Chirped coherent Rayleigh–Brillouin scattering (CRBS) is a flow diagnostic technique that offers high signal-to-noise ratios and nanosecond temporal resolution. To extract information of dilute gas flow, experimental spectra must be compared with theoretical predictions derived from the Boltzmann equation. In this work, we develop a MATLAB code that deterministically solves the Boltzmann equation (with a modelled collision kernel for the inverse power-law potential) to compute CRBS spectra, enabling each line shape to be obtained in approximately one minute. We find that the CRBS spectrum is highly sensitive to the intermolecular potential and that rapid chirping generates fine ripples around the Rayleigh peak along with spectral asymmetries.
The present paper describes, in a theoretical fashion, a variational approach to formulate fourth-order dynamical systems on differentiable manifolds on the basis of the Hamilton–d’Alembert principle of analytic mechanics. The discussed approach relies on the introduction of a Lagrangian function that depends on the kinetic energy and the covariant acceleration energy, as well as a potential energy function that accounts for conservative forces. In addition, the present paper introduces the notion of Rayleigh differential form to account for non-conservative forces. The corresponding fourth-order equation of motion is derived, and an interpretation of the obtained terms is provided from a system and control theoretic viewpoint. A specific form of the Rayleigh differential form is introduced, which yields non-conservative forcing terms assimilable to linear friction and jerk-type friction. The general theoretical discussion is complemented by a brief excursus about the numerical simulation of the introduced differential model.
Cesàro polynomials have been extended in various ways and applied in diverse areas. In this paper, we aim to introduce a multivariable and multiparameter generalization of Cesàro polynomials. Then we explore several generating functions, an addition formula, a differential-recurrence relation, a multiple integral formula for this extended Cesàro polynomial, as well as a multiple integral formula whose kernel is this extended Cesàro polynomial. Also we present several bilinear and bilateral generating functions for this extended Ces`aro polynomial, two of whose examples are demonstrated.
In this work, we study two boundary value problems for involutary parabolic equation with the first and second kind conditions. We propose absolute stable difference schemes for numerical solutions of these boundary value problems. Actually the stability estimates for solutions of difference schemes are proved. Later error analysis for the numerical solution of both difference schemes are illustrated by test examples.
The importance of energy-saving and correct design is obvious for energy efficiency. Correct design means that before construction considerable things, such as orientation or isolation decisions, need to be made. This study gives a mathematical model of the nonstationary energy consumption calculation problems. The model is well-posedness in Holder spaces of the mixed one-dimensional parabolic problem with Robin boundary conditions. In this study, an effective numerical method is also developed for energy consumption calculation which is related to this mathematical model. The three case problems are taken to test this numerical method. The dynamic model results have been compared with the previous finite-difference or steady-state solutions. The study also aims to develop a mathematical model in which the result can be found at any time.
This article is devoted to the solvability of degenerate nonlinear equations of pseudoparabolic type. Such problems appear naturally in physical and biological models. The article aims to study the solvability in the classes of regular solutions of (all derivatives generalized in the sense of S.L. Sobolev included in the equation) initial-boundary value problems for differential equations. For the problems under consideration, we have found conditions on parameters ensuring the existence of solutions and we have proved existence and uniqueness theorems. The main method for proving the solvability of boundary value problems is the regularization method.
Partial differential equations of the parabolic type with discontinuous coefficients and the heat equation degenerating in time, each separately, have been well studied by many authors. Conjugation problems for time-degenerate equations of the parabolic type with discontinuous coefficients are practically not studied. In this work, in an n-dimensional space, a conjugation problem is considered for a heat equation with discontinuous coefficients which degenerates at the initial moment of time. A fundamental solution to the set problem has been constructed and estimates of its derivatives have been found. With the help of these estimates, in the Sobolev classes, the estimate of the solution to the set problem was obtained.
We consider discrete analogue for simplest boundary value problem for elliptic pseudo-differential equation in a half-space with Dirichlet boundary condition in Sobolev–Slobodetskii spaces. Based on the theory of discrete boundary value problems for elliptic pseudo-differential equations we give a comparison between discrete and continuous solutions for certain model boundary value problem.
This paper deals with a fully fuzzy linear programming problem (FFLP) in which the coefficients of decision variables, the right-hand coefficients and variables are characterized by fuzzy numbers. A method of obtaining optimal fuzzy solutions is proposed by controlling the left and right sides of the fuzzy variables according to the fuzzy parameters. By using fuzzy controlled solutions, we avoid unexpected answers. Finally, two numerical examples are solved to demonstrate how the proposed model can provide a better optimal solution than that of other methods using several ranking functions.
The stable difference scheme for the approximate solution of the initial boundary value problem for the telegraph equation with time delay in a Hilbert space is presented. The main theorem on stability of the difference scheme is established. In applications, stability estimates for the solution of difference schemes for the two type of the time delay telegraph equations are obtained. As a test problem, one-dimensional delay telegraph equation with nonlocal boundary conditions is considered. Numerical results are provided.
A nonlocal problem for the fourth order system of loaded partial differential equations is considered. The questions of a existence unique solution of the considered problem and ways of its construction are investigated. The nonlocal problem for the loaded partial differential equation of fourth order is reduced to a nonlocal problem for a system of loaded hyperbolic equations of second order with integral conditions by introducing new functions. As a result of solving nonlocal problem with integral conditions is applied a method of introduction functional parameters. The algorithms of finding the approximate solution to the nonlocal problem with integral conditions for the system of loaded hyperbolic equations are proposed and their convergence is proved. The conditions of the unique solvability of the nonlocal problem for the loaded hyperbolic equations are obtained in the terms of initial data. The results also formulated relative to the original problem.
When studying Jonsson theories, which are a wide subclass of inductive theories, it becomes necessary to study the so - called Jonsson sets. Similar problems are considered both in model theory and in universal algebra. This topic is related to the study of model - theoretical properties of positive fragments. These fragments are a definable closure of special subsets of the semantic model of a fixed Jonsson theory. In this article are considered model - theoretical properties of a new class of theories, namely ∆ - PJ theories of countable first - order language. These are theories that are obtained from ∆ - PJ theories by replacing in the definition of ∆- PJ theories of morphisms (∆ - continuities) with morphisms (∆ - immersions). A number of results were obtained, ∆ - PJ fragments, ∆ - PJ sets, hybrids of ∆ - PJ theories. All questions considered in this article are relevant in the study of Jonsson theories and their model classes.
In this article, we consider weighted spaces of numerical sequences λp,q, which are defined as sets of sequences a = {ak}∞k=1, for which the norm ||a||λp,q := (∞Σk=1|ak|qkq/p −1)1/q < ∞ is finite. In the case of non-increasing sequences, the norm of the space λp,q coincides with the norm of the classical Lorentz space lp,q. Necessary and sufficient conditions are obtained for embeddings of the space λp,q into the space λp1,q1. The interpolation properties of these spaces with respect to the real interpolation method are studied. It is shown that the scale of spaces λp,q is closed in the relative real interpolation method, as well as in relative to the complex interpolation method. A description of the dual space to the weighted space λp,q is obtained. Specifically, it is shown that the space is reflective, where p', q' are conjugate to the parameters p and q. The paper also studies the properties of the convolution operator in these spaces. The main result of this work is an O’Neil type inequality. The resulting inequality generalizes the classical Young-O’Neil inequality. The research methods are based on the interpolation theorems proved in this paper for the spaces λp,q.
Nowadays, much attention is paid to the creation of favorable conditions for the health and work of people and the microclimate in the design and further operation of buildings. It is necessary to investigate the processes of forming a microclimate in the room to assess the comfort of the microclimate, as well as to determine the required capacity of the equipment of engineering systems. Analytical research methods, methods of mathematical and computer modeling are used as research methods. The methods of analysis and synthesis of automatic control systems, mathematical apparatus of theory of fuzzy logic, the Matlab software environment, the system of visual modeling Simulink, and the system for designing fuzzy systems Fuzzy Logic Toolbox are used for research. The paper presents a model of the room, taking into account the heat loss through the enclosing structures, the model of the air conditioner, the structure of the fuzzy control system, the algorithm of its functioning, input and output variables of the fuzzy controller, the composition of their terms, membership functions, the formed complete database of rules. On the basis of research methods, the actual scientific and practical problem of developing an intelligent control system for the formation of the comfort microclimate of the building is solved.