System Of Parabolic Differential Equations Research Articles

BIFURCATION OF TORI FOR PARABOLIC SYSTEMS OF DIFFERENTIAL EQUATIONS WITH WEAK DIFFUSION

The aim of the present article is to investigate of some properties of quasiperiodic solutions of nonlinear autonomous parabolic systems with the periodic condition. The research is devoted to the investigation of parabolic systems of differential equations with the help of integral manifolds method in the theory of nonlinear oscillations. We prove the existence of quasiperiodic solutions in autonomous parabolic system of differential equations with weak diffusion on the circle. We study existence and stability of an arbitrarily large finite number of tori for a parabolic system with weak diffusion. The quasiperiodic solution of parabolic system is sought in the form of traveling wave. A representation of the integral manifold is obtained. We seek a solution of parabolic system with the periodic condition in the form of a Fourier series in the complex form and introduce the norm in the space of the coefficients in the Fourier expansion. We use the normal forms method in the general parabolic system of differential equations with weak diffusion. We use bifurcation theory for ordinary differential equations and quasilinear parabolic equations. The existence of quasiperiodic solutions in an autonomous parabolic system of differential equations on the circle with small diffusion is proved. The problems of existence and stability of traveling waves in the parabolic system with weak diffusion are investigated.

Construction of a numerical model and algorithm for solving two-dimensional problems of filtration of multicomponent liquids, taking into account the moving “oil-water” interface

The paper considers a two-dimensional mathematical model of the filtration of a viscous, incompressible fluid in a deformable porous medium. The article describes a mathematical model of the problem at the “oil-water” interface with a system of parabolic differential equations. The problem posed is solved numerically using the differential-difference method based on the longitudinal-transverse scheme and the differential sweep method to determine the unknown boundary. To obtain a differential-difference problem, an algorithmic representation of a hidden scheme of alternating directions (longitudinal-transverse scheme) is used. The resulting system of differential-difference equations with initial conditions is known for each time layer along straight lines along the x axis, and then along each y axis it is solved by a differential sweep along a straight line, where the values just found correspond to time, i.e., the layer is taken as initial conditions. By approximating the differential equations in time, the position of the interface is determined for each time layer.

Existence and stability of traveling waves in parabolic systems of differential equations with weak diffusion

The aim of the present paper is to investigate of some properties of periodic solutions of a nonlinear autonomous parabolic systems with a periodic condition. We investigate parabolic systems of differential equations using an integral manifolds method of the theory of nonlinear oscillations. We prove the existence of periodic solutions in an autonomous parabolic system of differential equations with weak diffusion on the circle. We study the existence and stability of an arbitrarily large finite number of cycles for a parabolic system with weak diffusion. The periodic solution of parabolic equation is sought in the form of traveling wave. A representation of the integral manifold is obtained. We seek a solution of parabolic system with the periodic condition in the form of a Fourier series in the complex form and introduce a norm in the space of the coefficients in the Fourier expansion. We use the normal forms method in the general parabolic system of differential equations with retarded argument and weak diffusion. We use bifurcation theory for delay differential equations and quasilinear parabolic equations. The existence of periodic solutions in an autonomous parabolic system of differential equations on the circle with retarded argument and small diffusion is proved. The problems of existence and stability of traveling waves in the parabolic system with retarded argument and weak diffusion are investigated.

Regularized asymptotics of the solution of systems of parabolic differential equations

The regularization method for singularly perturbed problems of S. A. Lomov is generalized to constructing the asymptotics of the solution of the first boundary value problem for systems of differential equations of parabolic type with a small parameter at all derivatives.It is shown that the asymptotics of the solution of the problem contains n exponential, 2n parabolic and 2n angle boundary layer functions. The exponential boundary layer function describes the boundary layer along t = 0, the boundary layer along x = 0 and x = 1 is described by parabolic boundary layer functions.

Квазiлiнiйнi системи параболiчних диференцiальних рiв- нянь в дивергентнi формi з форм-обмеженими коефiцiєнтами

В роботi дослiджуються квазiлiнiйнi системи параболiчних диференцiальних рiвнянь в дивергентнi формi другого порядку з сингулярними коефiцiєнтами за умов форм-обмеженостi i лiнiйного росту нелiнiйного збурення. Встановлюється iснування розв’язку першої крайової задачi для квазiлiнiйної системи параболiчних диференцiальних рiвнянь за умов форм-обмеженостi i лiнiйного росту в просторi Соболева. Розглядаються умови за яких нелiнiйне збурення параболiчного диференцiального оператору обмежене лiнiйною функцiєю з коефiцiєнтами, якi можуть бути сингулярними за просторовою змiною, в лiнiйному випадку цi коефiцiєнти належать функцiональним класам Като та Неша

Transient Thermal Field Analysis in ACCC Power Lines by the Green’s Function Method

The paper investigates the dynamics of the thermal field of the ACCC (aluminum conductor composite core) line. The system was heated by solar radiation and current flow. Conductor cooling was modeled using the total heat transfer coefficient as the sum of convective and radiative components. The temperature increase generated by the current is described by a system of parabolic differential equations with an appropriate set of boundary, initial and continuity condition. The mentioned boundary-initial problem was solved by a modified Green’s method, adapted to the layered structure of the system. For this purpose, Green’s functions, as the kernels of integral operators inverse to differential ones, were determined. Aluminum resistivity and heat transfer coefficient change significantly with temperature. For this reason, the solution to the problem is presented in the form of a lower and upper estimation of the heating curve and local time constant. A steady-state current rating was also determined. The results are presented graphically and verified by other methods (power balance and finite element). The physical interpretation of the presented solution is also given.

Existence of positive solutions of (p(x),q(x))‐Laplacian parabolic systems with right hand side defined as a multiplication of two separate functions

The paper deals with the study of the existence of weak positive solutions for a new class of the system of parabolic differential equations with respect to the symmetry conditions by using sub‐super solutions method. Our results are natural extensions from our previous ones in (Bol. Soc. Mat. Mex. 2019; 25:145‐162) and (Appl. Math E‐Notes. 2018; 18:209‐218), where we have already studied the existence of positive solutions for some classes of Laplacian elliptic problems by using one classical method.

Model of Blood Circulation in Human Lower Limbs

A mathematical model of blood circulation in human lower limbs is proposed. The model is based on the laws of motion (filtration) of a viscous fluid in a heterogeneous medium consisting of two or more interpenetrating continua. It is assumed that the blood system consists of a distribution network of comparatively large vessels (arterioles) connected to small capillaries and a similarly structured collection network of small capillaries united into larger veins. A system of parabolic differential equations is derived, for which a problem with no initial data is posed. A periodic (in time) solution of the system corresponding to harmonic oscillations defined by the cardiac rhythm is found. Analytical solutions for particular cases of problems following from the general model of blood circulation are obtained. Numerical calculations are performed, and a numerical solution is found for a one-dimensional problem with parameters similar to those corresponding to real conditions of blood circulation with allowance for the cross-sectional area of the muscular tissue of the lower limb.

INVARIANT MEASURES FOR SYSTEMS OF KOLMOGOROV EQUATIONS

In this paper we provide sufficient conditions which guarantee the existence of a system of invariant measures for semigroups associated to systems of parabolic differential equations with unbounded coefficients. We prove that these measures are absolutely continuous with respect to the Lebesgue measure and study some of their main properties. Finally, we show that they characterize the asymptotic behaviour of the semigroup at infinity.

Bifurcation of Self-Excited Vibrations for Parabolic Systems with Retarded Argument and Weak Diffusion

We prove the existence of periodic solutions of an autonomous parabolic system of differential equations with retarded argument and weak diffusion on a circle. The problems of existence and stability of traveling waves in the equation of spin combustion with delay are investigated.

Acoustic radiation frequency of a cylindrical vortex

Acoustic radiation frequency of a cylindrical vortex in the air has been calculated. Calculations are based on the Navier-Stokes equations using the expansion of the functions in powers of small parameter characterizing the initial vorticity. Non-uniform system of parabolic differential equations with constant coefficients is obtained. The initial radius of the cylinder varies over a wide range. The problem is considered for the case of plane flow. In contrast to previously results, it is shown that at small values of vorticity, the frequency of the acoustic radiation depends only on the geometric size (the initial radius of the vortex).

Application of the moving mesh method to the porous medium equation with variable exponent

In this paper, we investigate the applicability of a moving mesh method when applied to the porous medium equation with variable exponent in 2D with free boundaries.The movement of the boundary is derived by a discretization of the Darcy law. The spatial discretization is defined by a partition in triangles of the domain, and the solution is approximated by piecewise polynomial functions of degree r≥1 using Lagrange interpolating polynomials in area coordinates. A parabolic system of differential equations to control the movement of the interior vertices of the triangles is added to the equations of the problem. The resulting ordinary differential system of equations in time variable is solved using a suitable integrator. The integrals that arise in the system of ordinary differential equations are calculated using the Gaussian quadrature of appropriate order. Finally, we present a numerical application of this technique and compare the results with a fixed mesh method.

Mean curvature flows and isotopy problems

In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differential equations. Several global existence theorems and applications to isotopy problems in geometry and topology will be presented. The results are based on joint works of the author with his collaborators I. Medoys, K. Smoczyk, and M.-P. Tsui.

Relay control of an evolution process in a Banach space in a neighborhood of an unstable stationary regime

We study a controlled evolution process whose states are characterized by points of a real Banach space and which is described by a family of semigroups in a neighborhood of an unstable stationary regime. The goal is to find control to keep the state of the process however long in the neighborhood. We find conditions under which this control is possible, and prove an existence theorem. Also, we apply the theorem to the processes described by systems of parabolic differential equations.

Oscillation Tests for Certain Systems of Parabolic Differential Equations with Neutral Type

Oscillation Tests for Certain Systems of Parabolic Differential Equations with Neutral Type

Evolution of curves on a surface driven by the geodesic curvature and external force

We study a flow of closed curves on a given graph surface driven by the geodesic curvature and external force. Using vertical projection of surface curves to the plane we show how the geodesic curvature-driven flow can be reduced to a solution of a fully nonlinear system of parabolic differential equations. We show that the flow of surface curves is gradient-like, i.e. there exists a Lyapunov functional nonincreasing along trajectories. Special attention is placed on the analysis of closed stationary surface curves. We present sufficient conditions for their dynamic stability. Several computational examples of evolution of surface curves driven by the geodesic curvature and external force on various surfaces are presented in this article. We also discuss a link between the geodesic flow and the edge detection problem arising from the image segmentation theory.

Surface diffusion including adatoms

The aim of this paper is to study continuum models for surface diffusion taking into account free adatoms on the surface, which is of particular importance in the (self-assembled) growth of nanostructures. The extended model yields a coupled system of parabolic differential equations for the surface morphology and the adatom density, involving a cross-diffusion structure. We investigate two different situations, namely the growth of a film on a substrate and the growth of a crystal-like structure (a closed curve or surface). An investigation of the equilibrium situation, which can be phrased as an energy minimization problem subject to a mass constraint, shows a different behaviour in both situations: for the film the equilibrium is attained when all atoms are attached to the surface, while for a crystal the adatom density does not vanish on the surface. The latter is also a deviation from the usual equilibrium theory, since the equilibrium shape will be strictly included in the Wulff shape. Moreover, it turns out that the total energy is not lower semicontinuous and non-convex for large adatom densities and rough surfaces. The dynamics of the adatom surface diffusion model is investigated in detail for situations close to a flat surface in the film case and the situation close to a radially symmetric curve, both with an almost spatially homogeneous adatom density, where the cross-diffusion structure of the model and the decay to equilibrium can be studied in detail. Finally, we discuss the numerical solution of the adatom surface diffusion model in the film case and provide various simulation results.

Analysis of free convection about a horizontal cylinder in a porous media using a thermal non-equilibrium model

The thermally non-equilibrium model is used to study the free convection from a horizontal cylinder immersed in porous media. The governing equations are transformed to dimensionless form by introducing the boundary layer dimensionless variables. The resultant parabolic system of differential equations is solved by using an implicit finite difference method based on Keller box algorithm. The results of the developed code are validated with different mesh sizes, which can be used as benchmark results for thermally equilibrium condition. Numerical results are obtained for non-equilibrium model to analyze the effect of the governing parameters, which are the heat transfer coefficient between the solid and fluid phases H and the porosity scaled thermal conductivity ratio K r . The results show that increasing H or K r leads to increase in the total average Nusselt number. The value of the average Nusselt number for both fluid and solid phases as well as the total average Nusselt number have approached the corresponding values of the thermally equilibrium model at high value of H × K r .

A Primal-Dual Active Set Strategy for Optimal Boundary Control of a Nonlinear Reaction-Diffusion System

This paper is concerned with optimal boundary control of an instationary reaction-diffusion system in three spatial dimensions. This problem involves a coupled nonlinear system of parabolic differential equations with bilateral as well as integral control constraints. We include the integral constraint in the cost by a penalty term whereas the bilateral control constraints are handled explicitly. First- and second-order conditions for the optimization problem are analyzed. A primal-dual active set strategy is utilized to compute optimal solutions numerically. The algorithm is compared to a semismooth Newton method.

ANALYTICAL SOLUTION OF TWO-DIMENSIONAL CONTACT PROBLEMS OF UNSTEADY HEAT CONDUCTION IN THE PRESENCE OF MIXED BOUNDARY CONDITIONS IN THE CONTACT PLANE

A method for solution of systems of parabolic differential equations of heat conduction on the model of thermal contact between two bodies with different thermophysical characteristics in the presence of mixed boundary conditions in the plane of their contact has been suggested for the first time. The case of contact of two semibounded bodies has been considered. In this case, a heat source of low heat capacity acts in a circular region of finite radius on the contact surface, and beyond this region the initial temperature is maintained during the whole period of heat transfer.