Boundary Problems Of Mathematical Physics Research Articles

Examining the temperature fields in flat piecewise- uniform structures

The paper considers linear and non-linear mathematical models for the thermal conductivity process in designs that are described by a plate and a layered plate with a foreign parallelepiped-shaped through-inclusion on whose one boundary surface heat flux is concentrated. Classic methods do not make it possible to solve the boundary problems of mathematical physics that match these models in a closed form. Given this, in the present work we propose an approach that is based on the fact that the thermal-physical parameters for the piecewise uniform environments are described using generalized functions as a single entity for the whole system. As a result, we obtained one equation of thermal conductivity with generalized derivatives for the entire system with boundary conditions at the boundary surfaces of non-uniform environments. In a classic case, the process of thermal conductivity would be described by a system of equations of thermal conductivity for each of the elements of a non-uniform environment with conditions for an ideal thermal contact at the interface surfaces of non-uniform elements and boundary conditions on boundary surfaces of non-uniform environments. For the case of non-linear models, the condition of temperature equality at the interface surfaces of non-uniform elements of the designs is not applicable. With regard to the aforementioned, this work proposed yet another approach, which is in the introduction of linearizing functions that make it possible to linearize corresponding nonlinear boundary problems for these designs, which, as a result, allows us to solve this kind of boundary problems in mathematical physics. We received calculation formulas for determining the temperature field in the examined thermosensitive systems in the case of linearly variable coefficient of thermal conductivity of design materials. By using the obtained analytical-numerical solutions of linear and nonlinear boundary problems for the given piecewise-uniform structures, we created computational programs that make it possible to obtain the numerical values of temperature distribution and analyze the structures in terms of thermostability. As a result, it becomes possible to improve thermal stability of these designs and thus protect them from overheating, which can cause destruction of separate elements and even entire systems.

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.

One-, two-, and three-dimensional ising model in the static fluctuation approximation

Viewed as a prototype for strongly interacting many-body systems, the spin-1/2n-dimensional Ising model (n = 1, 2, 3) is studied within the so-calledstatic fluctuation approximation (SFA). The underlying physical picture is that the local fieldoperator σfz withquadratic fluctuations is replaced with its mean value [(σfz)2 ≃ 〈(σfz)2〉]. This means that the true quantum mechanical spectrum of the operator σfz is replaced with a distribution; along with the calculation of its mean value, we take into accountself-consistently the moments of this distribution. It is shown that this sole approximation is sufficient for deducing the equilibrium correlation functions and the main thermodynamic characteristics of the system. Special new features of this study include an analysis of the two-dimensional modelwithout periodic boundary conditions, and the demonstration that the phase-transition scenario is quite sensitive to the boundary conditions in the two-and three-dimensional cases. In passing, new boundary problems in mathematical physics are emphasized.

Book reviews

Book reviews

Difference analogues of orthogonal decompositions, basic differential operators and some boundary problems of mathematical physics. I

In recent times, in connection with the development of the theory of generalized functions and the theory of generalized solutions of boundary value problems for differential equations, the approach to the construction of difference operators, which approximate the differential problem, has essentially changed. First of all we have in mind the following: 1. (1) the rejection, because of results obtained, of the attempt to approximate (in more or less explicit form) a differential equation, with continuity conditions and boundary conditions, which must be done locally and independently of each other, by systems of finite difference equations; 2. (2) the development of numerical methods for a group of non-stationary problems, in which the form of the difference schemes is determined, not by the requirements of the best local approximation, but by conditions of stability and simplicity of calculation; 3. (3) the construction and basis of difference operators which possess the integral properties of differential operators of boundary value problems. We shall dwell only on the last group of methods. Difference equations for whose investigation integral characteristics were used, have been considered, for example, in works [1]–[13]. For the study of the properties of difference operators, which keep the integral properties of the differential problems, we can successfully use functional methods of investigation. It is well known with what success functional methods are applied to the study of problems of mathematical physics, in particular the method of orthogonal projections, which form a considerable part of the theory of generalized solutions (see [14]–[17]). A significant advance in the theory of difference equations has been marked by the fact that functional methods of directly investigating their inner properties (see [18]–[20]) have been applied to equations on a network. In the present work we put forward and systematically apply a method of orthogonal projections for network vector-functions. It permits us 1. (1)to investigate the structural properties of subspaces of a space of network functions, 2. (2) to compose difference equations for some problems of mathematical physics, 3. (3) to solve the question of difference analogues of boundary conditions by using the fact that the solution is a member of a definite subspace of functions, 4. (4) to investigate the properties of the solution of difference equations, 5. (5) to give optimal evaluations for the solutions of difference equations by means of projectional operators on the right-hand sides of these equations. The difference equations obtained for boundary value problems of types I, II and III, for self-conjugate elliptic equations of the second order, possess a regular closure of the computational algorithm; with symmetrization of these systems of equations they are converted into known systems of equations of a divergent type. We have called the difference equations obtained “difference analogues” of the problems considered, wishing thereby to stress the fact that in conclusions obtained from them we did not make use of the idea of local approximation of the differential operator by a difference operator, but obtained these equations by using functional (integral) methods. Local approximation was used only in replacing the first derivative of a function by a centrally divided difference. With the selected approach to the construction of difference operators we shall consider as a natural form of approximation an approximation, in the weak sense, which characterizes the nearness of two functionals, with which we may meet not only the case of local smallness of the discrepancy in the approximation of the differential operator by a difference, but also the case of its rapid oscillation. Both cases are equally well represented in the chosen method of evaluation of the errors of solutions. For clearness and simplicity of exposition we have limited ourselves to the case of three-dimensional space and have considered problems involving the Laplace operator.

Difference analogues of orthogonal decompositions, basic differential operators and some boundary problems of mathematical physics. II

Difference analogues of orthogonal decompositions, basic differential operators and some boundary problems of mathematical physics. II

LXII.On the solution of boundary problems in mathematical physics

LXII.<i>On the solution of boundary problems in mathematical physics</i>

LXXIX.On the solution of boundary problems in mathematical physics

LXXIX.<i>On the solution of boundary problems in mathematical physics</i>

A Class of Real Quadratic Forms in Infinitely Many Variables

condition x = 1; it was shown by Carlemant that the theory of these forms may be brought under the theory of integral equations, but the possible applications of the theory to the boundary problems of mathematical physics made it desirable to develop the theory in as direct a manner as possible. In the first four sections the spectral form corresponding to a given real quadratic form is discussed, with certain properties of related linear forms; in the fifth and sixth sections, respectively, a representation is given of the solutions of an infinite system of ordinary linear equations, and of an infinite system of differential equations of the second order, with constant coefficients. In the remaining sections the preceding theory is applied to the formal solution of certain boundary problems for a system of partial differential equations. 1. Two theorems. In the following developments two theorems will be used constantly, of which the first is the following:4 THEOREM I: If f(x) is continuous in the interval a _ x _ b, while g(x, n) has its total variation in this interval less than a constant B for all positive integral values of n, and for every x lim g (x, n) = g (x), then 71 -_ 00

Some Boundary Problems of Mathematical Physics

Proceedings of the London Mathematical SocietyVolume s2-24, Issue 1 p. 435-458 Papers Some Boundary Problems of Mathematical Physics D. M. Wrinch, D. M. WrinchSearch for more papers by this author D. M. Wrinch, D. M. WrinchSearch for more papers by this author First published: 1926 https://doi.org/10.1112/plms/s2-24.1.435AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volumes2-24, Issue11926Pages 435-458 RelatedInformation