Types Of Boundary Problems Research Articles

A deterministic game interpretation for fully nonlinear parabolic equations with dynamic boundary conditions

This paper is devoted to deterministic discrete game-theoretic interpretations for fully nonlinear parabolic and elliptic equations with nonlinear dynamic boundary conditions. It is known that the classical Neumann boundary condition for general parabolic or elliptic equations can be generated by including reflections on the boundary to the interior optimal control or game interpretations. We study a dynamic version of such type of boundary problems, generalizing the discrete game-theoretic approach proposed by Kohn-Serfaty (2006, 2010) for Cauchy problems and later developed by Giga-Liu (2009) and Daniel (2013) for Neumann type boundary problems.

Investigation of force factors and stresses at singular points of plate elements in special cranes

The work addresses studying the possibilities of a numerical-analytical variant of the boundary elements method (BEM) in determining the internal force factors and stresses at singular points when bending thin isotropic plates. The simplest type of singularity has been investigated, a point of application of external concentrated forces and moments. The importance of a given problem is due to the fact that at these points the internal force factors tend towards infinity and it is not possible to determine the size using elementary methods. At the same time, these singular points are the significant stress concentrators (both tangential and normal), which is why calculating the limits to which the internal forces and moments tend is essential to analyze the strength of plate structures. In order to describe an external load, it is proposed to apply the Dirac delta function of two variables. The models of external loads are presented. A given proposal makes it possible to accurately calculate the limits to which the transverse forces, as well as bending and torsional moments, tend at singular points of thin plates. We simulated plate bending using the variational Kantorovich-Vlasov method, which is fully compatible with the models of external load. The internal force factors at the singular points of plates were determined while solving the boundary value problems, formed based on the algorithm of BEM. The MATLAB environment was used for programming and computation. Results of the calculations are characterized by high accuracy and reliability, in particular the errors in determining the deflections of plates at singular points do not exceed 2.0 % and the errors for bending moments are not above 3.0 %. Recommendations have been given to solving different types of boundary problems on bending the plates with singular points based on the proposed approach. It has been established that an accurate model of the external load in the form of concentrated forces and moments fundamentally enables determining the internal forces and moments at the singular points of thin plates applying an algorithm of the variational Kantorovich-Vlasov method. Up to now, there are no data on the importance of internal forces and moments at the singular points of plates. It is also shown that when calculating the internal forces and moments of plates, it is inappropriate to apply a single term from a series of the Kantorovich-Vlasov method; the errors amount to significant magnitudes of the order of 43‒44 %

Development and analysis of mathematical models for the process of thermal conductivity for piecewise uniform elements of electronic systems

We examined linear and non-linear mathematical models for the thermal conductivity process in the elements of electronic systems, which are described by a layer and a piecewise uniform layer with a through foreign cylindrical inclusion, with a concentrated heat flow at one of their boundary surfaces. Classical methods cannot resolve boundary problems of mathematical physics, which correspond to these models, in a closed form. In this connection, thermophysical parameters for piecewise uniform media are described by using generalized functions as a single entity for the entire system. As a result of this approach, we obtain one equation of thermal conductivity with generalized derivatives for the entire system with boundary conditions at the boundary surfaces of inhomogeneous media. In the classic case, the process of thermal conductivity would be described by a system of equations on thermal conductivity for each of the elements of heterogeneous medium under conditions of perfect thermal contact at the conjugating surfaces of dissimilar elements and boundary conditions at the boundary surfaces of non-uniform media. For a case of nonlinear models, which are more accurate than the linear ones, one of the conditions of a perfect thermal contact, namely equality of temperatures at the conjugating surfaces of dissimilar elements of the structure, cannot be used in the process of linearization of nonlinear boundary problems that correspond to these models. In this regard, in the present study we propose approaches that make it possible to solve such type of boundary problems in mathematical physics.

A VARIATIONAL FORMULATION OF ELECTRODYNAMICS WITH EXTERNAL SOURCES

We present a variational formulation of electrodynamics using de Rham even and odd differential forms. Our formulation relies on a variational principle more complete than the Hamilton principle and thus leads to field equations with external sources and permits the derivation of the constitutive relations. We interpret a domain in space-time as an odd de Rham 4-current. This permits a treatment of different types of boundary problems in an unified way. In particular we obtain a smooth transition to the infinitesimal version by using a current with a one point support.

Solving parabolic PDE's by a decomposition method

The paper deals with the application of the hybrid method of decomposition for the solution of two-dimensional parabolic partial differential equations with constant coefficients. The convergence and accuracy of the CSDT method are analysed, deriving the form of the relationship between ‘eigenvalues’ and the time step. Theoretical problems of this method of decomposition are mentioned. Also, the expression for computing the coefficients for various types of boundary problems has been derived. Practical applications of the method are presented. The cubic spline function interpolation is used for the continuous signal reproduction in the hybrid system. Some results obtained are presented at the end.

On a Particular Class of Similar Solutions of the Equations of Motion and Energy of a Viscous Fluid

By introducing the similarity concept to the two-dimensional, incompressible Navier-Stokes equations and energy equation, a particular class of solutions is found. Two general types of flows are considered: (1) laminar free convection—i.e., flows which take place due to a body force—and (2) laminar forced convection. For free convection on vertical plates, similar solutions are obtained for two different power-law surface temperature variations, and it is shown that one of these solutions constitutes a new type of boundary problem. Results of numerical integrations of the equations are compared with solutions of the similar boundary-layer equations for free convection, and it is demonstrated that a range of surface temperature variations exists for which the boundary layer equations are no longer valid. For forced convection, it is shown that the use of similarity transformations provides an alternate method of deriving the ordinary differential equations for some well-known solutions, such as Couette and stagnation point flows. Solutions are obtained for radial converging or diverging flows between plane surfaces when the temperatures of the surfaces vary as arbitrary powers of the distance from the origin. Results of numerical integrations of the ordinary differential equations are presented for Prandtl numbers of 0.01 and 1.0 and for linear surface temperature variations. Some rather surprising results are obtained for diverging flows when separation occurs and some revealing comparisons with results from boundary-layer theory are made.