One-dimensional Initial-boundary Value Problem Research Articles

Well-posedness and exponential stability of piezoelectric beam with thermal effect and boundary or internal distributed delay

In this work, we focus on a one-dimensional initial-boundary value problem of a thermoelastic piezoelectric beam in the presence of magnetic effects with distributed delay acting in the mechanical equation. The well-posedness of the system is initially demonstrated by applying semigroup theory (Hille-Yosida theorem). Also, based on the construction of a Lyapunov functional, which is equivalent to the energy functional of the problem through multiplier techniques, we show that a unique dissipation through frictional damping is strong enough to ensure exponential stabilization of the model. Next, the results are compared to those of the electrostatic or quasi-static approaches. Moreover, we consider the fully dynamic and electrostatic or quasi-static piezoelectric beams with thermal effects, boundary feedback controllers and boundary distributed delays and using the multipliers technique, we establish an exponential stability result of the solution. Our results are related to the distributed delay weights. Finally, we give some numerical tests to illustrate the theoretical result.

Математичне моделювання та комп’ютерний аналіз теплових режимів електропровідної панелі при індукційній термообробці

An electrically conductive panel of rectangular cross-section is considered. During induction heat treatment with a quasi-steady electromagnetic field, unsteady volumetrically distributed Joule heat sources arise in it. With appropriate parameters of the external electromagnetic field, thermal processes in the panel can occur under conditions of near-surface or continuous heating. A two-dimensional physical and mathematical model is proposed to study the regularities of its thermal modes under the above-mentioned heating conditions. This model consists of two stages. At the first stage, the component of the magnetic field intensity vector tangent to the panel bases is determined from Maxwell's relations. At the second stage, the Joule heat is calculated from this component. To construct a solution to the electrodynamics problem, we use a cubic polynomial approximation of the distribution of the magnetic field intensity vector component tangent to the panel bases along the thickness coordinate. The coefficients of the approximation polynomial are represented as a linear combination of the thickness-integral characteristics of the tangential component of the magnetic field intensity vector and the specified boundary values of this component on the panel bases. As a result, the original two-dimensional initial-boundary value problem on the determining function (the component under consideration) is reduced to a one-dimensional initial-boundary value problem on the integral characteristics of the determining function. The solutions to this problem are found by applying a finite integral transformation along the transverse coordinate of the panel using the given boundary conditions on the determining function at the end sections of the panel and the Laplace transform in the time variable. Numerical studies have been performed for a copper panel during its induction heat treatment with a homogeneous quasi-steady-state electromagnetic field. The thermal modes of the panel are analyzed numerically depending on the parameter characterizing the relative depth of induction current penetration for two typical cases of near-surface and continuous heating.

Analysis of Varying Temperature Regimes in a Conductive Strip during Induction Heating under a Quasi-Steady Electromagnetic Field

Transition processes in a steel conductive strip are analyzed during its induction heating under a quasi-steady electromagnetic field. In particular, the temperature field in the strip is studied. A method of solving corresponding initial boundary problems in a two-dimensional mathematical model for differential equations of electrodynamics and heat conduction is developed. The Joule heat and the temperature are determined with a high level of accuracy. The defining functions are the temperature and component of the magnetic field intensity vector tangent to the bases and end planes of the strip. To find them, we use cubic approximation of the defining functions’ distribution along the thickness coordinate. The original two-dimensional initial boundary value problems for the defining functions are reduced to one-dimensional initial boundary value problems on their integral characteristics. General solutions for these problems are obtained using the finite integral transformation by the transverse variable and the Laplace transform of the integral by time. Integral characteristics’ expressions are represented as convolutions for functions that describe homogeneous solutions of one-dimensional initial boundary value problems and limiting values of defining functions on the bases and end planes of the strip. The change of temperature under a varying regime in the dimensionless Fourier time and temperature distribution over the strip cross-section in a steady state depending on the parameters of induction heating and the Biot number are numerically analyzed. Varying and constant temperature regimes of the strip under conditions of the near-surface and continuous induction heating are studied.

Визначення двовимірних нестаціонарних температурних полів у плитах та панелях з плоскопаралельними межами за наявності джерел тепла

Non-stationary two-dimensional problems of thermal conductivity for plates and panels with plane-parallel boundaries in the presence of volume-distributed non-stationary heat sources are formulated. A method of constructing a solution to the formulated heat conduction problems for the considered bodies is proposed. The technique uses the approximation of the temperature distribution in both elements by the thickness variable by a cubic polynomial. The coefficients of the approximation polynomial are given through the integral over the thickness variable temperature characteristics and the conditions for the boundary values of the temperature on the outer surfaces of the plate and panel. As a result, the initial two-dimensional initial boundary value problems for the temperature for the plate and panel are reduced to one-dimensional initial boundary value problems for the integral temperature characteristics. To construct the solution of the initial-boundary value problem for the integral characteristics of the temperature in the case of a plate infinite in longitudinal and transverse coordinates, the integral Laplace transform in time and the integral Fourier transform in the longitudinal coordinate were used. The solution of the problem on the integral temperature characteristics in the case of the panel is found using the integral Laplace transform in time and the finite integral transform in the transverse coordinate. Expressions of integral temperature characteristics for the plate and panel are obtained in the form of convolutions of functions corresponding to homogeneous solutions of initial-boundary value problems for integral temperature characteristics and functions describing the available non-stationary heat sources in these bodies and given surface temperature values. The general solutions of the two-dimensional initial boundary value problems of thermal conductivity for slabs and panels are recorded for the presence of arbitrarily variable spatial coordinates of non-stationary heat sources and the conditions of convective heat exchange with the external environment on the surfaces of the considered bodies.

Mapping of Western Siberian heat flow (southeast)

This paper maps the heat flow density from the pre-Jurassic basement within a large territory of oil accumulation (circa 120 thousand sq. km) in the South-East of Western Siberia. Values of heat flow, calculated for 201 deep wells, comprise a dataset for developing the map with contour lines for every 2 mW/m2. These values were calculated by solving the inverse problem of Geothermy — one-dimensional initial-boundary value problem for equation of thermal conductivity in a solid with the moving upper boundary. The accepted mathematical statement is sufficiently accurate for modeling near horizontal bedded sedimentary section: existence of heat flow convectional component is taken into account via calculation of an effective heat flow value. The map shows different types of anomalous features of heat flow density distribution. Previously it was stated for Western Siberian Plate that values of deep heat flow within positive tectonic structures of sedimentary cover are 5—20 % higher than within negative tectonic structures. As it is, combined analysis of deep heat flow density distribution and location of tectonic structures (Kaimysov arch, Parabelmegaarch, especially Alexandrov arch and Pudino mega swell) shows tendency among positive tectonic structures for increasing deep heat flow value. However, it is not always so. For example, there is the utterly different correlation for the Srednevasyugan mega swell. This structure is characterized with lower heat flow. The location of the Nizhnevartovsk arch is almost untraceable in the deep heat flow density distribution. Concerning oil-and-gas potential ... (?) A large positive anomaly has formed in the zone around the Traigorodsko-Kondakov field in the north. Two positive anomalies are in the central part of the map: around the Snezhnoe field and close by the Lomovoe, the Ozernoe and the Katylgin fields. Such fields as the Rybalnoe, the Pindzhin and the Mirnoe surround a positive anomaly in the southeastern part of the map.
 This paper contains a catalogue of discrete values (by wells) and a map of heat flow, which may be used as a «framework» in basin modeling. Upcoming research concerning origin of heat flow density anomalies — graded assessment of possible influence of tectonics, material composition and oil-and-gas potential of basement rocks has theoretical and practical significance.

Bounded Solutions of Second Order of Accuracy Difference Schemes for Semilinear Fractional Schrödinger Equations

AbstractThe present paper deals with initial value problem (IVP) for semilinear fractional Schrödinger integro-differential equation$$\begin{array}{} \displaystyle i\!\frac{du}{dt}+Au = \int\limits_{0}^{t}f\left( s,D_{s}^{\alpha }u(s)\right) ds,\, \, \, 0 \lt t \lt T,\, u\left( 0\right) = 0 \end{array} $$in a Hilbert space H with a self-adjoint positive definite (SAPD) operator A. Stable difference schemes (DSs) have significant interest in investigations of fractional partial differential equations. The main theorem concerns the existence and uniqueness of the uniformly bounded solutions (UBSs) with respect to step time of second order of accuracy DSs for this semilinear fractional Schrödinger differential problem. In practice, existence and uniqueness theorems for a UBS of the one-dimensional initial boundary value problem (BVP) with nonlocal condition and multi-dimensional problem with local condition on the boundary are proved. Numerical results and explanatory illustrations are presented to show the validation of the theoretical results.

Asymptotically Stable Stationary Solutions of the Reaction–Diffusion–Advection Equation with Discontinuous Reaction and Advection Terms

We study the Lyapunov asymptotic stability of the stationary solution of the spatially one-dimensional initial–boundary value problem for a nonlinear singularly perturbed differential equation of the reaction–diffusion–advection (RDA) type in the case where the advection and reaction terms undergo a discontinuity of the first kind at some interior point of an interval. Sufficient conditions are derived for the existence of a stable stationary solution with a large gradient near the point of discontinuity. An asymptotic method of differential inequalities is used to prove the existence and stability theorems. The resulting stability conditions can be employed to create mathematical models and develop numerical methods for solving “stiff” problems arising in various applications, for example, when simulating combustion processes and in nonlinear wave theory.

Методика побудови розв’язку двовимірної задачі електродинаміки для електропровідного тіла з плоскопаралельними межами за дії зов-нішнього нестаціонарного електромагнітного поля

A two-dimensional initial-boundary value problem of electrodynamics for an electroconductive non-ferromagnetic body with planar-parallel boundaries is formulated under the action of an external non-stationary electromagnetic field. The electromagnetic field is given by the values of the inhomogeneous longitudinal coordinate of the component of the magnetic field intensity vector at its bases. To construct a general solution of the formulated initial boundary value problem under such electromagnetic action, a cubic approximation by thickness coordinate and an integral Fourier transform along a longitudinal coordinate was used for a key function — a given component of the magnetic field intensity vector. As a result, the initial two-dimensional initial boundary value problem for the key function is reduced to the one-dimensional initial boundary value problem. This initial boundary value problem is on the integral key function. These characteristics are the functions of time variable and the thickness coordinate. The coefficients of the polynomial approximating the key function are given by the transformants of the integral characteristics of the key function and given its values on the bases of the body as corresponding functions of time and longitudinal coordinate. The general solutions of the one-dimensional initial boundary value problem are obtained as a convolution of functions. These functions describe homogeneous solutions and key function boundary values on the bases of the body. Using the inverse Fourier transforms, the solution of the original electrodynamics problem under the action of an arbitrary variable in time and by the longitudinal coordinate of a non-stationary electromagnetic field is written. On the basis of such a common solution, as a partial case, we also write the solutions of the original two-dimensional initial boundary value problem under the action of a stationary harmonic in time variable electromagnetic field. In order to improve the accuracy of the approximate solution of the problem under consideration, an independent approximation of the corresponding component of the electric field intensity vector along the thickness coordinate is proposed. Systems of equations are formulated to determine the integral characteristics of this component, both under the action of an arbitrary variable in time and for a longitudinal coordinate non-stationary and the stationary electromagnetic field.

Reproducing Kernel Method for Singularly Perturbed One-Dimensional Initial-Boundary Value Problems with Exponential Initial Layers

In this paper, a meshless numerical method is proposed for singularly perturbed one-dimensional initial-boundary value problems with exponential initial layers. The method is a combination of the domain decomposition method and the reproducing kernel method. A fitted reproducing kernel is constructed for initial layer domain problem. Some numerical results confirming the expected behavior of the method are shown.

A new proof of the existence of weak solutions to a model for phase evolution driven by material forces

We prove the existence of weak solutions to a one-dimensional initial-boundary value problem for a model system of partial differential equations, which consists of a sub-system of linear elasticity and a nonlinear non-uniformly parabolic equation of second order. To simplify the existence proof of weak solutions in the 2006 paper of Alber and Zhu, we replace the function |p|κ:=|p|2/|p|2+κ2 in that work by |p|κ:=|p|2+κ2. The model is formulated by using a sharp interface model for phase transformations that are driven by material forces. Copyright © 2017 John Wiley & Sons, Ltd.

Global solutions for a one-dimensional problem in conducting fluids

A mathematical model of the motion of conducting fluids is studied in this paper. The dynamics of such fluids is described by the equations of compressible fluids coupled to the Maxwell’s equations. We prove global existence of strong solution for a one-dimensional initial-boundary value problem of this model (plane conducting flows) with general large data.

Exponential stability in the theory of thermoelastic diffusion mixtures

In this article, we investigate the asymptotic behaviour of solutions to the one-dimensional initial-boundary value problem in the linear theory of thermoelastic diffusion mixtures recently developed by Aouadi [M. Aouadi, Qualitative results in the theory of thermoelastic diffusion mixtures, J. Thermal Stresses 33 (2010), pp. 595–615]. Our main result is to establish the necessary and sufficient conditions over the coefficients of the system to get the exponential stability of the corresponding semigroup.

Low Mach number limit for the non-isentropic Navier–Stokes equations

We study the low Mach number limit of the local in time solutions to the compressible Navier–Stokes equations with zero heat conductivity coefficient as the Mach number tends to zero. A uniform existence result for the one-dimensional initial–boundary value problem is proved provided that the initial data are “well-prepared” in the sense that the temporal derivatives up to order two are bounded initially.

Exponential Decay for a Thermoelastic Beam Between two Stops

A one-dimensional initial-boundary value problem for a thermoelastic beam that is clamped at one end and moves freely between two obstacles at the other, is studied using a method of functional analysis. The boundary conditions at the free end are represented either by classic Signorini's conditions or a normal compliance condition. The main result is that the energy associated with the system decays exponentially as time goes to infinity. Numerical examples are included.

A parameter-uniform numerical method for a Sobolev problem with initial layer

The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a linear Sobolev or pseudo-parabolic equation with initial jump. In order to obtain an efficient method, to provide good approximations with independence of the perturbation parameter, we have developed a numerical method which combines a finite difference spatial discretization on uniform mesh and the implicit rule on Shishkin mesh(S-mesh) for the time variable. The fully discrete scheme is shown to be convergent of order two in space and of order one expect for a logarithmic factor in time, uniformly in the singular perturbation parameter. Some numerical results confirming the expected behavior of the method are shown.

A grid-free, nonlinear shallow-water model with moving boundary

A meshless method based on the radial basis function is derived to solve the nonlinear, nondispersive shallow-water equations. The one-dimensional initial boundary value problem has a fixed boundary at one end and a free boundary at the other. The formulation employs a Lagrangian–Eulerian scheme to track the movement of the free boundary and transform the problem to a time-independent domain. The radial basis function evaluates the spatial derivatives in the numerical solution, while the Wilson- θ method integrates the development of the flow in time. The model is applied to calculate the flow of floodwater resulting from dam collapse and the run-up of waves on a plane beach. Comparisons of the computed results with analytical and finite difference solutions demonstrate the accuracy and capability of this meshless model in engineering applications.

On the Convergence of Difference Schemes for Hyperbolic Problems with Concentrated Data

Hyperbolic equations with unbounded coefficients and even generalized functions (in particular, Dirac-delta functions) occur both naturally and artificially and must be treated in numerical schemes. An abstract operator method is proposed for studying these equations. For finite difference schemes approximating several one-dimensional initial-boundary value problems convergence rate estimates in special discrete energetic Sobolev's norms, compatible with the smoothness of the solutions, are obtained.

Long-time behavior for the full one-dimensional Frémond model for shape memory alloys

This note deals with a well-posed one-dimensional initial-boundary value problem related to the Fremond thermomechanical model of structural phase transitions in shape memory materials. The long-time behavior is investigated and it is proved that the omega-limit set in a suitable topology only contains stationary states.

Multi-level adaptive solutions to initial-value problems

A multigrid algorithm is developed for solving the one-dimensional initial boundary value problems. The numerical solutions of linear and nonlinear Burgers’ equation for various initial conditions are studied. The stability conditions are derived by Von-Neumann analysis. Numerical results are presented.

SAINT-VENANT TYPE DECAY ESTIMATES FOR TRANSIENT HEAT CONDUCTION IN A COMPOSITE RIGID SEMISPACE

The global and pointwise spatial decay estimates for a solution to the one-dimensional initial-boundary value problem of heat conduction in a composite rigid semispace are presented. Vie global estimates cover: (i) a transient heat conduction problem for an isotropic nonhomogeneous semispace with arbitrary variation with depth of the density, specific heat, and thermal conductivity; (ii) a transient heat conduction problem for an isotropic semispace made of a functionally graded material that composition changes from a ceramic surface at x = 0 to a metal semispacex≥.x0> 0; and (iii) a transient heat conduction problem for a microperiodic layered semispace. The pointwise spatial decay estimates are confined to homogenized transient heat conduction problems for a microperiodic layered semispace described by an effective modulus theory as well as by a refined averaged theory