The Problem of Diffusion Wave Initiation for a Nonlinear Second-Order Parabolic System

Kazakov A.L., Spevak L.F.

Bekezhanova V.B., Stepanova I.V.

The theoretical approach for mathematical modeling of the evaporative convection in a multiphase system with interface based on the use of an exact solution of governing equations is discussed. The mathematical model builds on the “diffusive” laws of the transfer of matter, momentum and energy and includes the interface boundary conditions formulated with respect to the conservation laws. The carried out compatibility analysis of the equations concludes that there are three classes of exact solutions of the system under consideration. One of the possible solutions is circumstantially studied in the framework of the evaporative convection problem in a bilayer liquid – gas system, where both phases are the binary mixtures. The convection-diffusion equations are used to govern the transfer of one selected component and its vapor in the liquid and gas layers, respectively. The thermodiffusion effect is taken into account additionally for more precise description of heat transfer processes. The impact of this effect on the concentration and thermal characteristics as well as on the mass evaporation flow rate is investigated. It is shown that the utilized solution can describe convective regimes appearing on a working area of a long plane channel under thermal load distributed with respect to longitudinal coordinate by means of quadratic law. The solution correctly predicts hydrodynamical, temperature and concentration parameters of convective flows arising in the bilayer system. Basic characteristics calculated by this solution are feasible when the system is slightly deviated from the thermodynamic equilibrium state, and mass transfer through the interface is weak.

Kazakov A.L., Kuznetsov P.A., Spevak L.F.

We consider a system of two nonlinear second-order parabolic equations with singularity. Systems of this type are applied in chemical kinetics to describe reaction-diffusion processes. We prove the existence and uniqueness theorem of an analytic solution having the diffusion-wave type at a given wave front. The proof is constructive, and the solution had the form of a power series with recursively calculated coefficients. Moreover, we propose some numerical algorithm based on the boundary element method whose verification uses the segments of analytic solutions.

Kazakov A.L., Spevak L.F.

We consider the problem of constructing exact solutions to a system of two coupled nonlinear parabolic reaction–diffusion equations. We study solutions in the form of diffusion waves propagating over zero background with a finite speed. The theorem on the construction of exact solutions by reducing to the Cauchy problem for a system of ordinary differential equations (ODEs) is proved. A time-step numerical technique for solving the reaction-diffusion system using radial basis function expansion is proposed. The same technique is used to solve the systems of ordinary differential equations defining exact solutions to the reaction–diffusion system. Numerical analysis and estimation of the accuracy of solutions to the system of ODEs are carried out. These solutions are used to verify the obtained time-step solutions of the original system..

Kosov A., Semenov E.

Abstract A system of nonlinear parabolic differential equations is being studied, considered as a distributed mathematical model of the process of examining three-dimensional space by interacting robots of two types. Parametric families of exact solutions have been built that can be used to form the control of the survey process by creating the necessary densities at the border of the area that is the base for robots.

Kazakov A., Kuznetsov P., Lempert A.

The paper deals with a system of two nonlinear second-order parabolic equations. Similar systems, also known as reaction-diffusion systems, describe different chemical processes. In particular, two unknown functions can represent concentrations of effectors (the activator and the inhibitor respectively), which participate in the reaction. Diffusion waves propagating over zero background with finite velocity form an essential class of solutions of these systems. The existence of such solutions is possible because the parabolic type of equations degenerates if unknown functions are equal to zero. We study the analytic solvability of a boundary value problem with the degeneration for the reaction-diffusion system. The diffusion wave front is known. We prove the theorem of existence of the analytic solution in the general case. We construct a solution in the form of power series and suggest recurrent formulas for coefficients. Since, generally speaking, the solution is not unique, we consider some cases not covered by the proved theorem and present the example similar to the classic example of S.V. Kovalevskaya.

Filimonov M.Y.

Abstract One of the analytical methods of presenting solutions of nonlinear partial differential equations is the method of special series in powers of specially selected functions called basic functions. The coefficients of such series are found successively as solutions of linear differential equations. The basic functions can also contain arbitrary functions. By using such functional arbitrariness allows in some cases, to prove the global convergence of the corresponding constructed series, and also allows us to prove the solvability of the boundary value problem for the Korteweg-de Vries equation. In the paper for a nonlinear wave equation a theorem on the possibility of satisfying a given boundary condition using an arbitrary function contained in the basic function is proved.

Kazakov A.L., Nefedova O.A., Spevak L.F.

The paper is devoted to constructing approximate heat wave solutions propagating along the cold front at a finite speed for a nonlinear (quasi-linear) heat conduction equation with a power nonlinearity. The coefficient of the higher derivatives vanishes on the front of the heat wave, i.e., the equation degenerates. One- and two-dimensional problems about the initiation of a heat wave by the boundary mode specified on a given fixed manifold are studied. Algorithms for solving this problem based on the boundary element method and a special change of variables as a result of which the unknown function and the independent spatial variable exchange their roles are proposed. The solution of the transformed problem in the form of a converging power series is constructed. These algorithms are implemented in computer programs, and test computations are performed. Their results are compared with truncated power series mentioned above and with the known exact solutions; the results are in good agreement.

Kazakov A.L., Kuznetsov P.A.

The paper addresses a nonlinear heat equation (the porous medium equation) in the case of a power-law dependence of the heat conductivity coefficient on temperature. The equation is used for describing high-temperature processes, filtration of gases and fluids, groundwater infiltration, migration of biological populations, etc. The heat waves (waves of filtration) with a finite velocity of propagation over a cold background form an important class of solutions to the equation under consideration. A special boundary value problem having solutions of such type is studied. The boundary condition of the problem is given on a sufficiently smooth closed curve with variable geometry. The new theorem of existence and uniqueness of the analytic solution is proved.

Fornberg B., Flyer N.

Finite differences provided the first numerical approach that permitted large-scale simulations in many applications areas, such as geophysical fluid dynamics. As accuracy and integration time requirements gradually increased, the focus shifted from finite differences to a variety of different spectral methods. During the last few years, radial basis functions, in particular in their ‘local’ RBF-FD form, have taken the major step from being mostly a curiosity approach for small-scale PDE ‘toy problems’ to becoming a major contender also for very large simulations on advanced distributed memory computer systems. Being entirely mesh-free, RBF-FD discretizations are also particularly easy to implement, even when local refinements are needed. This article gives some background to this development, and highlights some recent results.

Antontsev S., Shmarev S.

Chen W., Fu Z., Chen C.S.

Stepanova E.V., Shishkov A.E.

The propagation of supports of solutions of second-order quasilinear parabolic equations is studied; the equations are of the type of nonstationary diffusion, having semilinear absorption with an absorption potential which degenerates on the initial plane. We find sufficient conditions, which are sharp in a certain sense, on the relationship between the boundary regime and the type of degeneration of the potential to ensure the strong localization of solutions. We also establish a weak localization of solutions for an arbitrary potential which degenerates only on the initial plane. Bibliography: 12 titles.

Kazakov A.L., Lempert A.A.

The problem of the motion of a filtration front in a zero background in the case of a power-law dependence of the filtration coefficient on gas density is considered, and the existence and uniqueness theorem for solutions in the class of analytic functions is proved. The solution is constructed in explicit form, recurrence formulas for computing the coefficients in the series are obtained, and the convergence of the series is proved by the majorant method. The filtration front construction procedure is proposed.

Rubina L.I., Ul’yanov O.N.

Some exact solutions to a nonlinear heat equation are constructed. An initial-boundary value problem is examined for a nonlinear heat equation. To construct solutions, the problem for a partial differential equation of the second order is reduced to a similar problem for a first order partial differential equation.