A domain of influence theorem for generalized thermoelasticity with memory-dependent derivative

Enlightened by the Caputo fractional derivative, the present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of a non-Gaussian pulsed laser type heat source in a stress free isothermal half-space in the context of Lord–Shulman (LS), dual-phase lag (DPL), and three-phase lag (TPL) theories of thermoelasticity simultaneously. The memory-dependent derivative is defined in an integral form of a common derivative on a slipping interval by incorporating the memory-dependent heat transfer. Employing Laplace transform as a tool, the problem has been transformed to the space-domain, and it is then solved analytically. To get back all the thermophysical quantities as a function of real time, we use two Laplace inversion formulas, viz. Fourier series expansion technique (Honig in J Comput Appl Math10(1):113–132, 1984) and Zakian method (Electron Lett 6(21):677–679, 1970). According to the graphical representations corresponding to the numerical results, a comparison among LS, DPL, and TPL model has been studied in the presence and absence of a memory effect simultaneously. Moreover, the effects of a laser pulse have been studied in all the thermophysical quantities for different kernels (randomly chosen) and different delay times. Then, the results are depicted graphically. Finally, a comparison of results, deriving from the two numerical inversion formulas, has been made.

Memory-dependent generalized thermoelasticity with finite wave speeds

This article is focused on developing a new mathematical model on the temperature-rate-dependent thermoelasticity theory (Green–Lindsay), using the methodology of memory-dependent derivative (MDD). First, the energy theorem of this model associated with two relaxation times in the context of MDD is derived for homogeneous, isotropic thermoelastic medium. Second, a uniqueness theorem for this model is proved using the Laplace transform technique. A variational principle for this model is also established. Finally, the results for Green–Lindsay model without MDD and coupled theory are obtained from the considered model.

This article theoretically demonstrates the reciprocity and uniqueness theorems for generalized theory of thermo-viscoelasticity involving memory-dependent derivative (MDD). To prove the theorems, a thermo-viscoelastic initial-boundary value problem under the domain of three-phase-lag (TPL) model is taken into consideration for an isotropic, homogeneous medium. The theorems are proved with the help of the Laplace transform of the thermophysical quantities. Finally, a few special cases in the generalized theory of thermo-elasticity and thermo-viscoelasticity with MDD and without MDD are derived from the present model.

This study is dedicated to some basic theorems in the thermoelastodynamics of microstretch bodies. Our intention is to show that the presence of the microstretch does not affect the main characteristics of the mixed initial boundary value problem for thermoelastic bodies. The result regarding the uniqueness theorem is derived with no definiteness assumptions on the elastic coefficients and in the absence of the restriction that the conductivity tensor is positive definite. In the last part of the paper we establish a basic relation which leads to the reciprocal theorem and to another uniqueness result. MSC:35M30, 35Q74, 74A15, 74A60, 74M25.

First, we formulate the mixed initial boundary value problem in the theory of thermoelastic microstretch bodies having certain internal state variables. Then by using some approachable computing techniques and the known Gronwall's inequality we will prove that the presence of internal state variables do not influence the uniqueness of solution of the mixed problem.

A time domain method building on the concept of wave splitting is used to study direct wave propagation phenomena in weakly nonlinear media. The starting point is the linear wave equation with time-dependent coefficients. This means that the studied nonlinear medium in some sense has to be approximated with a nonstationary medium which changes while the wave passes through. For the nonstationary equation homogeneous as well as particular solutions can be obtained. Two different iterative procedures to find the nonlinear solutions are discussed. They are illustrated by two problems fetched from different research fields of current interest. In the first case, the nonlinear term is linearized using the Fréchet derivative. This leads into a truly nonstationary, mixed initial boundary value problem with a linear equation characterized by both time-dependent coefficients and source terms. In this example a semiconductor device used for switching in high-frequency applications is considered. It can be described as a coplanar waveguide loaded with distributed resonant tunnel diodes. In the other example, wave propagation in Kerr media is considered. Then Taylor expansion transforms the nonlinear equation into a linear one with nonstationary source terms. In this case the nonlinearity does not lead to time-depending coefficients in the equation. The way to obtain the solution is a nonlinear variant of the Born approximation.

A procedure for calculating sensitivity coefficients of one-dimensional parabolic mixed initial-boundary value problems is developed. The method is based upon the use of implicit time integration and Newton's method in solving the governing equations. The link between the sensitivity coefficients and the solution method is studied with particular emphasis on the use of adaptive gridding and variable time stepping. The method is employed in the analysis of two model freely propagating premixed laminar flames. The numerical accuracy of the sensitivities is verified and their values are utilized for interpretation of the model results.

This study is dedicated to a micropolar porous body, including voidage time derivative among the independent constitutive variables. We formulate a heat-flux theory in this context taking into account a new set of state variables that includes the heat-flux vector and an evolution equation for it. The mixed initial boundary value problems have the specific feature that the governing system of equations consists of hyperbolic field equations only. Consistency of the theory is supported by a result of uniqueness.

Nonstationary plane contact problem in theory of elasticity for conformal cylindrical surfaces

There are considered controlled systems with distributed parameters in one dimension. Their mathematical model is given by mixed initial boundary value problems for hyperbolic partial differential equations in two dimensions. The control is performed at the boundaries, being described by ordinary differential equations accounting for the dynamics of the controllers. Two are the standard problems for such systems, especially in the nonlinear case: well posedness in the sense of J. Hadamard (existence,uniqueness and smooth dependence on data) and stability of the steady state solutions - a control oriented property. While feedback control synthesis may be performed at the formal level, validation of the resulting closed loop model as well as stability have to be analyzed with the mathematical rigor required by the theory of partial differential equations. The present paper is focused on these topics starting from the basic model of the flexible beam, accounting for the models of the flexible manipulator arm, of the overhead crane and of the oilwell drillstring. It is shown how various approaches that are known from the fundamentals of partial differential equations can be used to obtain the necessary properties.

A partition of energy in thermoelasticity of microstretch bodies

The Navier–Stokes equations at high Reynolds numbers can be viewed as an incompletely elliptic perturbation of the Euler equations. By using the entropy function for the Euler equations as a measure of “energy” for the Navier–Stokes equations we are able to obtain nonlinear “energy” estimates for the mixed initial boundary value problem. These estimates are used to derive boundary conditions that guarantee $L^2 $ boundedness even when the Reynolds number tends to infinity. Finally, we propose a new difference scheme for modeling the Navier–Stokes equations in multidimensions for which we are able to obtain discrete energy estimates exactly analogous to those we obtained for the differential equation.

On energy, uniqueness theorems and variational principle for generalized thermoelasticity with memory-dependent derivative

In our study we formulated the mixed initial boundary value problem corresponding to the thermoelasticity of type III for bodies with dipolar structure. In main section we approached four qualitative results regarding the solutions for this problem. In two of these (in the first two theorems) we obtained two results of uniqueness, proved in different ways. Also, we proven two results which show that the solutions of the considered problem depend continuously with respect to the supply terms. We use different procedures in the two theorems on continuous dependence, but we essentially rely on the auxiliary results from Section 3 and Gronwall-type inequalities. It is important to emphasize that all results are obtained by imposing on the basic equations and basic conditions, average constraints that are common in the mechanics of continuous solids.