized solution are established by using the Galerkin method These problems can be encountered in many scientific domains and many engineering models (see previous works [5, 25–32, 35, 36, 40, 41]). Boulaaras et al investigated the Moore–Gibson–Thompson equation with the integral condition in [4] Motivated by these outcomes, we improve the existence and uniqueness by the Galerkin method of the fourth-order equation of the Moore–. + αuttt + βutt − ρΔu − δΔut − γΔutt = 0, 0Þ = u0ðxÞ, utðx, 0Þ = u1ðxÞ, uttðx, 0Þ = u2ðxÞ, uttt ðx, 0Þ The aim of this manuscript is to consider the following nonlocal mixed boundary value problem for the Moore–Gibson–Thompson (MGT) equation for all ðx ; tÞ ∈ QT = ð0, TÞ, where Ω ⊂ Rn is a bounded domain with sufficiently smooth boundary ∂Ω. In “Solvability of the Problem,” we use Galerkin’s method to prove the existence, and in “Uniqueness of Solution,” we demonstrate the uniqueness

Preliminaries

Solvability of the Problem

Uniqueness of Solution

Conclusion