Initial-boundary Problem Research Articles

Asymptotic estimate of weak solutions in a fourth-order parabolic equation with logarithm

This paper deals with an initial-boundary problem of the fourth-order parabolic equation involving two logarithm terms. First, we give some results for blow-up or global solutions through classifying the initial energy and the Nehari energy. Second, we show asymptotic estimates about blow-up time and a large time estimate of solutions, respectively.

Mathematical modeling and analysis of the thermomechanical behavior of a bimetallic spherical element during technological processing with an electromagnetic impulse

The initial-boundary problem of thermomechanics for a bimetallic spherical element under homogenous non-stationary electromagnetic action is formulated. The azimuthal component of the magnetic field strength vector, temperature, and the radial component of the displacement vector were selected as key functions. To find the key functions, the approximation of their distributions in the constituent layers of the spherical element by quadratic polynomials in the radial variable is proposed. This approximation makes it possible to accurately satisfy the specified boundary conditions both on the surfaces of this element and on the surface of the connection of the component layers, where ideal electromagnetic, thermal and mechanical contacts take place. The coefficients of the polynomials approximating the key functions in the component layers of the spherical element are given by a linear combination of functions describing all given boundary conditions and integral characteristics of the key functions on both component layers. As a result, the original initial-boundary value problems for the defining functions are reduced to Cauchy problems for their integral (summed over the package of layers) characteristics. With the use of Laplace transformation in time, the general solutions of these problems under homogeneous non-stationary electromagnetic action are recorded. The change in time of radial and azimuthal stresses and stress intensity was numerically analyzed by technological processing with an electromagnetic impulse, as well as the performance and properties of the contact connection of the bimetallic spherical element were investigated.

Начально-краевая задача о течении в кровеносном сосуде

The solution of the initial boundary value problem for a system of two quasi-linear hyperbolic equations, which describes a rotationally symmetric vortex-free flow of a viscous incompressible fluid in an infinite cylindrical domain (pipe, blood vessel) is constructed. It is assumed that the side surface of the domain is a free (soft, compliant) boundary on which the kinematic condition is set. The dynamic condition at the boundary is modeled by some constitutive relation – the dependence of the pressure P on the cross-sectional area S of the cylindrical domain. To study the flow an asymptotic model based on the shallow water theory is used. When constructing the solution of the initial-boundary (as well as initial and boundary) problem for a system of two quasilinear partial differential equations of the first order, the hodograph method based on the conservation law is used. A variant of the polynomial constitutive relation is chosen for the study: P~S2β (β>0). In the case of the initial data problem which specified at the initial time the Riemann-Green function, allowing to construct an implicit analytical solution, and an algorithm (numerical) for constructing an explicit solution of the original problem are given. The main attention is focused on the special case P~S2, for which all the formulas necessary to construct a solution are written in explicit form. For P~S2, several variants of conditions (initial and boundary value) are considered for the initial boundary value problem, which allow us to trace the evolution of the solution in detail. Numerical calculations demonstrating the motion of shock waves and wave fronts are given. The results obtained, in practice, can be used to describe flows in blood vessels, as well as reliable tests for testing compu tational algorithms intended to solve such problems, in particular, systems of quasi-linear hyperbolic equations.

An upwind-block-centered finite difference method for a semiconductor device of heat conduction and its numerical analysis

The mathematical model is formulated by a nonlinear system of initial–boundary problem including four partial differential equations: an elliptic equation for electrostatic potential, two convection–diffusion equations for electron concentration and hole concentration, a heat conduction equation for temperature. The electric field potential is solved by the conservative block-centered method, and the first order of the accuracy is improved. The concentrations and temperature are computed by the upwind-block-centered difference method. The block-centered method is used to discretize the diffusion terms. The upwind difference is applied to approximate the convection parts without numerical dispersion or nonphysical oscillation. The block-centered difference simulates the concentrations, temperature and their adjoint vector functions simultaneously. The local conservation of mass is preserved. An optimal order error estimates is obtained. Finally, numerical examples show the effectiveness and viability of this method.

Аналіз теплового режиму електропровідного трубчастого елемента з тонким зовнішнім електропровідним покриттям за дії неусталеного електромагнітного поля

An electrically conductive tubular element with a thin external electri-cally conductive coating underthe action of an unsteady electromagnetic field of the radio frequency range is considered. To analyze the thermal re-gime, the initial-boundary problem of electrodynamics for a two-layer conductive non-ferromagnetic hollow cylinder under the action of an ex-ternal non-stationary electromagnetic field is formulated. The electromag-netic field is given by the values of the axial component of the magnetic field intensity vector on the inner and outer surfaces of the cylinder. To construct the general solution of the formulated initial-boundary value problem, a quadratic approximation of the key function (the axial compo-nent of the magnetic field intensity vector) along the radial coordinate in each constituent layer (base and cover) of the hollow cylinder was used. As a result, the original initial-boundary value problem for the key function is reduced to the Cauchy problem in terms of the time variable for the charac-teristics of the key function integral over the radial coordinate. The coeffi-cients of the approximation polynomials are presented through the integral characteristics of the key function and its given values on the inner and outer surfaces of the cylinder as a function of time. The solution of the Cauchy problem is obtained using the integral Laplacetransform. Expres-sions of the key function in each component layer of the cylinder are rec-orded in the form of quadratic polynomials, the coefficients of which are convolutions of functions describing the given limit values of the key func-tion on the inner and outer surfaces of the cylinder and homogeneous solu-tions of the Cauchy problem on integral characteristics. Expressions of the key function and Joule heat in the base and coating of the considered tubu-lar element under the action of an unsteady electromagnetic field in the mode with a pulsed modulating signal were obtained on the basis of gen-eral solutions for homogeneous non-stationary electromagnetic action. A computer analysis of the change in time and thickness of the constituent layers of this tubular element of the axial component of the vector of mag-netic field intensity and Joule heat was carried out.

Galerkin Approximations for an Initial Boundary Problem of Transient Flow in Fractured Porous Media

Galerkin Approximations for an Initial Boundary Problem of Transient Flow in Fractured Porous Media

Global solutions to an initial boundary problem for the compressible 3D MHD equations with Navier-slip and perfectly conducting boundary conditions in exterior domains

An initial boundary value problem for compressible magnetohydrodynamics (MHD) is considered on an exterior domain (with the vanishing first Betti number) in in this paper. The global existence of smooth solutions near a given constant state for compressible MHD with the boundary conditions of Navier-slip for the velocity filed and perfect conduction for the magnetic field is established. Moreover the explicit decay rate is given. In particular, the results obtained in this paper also imply the global existence of classical solutions for the full compressible Navier–Stokes equations with Navier-slip boundary conditions on exterior domains in three dimensions, which was not available in literature prior to the work in this paper, to the best of knowledge of the authors’.

Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension

This manuscript considers the no-flux initial-boundary problem for the migration-consumption system{ut=Δ(uϕ(v)),vt=Δv−uv,(⋆) in a smoothly bounded domain Ω⊂Rn, n≥1, where ϕ suitably generalizes the singular prototype given byϕ(ξ)=ξ−α,ξ>0, with α>0.It is firstly shown that for such diffusion singularities of arbitrary strength, and for any given initial data u0 and v0 from W1,∞(Ω) with u0≥0 and v0>0 in Ω‾, a so-called very weak-strong solution (u,v) with (u,v)|t=0=(u0,v0) can be constructed. Under the additional restrictions that 2≤n≤5 and α>n−26−n, and under an additional assumption on ϕ′, it is furthermore asserted that the flux ∇(uϕ(v)) belongs to some reflexive Lebesgue space, and that (⋆) is satisfied in a standard weak sense.Finally, in the one-dimensional case a statement on global classical solvability is derived under the mere condition that ϕ∈C3((0,∞)) be positive.

On one approach for the numerical solving of hyperbolic initial-boundary problems on an adaptive moving grids

An approach for the quality improvement of numerical modeling of the processes described by nonlinear hyperbolic equations, which have conservative form is proposed. The transition to synchronous moving coordinate system along with Godunov type numerical scheme for the problem on moving mesh is applied. The grid motion law relies heavily on Rankine–Hugoniot relations. In order to prevent the degeneration of moving computational mesh the regularization mechanism is implemented. It allows to almost automatically fit the computational mesh to the admissible width of the solution singularity resolution. The proposed algorithm is tested on several initial-boundary problems for the Hopf equation with well known exact solutions. The hyperbolic Buckley–Leverett equation is also considered as an example of application of the proposed approach.

Determination of pressure from nonlinear equations of nonhomogeneous non-newtonian fluids

In this paper, the initial-boundary problem for a system of nonlinear equations modified by p-laplacian diffusion and nonlinear damping term, in which describe the motion of a non-homogeneous (with unknown and non constant density ) incompressible viscoelastic non-Newtonian fluids is considered. Usually, a pressure does not include in the definition of weak generalized solution to equations of hydrodynamics for incompressible fluids, because weak solutions consider in the solenoidal space due to the incompressibility (continuity) equation. In the classical hydrodynamic equations, after determining a velocity and a density, a pressure can be determined by using the theorem of a representation of the space L2(Ω) into the direct sum of two orthogonal subspaces. The recovering of a pressure from the nonlinear equations of hydrodynamics modified by p-laplacian and nonlinear damping term requires a new approches. Here, under a suitable conditions on the data of the problem, a pressure has been uniquelly determined from the initial-boundary value problem for the system of nonlinear modified equations describing the motion of nonhomogeneous (with unknown and non constant density ) incompressible viscoelastic non-Newtonian fluids. The main functional spaces and necessary axuillary conclusions are introduced. A space of generalized weak solutions is identified. The pressure uniquelly recovered by using the known de-Ram lemma.

Nonlinear inverse problem of finding thermophysical characteristics

In this paper, a method has been developed for determining the nonlinear heat-conducting characteristics of the soil. Two-layer container complexes were created, the side faces of which are thermally insulated, so the 1D thermal conductivity equation is used. The temperature sensor is placed at the junction of two media, and a mixed boundary value problem is solved in each region. In order to provide the inverse coefficient problem with initial data, two temperature sensors are used: one sensor was placed at the open boundary of the container and recorded the soil temperature at this boundary, and the second sensor was placed a short distance from the boundary, which recorded the air temperature. The measurements were carried out in the time interval (0,4tmax). First, the initial-boundary problem of heat conduction with nonlinear coefficients is studied by the finite difference method. Two types of difference schemes are constructed: linearized and nonlinear. The linearized difference scheme is implemented numerically by the scalar Thomas method, and the nonlinear difference problem is solved by the Newton method. The solution of a linearized difference problem was taken as the initial approximation of Newton’s method. To find the thermophysical parameters, the corresponding functional is minimized using the gradient descent method. In addition, all thermophysical characteristics (8 coefficients) were found for a two-layer container with sand and chernozem.

The Neumann problem for one-dimensional parabolic equations with linear growth Lagrangian: evolution of singularities

In this paper, we obtain existence and uniqueness of strong solutions to the inhomogenous Neumann initial-boundary problem for a parabolic PDE which arises as a generalization of the time-dependent minimal surface equation. Existence and regularity in time of the solution are proved by means of a suitable pseudoparabolic relaxed approximation of the equation and the corresponding passage to the limit. Our main result is monotonicity in time of the positive and negative singular parts of the distributional space derivative for bounded variation initial data. Sufficient conditions for instantaneous L^1-W^{1,1}_{{mathrm{loc}}} or BV-W^{1,1}_{{mathrm{loc}}} regularizing effects are also discussed.

Numerical Solution of 2nd Order Boundary Value Problems with Dirichlet, Neumann and Robin Boundary Conditions using FDM

In many fields of science and engineering, to determine the harmonic motion, damped and forced variation, current from electric circuit, 2nd order ODE is required to solve. Solving the ODE with complicated boundary condition that occur in engineering problems is a great challenges analytically. Therefore, numerical technique finite difference method (FDM) is very popular and important for solving the boundary value problems. In this article three different conditions as Dirichlet, Neumann and Robin (mixed) boundary conditions are applied in initial-boundary problem. FDM is used to solve ODE boundary value problems. Error calculation, stability, convergence are also explained. To test the accuracy numerical solutions are verified with analytical solution and error is calculated at each point for different mesh grid size as mesh grid size is decreased result will give the accuracy.

Initial-boundary problem for 3D system of viscoelasticity

Initial-boundary problem for 3D system of viscoelasticity

Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values

Abstract In this paper, we study the initial boundary problem of fifth-order Korteweg-de Vries equation with nonlinear boundary values. First, we establish a so-called sharp boundary trace regularity associated with the linearized fifth-order Korteweg-de Vries equation. Then, aided by the sharp boundary trace regularity, we verify that initial-boundary value problem of fifth-order Korteweg-de Vries equation with nonlinear boundary conditions is locally well-posed when initial and boundary values are properly chosen.

Boundary layer of Hsieh's equation with conservative nonlinearity

In this paper, we consider an initial-boundary value problem of Hsieh's equation with conservative nonlinearity. The global unique solvability in the framework of Sobolev is established. In particular, one of our main motivations is to investigate the boundary layer (BL) effect and the convergence rates as the diffusion parameter $\beta$ goes zero. It is shown that the BL-thickness is of the order $O(\beta ^{\gamma })$ with $0<\gamma <\frac {1}{2}$. We need to point out that, different from the previous work on nonconservative form of Hsieh's equations, the conservative nonlinearity $(\psi ^{\beta }\theta ^{\beta })_x$ implies that new nonlinear term $\psi _x^{\beta }\theta ^{\beta }$ needs to be handled. It is important that more regularities on the solution to the limit problem are required to obtain the convergence rates and BL-thickness. It is more difficult for initial-boundary problem due to the lack of boundary conditions (especially, higher-order derivatives) prevents us from applying the integration by part to derive the energy estimates directly. Thus it is more complicated than the case of nonconservative form. Consequently more subtle mathematical analysis needs to be introduced to overcome the difficulties.

Solution of a nonlinear heat transfer problem based on experimental data

The paper develops a method for solving nonlinear equations of heat conduction. Two-layer container complexes have been created, the side faces of which are thermally insulated so that the 1D heat equation can be used. In order not to solve the boundary value problem with a contact discontinuity and lose the accuracy of the solution method, a temperature sensor was placed at the junction of two media, and a mixed boundary value problem is solved in each area (container). To provide initial data for the initial boundary value problem, three temperature sensors were used: two sensors measure air temperatures at the left and right boundaries of the container complex; the third sensor measures the temperature of the soil at the junction of two media. The paper numerically investigates the initial-boundary problem of heat conduction with nonlinear coefficients of heat conduction, heat capacity, heat transfer and density of the material. To solve a nonlinear initial-boundary value problem, the grid method is used. Two types of difference schemes are constructed: linearized and nonlinear. The linearized difference scheme is implemented numerically by the scalar sweep method, and the nonlinear difference problem is solved by the Newton method. On the basis of an a priori estimate of the solution of a nonlinear difference problem, we prove the convergence of the second degree of Newton’s method. The numerical calculations carried out show that, for small time intervals, the solutions of the linearized difference problem differ little from the solution of the nonlinear difference problem (1 - 3%). And for long periods of time, tens of days or months, the solutions of the two methods differ significantly, sometimes exceeding 20%

Operator Approach to the Problem on Small Motions of an Ideal Relaxing Fluid

In this paper, we study the problem on small motions of the ideal relaxing fluid that fills a uniformly rotating or fixed container. We prove a theorem on the uniform strong solvability of the corresponding initial-boundary problem. In the case where the system does not rotate, we find the asymptotic behavior of the solution under the stress of a special form. We investigate the spectral problem associated with the considered system. We obtain localization results for the spectrum, results on the essential and discrete spectrum, and results on asymptotic properties of the spectrum. For the nonrotating system in the weightlessness conditions, we prove the multiple basis property of a special system of elements. In this case, we find an expansion of the solution of the evolution problem in the special system of elements.

Small Motions of an Ideal Stratified Fluid in a Basin Covered by Ice

We study the problem on small motions of the ideal stratified fluid with a free surface partially covered by the crushed ice. The crushed ice is supposed to be ponderable particles of a matter floating on the free surface. These particles do not interact with each other during oscillations of the free boundary (or this interaction is negligible) and stay on the surface during these oscillations. Using the method of orthogonal projection of boundary-value conditions on the free surface and introducing auxiliary problems, we reduce the original initial-boundary problem to the equivalent Cauchy problem for a second-order differential equation in a Hilbert space. We obtain conditions providing the existence of a solution strong with respect to time of the initial-boundary problem describing the evolution of this hydraulic system.

Problems on Small Motions of Systems of Two Viscoelastic Fluids in Fixed Vessels

In this paper, we study the problem on small motions of two Oldroyd viscoelastic incompressible fluids contained in a fixed vessel. By means of the operator approach, we reduce the original initial-boundary problem to the Cauchy problem for an operator-differential equation in a Hilbert space and prove the well-posed solvability of the problem on an arbitrary time interval. We deduce the equation for normal oscillations of the considered hydraulic system (the Krein generalized operator pencil).