scholarly journals On the Partition of Energies for the Backward in Time Problem of Thermoelastic Materials with a Dipolar Structure

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 863 ◽  
Author(s):  
M. Marin  ◽  
S. Vlase ◽  
R. Ellahi ◽  
M.M. Bhatti
Keyword(s):  
Mixed Problem ◽  
Temporal Behavior ◽  
Time Problem ◽  
Uniqueness Of The Solution ◽  
Thermoelastic Body ◽  
Dipolar Structure ◽  
Integral Identities

We first formulate the mixed backward in time problem in the context of thermoelasticity for dipolar materials. To prove the consistency of this mixed problem, our first main result is regarding the uniqueness of the solution for this problem. This is obtained based on some auxiliary results, namely, four integral identities. The second main result is regarding the temporal behavior of our thermoelastic body with a dipolar structure. This behavior is studied by means of some relations on a partition of various parts of the energy associated to the solution of the problem.

scholarly journals Some Results in Green–Lindsay Thermoelasticity of Bodies with Dipolar Structure

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 497 ◽  
Author(s):  
Marin Marin ◽  
Eduard M. Craciun ◽  
Nicolae Pop
Keyword(s):  
Mixed Problem ◽  
Classical Theory ◽  
Theory Of Elasticity ◽  
Reciprocal Theorem ◽  
General Context ◽  
Main Concern ◽  
Dipolar Bodies ◽  
Uniqueness Of The Solution ◽  
Thermoelastic Body ◽  
Dipolar Structure

The main concern of this study is an extension of some results, proposed by Green and Lindsay in the classical theory of elasticity, in order to cover the theory of thermoelasticity for dipolar bodies. For dynamical mixed problem we prove a reciprocal theorem, in the general case of an anisotropic thermoelastic body. Furthermore, in this general context we have proven a result regarding the uniqueness of the solution of the mixed problem in the dynamical case. We must emphasize that these fundamental results are obtained under conditions that are not very restrictive.

scholarly journals An approach with Lagrange identity of the mixed problem in theory of strain gradient thermoelasticity

2020 ◽  
Vol 34 ◽  
pp. 01004
Author(s):  
Marin Marin
Keyword(s):  
Mixed Problem ◽  
Strain Gradient ◽  
Continuous Dependence ◽  
Initial Boundary ◽  
Boundary Values ◽  
Conservation Of Energy ◽  
Lagrange Identity ◽  
Uniqueness Of The Solution ◽  
Continuous Dependence Of Solutions ◽  
Elastic Coefficients

In our paper we first define the mixed initial-boundary values problem in the theory of strain gradient thermoelasticity. With the help of an identity of Lagrange’s type, we then prove some theorems regarding the uniqueness of the solution of this mixed problem and also two results regarding the continuous dependence of solutions on initial data and on the charges. We must ouline that we obtain these qualitative results without recourse to any laws of conservation of energy and without recourse to any boundedness assumptions on the coefficients. It is equally important to note that we do not impose restrictions on the elastic coefficients regarding their positive definition.

scholarly journals Classical solution of the mixed problem for the Klein – Gordon – Fock type equation in the half-strip with curve derivatives at boundary conditions

2019 ◽  
Vol 54 (4) ◽  
pp. 391-403 ◽  
Author(s):  
V. I. Korzyuk ◽  
I. I. Stolyarchuk
Keyword(s):  
Boundary Conditions ◽  
Mixed Problem ◽  
Type Equation ◽  
Approximate Solutions ◽  
Analytical Form ◽  
Problem Analysis ◽  
Matching Conditions ◽  
Half Strip ◽  
Uniqueness Of The Solution ◽  
Klein Gordon

The mixed problem for the one-dimensional Klein – Gordon – Fock type equation with curve derivatives at boundary conditions is considered in the half-strip. The solution of this problem is reduced to solving the second-type Volterra integral equations. Theorems of existence and uniqueness of the solution in the class of twice continuously differentiable functions were proven for these equations when initial functions are smooth enough. It is proven that the fulfillment of the matching conditions on the given functions is necessary and sufficient for the existence of the unique smooth solution when initial functions are smooth enough. The method of characteristics is used for the problem analysis. This method is reduced to splitting the original area of definition to the subdomains. The solution of the subproblem can be constructed in each subdomain with the help of the initial and boundary conditions. Then, the obtained solutions are glued in common points, and the obtained glued conditions are the matching conditions. This approach can be used in constructing as an analytical solution when a solution of the integral equation can be found in an explicit way, so an approximate solution. Moreover, approximate solutions can be constructed in numerical or analytical form. When a numerical solution is built, the matching conditions are essential and they need to be considered while developing numerical methods.

scholarly journals Study of a mixed problem for a nonlinear elasticity system by topological degree

2021 ◽  
Vol 66 (3) ◽  
pp. 537-551
Author(s):  
Zoubai Fayrouz ◽  
Merouani Boubakeur
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Variational Problem ◽  
Mixed Problem ◽  
Nonlinear Elasticity ◽  
Point Theorem ◽  
Topological Degree ◽  
General Behavior ◽  
Elasticity System ◽  
Uniqueness Of The Solution

"In this paper, we consider a mixed problem for a nonlinear elasticity system with laws of general behavior. The coefficients of elasticity depends on x meanwhile the density of the volumetric forces depends on the displacement. The main aim of this paper is to apply the Schauder's fixed point theorem and the techniques of topological degree to prove a theorem of the existence and the uniqueness of the solution of the corresponding variational problem."

scholarly journals Classical solution of the mixed problem for the Klein – Gordon – Fock type equation with characteristic oblique derivatives at boundary conditions

2019 ◽  
Vol 55 (1) ◽  
pp. 7-21
Author(s):  
V. I. Korzyuk ◽  
I. I. Stolyarchuk
Keyword(s):  
Boundary Conditions ◽  
Mixed Problem ◽  
Type Equation ◽  
Approximate Solutions ◽  
Analytical Form ◽  
Problem Analysis ◽  
Matching Conditions ◽  
Uniqueness Of The Solution ◽  
Klein Gordon ◽  
The One

The mixed problem for the one-dimensional Klein – Gordon – Fock type equation with oblique derivatives at boundary conditions in the half-strip is considered. The solution of this problem is reduced to solving the second-type Volterra integral equations. Theorems of existence and uniqueness of the solution in the class of twice continuously differentiable func tions were proven for these equations when initial functions are smooth enough. It is proven that fulfilling the matching conditions on the given functions is necessary and sufficient for existence of the unique smooth solution, when initial functions are smooth enough. The method of characteristics is used for the problem analysis. This method is reduced to splitting the ori ginal definition area into subdomains. The solution of the subproblem can be constructed in each subdomain with the help of the initial and boundary conditions. The obtained solutions are then glued in common points, and the obtained glued сonditions are the matching conditions. Intensification of smoothness requirements for source functions is proven when the di rections of the oblique derivatives at boundary conditions are matched with the directions of the characteristics. This approach can be used in constructing both the analytical solution, when the solution of the integral equation can be found explicitly, and the approximate solution. Moreover, approximate solutions can be constructed in numerical and analytical form. When a numerical solution is constructed, the matching conditions are significant and need to be considered while developing numerical methods.

scholarly journals A Mixed Problem with an Integral Two-Space-Variables Condition for Parabolic Equation with The Bessel Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Bouziani Abdelfatah ◽  
Oussaeif Taki-Eddine ◽  
Ben Aoua Leila
Keyword(s):  
Parabolic Equation ◽  
Mixed Problem ◽  
Existence And Uniqueness ◽  
Weighted Sobolev Space ◽  
A Priori ◽  
Energy Inequality ◽  
A Priori Estimate ◽  
Bessel Operator ◽  
Priori Estimate ◽  
Uniqueness Of The Solution

We study a mixed problem with an integral two-space-variables condition for parabolic equation with the Bessel operator. The existence and uniqueness of the solution in functional weighted Sobolev space are proved. The proof is based on a priori estimate “energy inequality” and the density of the range of the operator generated by the problem considered.

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