Search

This paper is devoted to approximate methods of solving direct and inverse problems for parabolic equations. An approximate method to solve the initial problem of a multidimensional nonlinear parabolic equation has been proposed. It is based on reducing the initial problem to a nonlinear multidimensional Fredholm intergral equation of the second kind, which is approximated by a system of nonlinear algebraic equatiions using a method of mechanical quadratures. In constructing a computational scheme, the points of local splines have been applied for optimal with respect to order approximation of a functional class that contains the solutions of parabolic equations. For the numerical implementation of the computational scheme, we have used the generalization of a continuous method of solving nonlinear operator equations that is described in the paper. In addition, the inverse problem of a parabolic equation with a fractional order derivative with respect to a time variable has been studied. Approximate methods of determining the fractional order of the time derivative and a coefficient at a spatial derivative have been proposed. [ABSTRACT FROM AUTHOR]

When solving many physical and technical problems, a situation arises when only operators (functionals) from the objects under study (signals, images, etc.) are available for observations (measurements). It is required to restore the object from its known operator (functional). In many cases, the correlation (autocorrelation) function acts as an operator. A large number of papers have been devoted to the study of the existence of a solution to the problem of signal reconstruction from its autocorrelation function and the uniqueness of this solution. Since the solution to the problem of restoring a function from its autocorrelation function is not known in an analytical form, the problem of developing approximate methods arises. This is relevant not only in the problems of signal and image recovery, but also in solving the phase problem. From the above, the relevance of the problem of restoring a function (images) from the autocorrelation function follows. The article is devoted to approximate methods for solving this problem. Materials and methods. The construction and justification of the computing scheme is based on a continuous method for solving nonlinear operator equations, based on the theory of stability of solutions to ordinary differential equation systems. The method is stable under perturbations of the parameters of the mathematical model and, when solving nonlinear operator equations, does not require the reversibility of the Gateaux (or Freshet) derivatives of nonlinear operators. Results. In this work, an approximate method of signal reconstruction from its autocorrelation function and calculation of the phase of its spectrum from the reconstructed signal is constructed and substantiated. Conclusions. An approximate method for reconstructing a signal from its autocorrelation function and calculating the phase of its spectrum from the recovered signal is constructed and substantiated. The method does not require additional information about the signal under study. The results of the work can be used in solving a number of problems in optics, crystallography, and biology. [ABSTRACT FROM AUTHOR]

One of the developing branches of the modern fluid and gas mechanics is the mechanics of multiphase and multicomponent media. Experimental study of many processes of dynamics of inhomogeneous media is difficult; therefore, mathematical modeling is of great importance. Moreover, many models are essentially nonlinear; for this reason, numerical methods are used to integrate such models. The aim of this work is to obtain an exact solution for one of the special cases, assuming a number of simplifications (one-dimensionality of the flow, incompressibility of the carrier and dispersed components, and the linear nature of inter-component momentum exchange). At the same time, to obtain an exact solution, a model has been used, which implements the continual approach of the mechanics of multiphase media, i.e., taking into account the inter-component exchange of momentum and heat transfer. The paper presents a mathematical model of a one-dimensional unsteady flow of an incompressible two-component medium. The equations of dynamics are derived from the equations of the dynamics of the flow of a two-component inhomogeneous medium, taking into account the inter-component exchange of momentum and heat. In the model under consideration, the inter-component force interaction takes into account the Stokes force. The system of partial differential equations is reduced to a nonlinear system of ordinary differential equations due to the condition of incompressibility of the flow. The system of nonlinear differential equations is reduced to the sequential solution of three linear systems of ordinary differential equations with respect to six unknown functions. The analytical solution of the mathematical model of the gas–suspension flow is implemented in the form of a computer program. The exact solution for the continual model of aerosol dynamics can be used to test numerical models of aerosol dynamics. [ABSTRACT FROM AUTHOR]

Ambartsumian equation and its generalizations are some of the main integral equations of astrophysics, which have found wide application in many areas of physics and technology. An analytical solution to this equation is currently unknown, and the development of approximate methods is urgent. To solve the Ambartsumian equation, several iterative methods are proposed that are used in solving practical problems. Methods of collocations and mechanical quadratures have also been constructed and substantiated under rather severe conditions. It is of considerable interest to construct an iterative method adapted to the coefficients and kernels of the equation. This paper is devoted to the construction of such method. The construction of the iterative method is based on a continuous method for solving nonlinear operator equations. The method is based on the Lyapunov stability theory and is stable against perturbation of the initial conditions, coefficients, and kernels of the equations being solved. An additional advantage of the continuous method for solving nonlinear operator equations is that its implementation does not require the reversibility of the Gateaux derivative of the nonlinear operator. An iterative method for solving the Ambartsumian equation is constructed and substantiated. Model examples were solved to illustrate the effectiveness of the method. Equations generalizing the classical Ambartsumian equation are considered. To solve them, computational schemes of collocation and mechanical quadrature methods are constructed, which are implemented by a continuous method for solving nonlinear operator equations. [ABSTRACT FROM AUTHOR]

The work is devoted to the analysis of Lyapunov stability of Cohen–Grossberg neural networks with time-dependent delays. For this, the stability of steady solutions of systems of linear differential equations with time-dependent coefficients and time-dependent delays is analyzed. The cases of continuous and pulsed perturbations are considered. The relevance of the study is due to two circumstances. Firstly, Cohen–Grossberg neural networks find numerous applications in various fields of mathematics, physics, and technology, and it is necessary to determine the limits of their possible application in solving each specific problem. Secondly, the currently known conditions for the stability of the Cohen–Grossberg neural networks are rather cumbersome. The article is devoted to finding the conditions for the stability of the Cohen–Grossberg neural networks, expressed via the coefficients of the systems of differential equations simulating the networks. The analysis of stability is based on the method of "freezing" time-dependent coefficients and the subsequent analysis of the stability of the solution in a vicinity of the freezing point. The analysis of systems of differential equations thus transformed uses the properties of logarithmic norms. A method is proposed making it possible to obtain sufficient stability conditions for solutions of finite systems of nonlinear differential equations with time-dependent coefficients and delays. The algorithms are efficient both in the case of continuous and pulsed perturbations. The method proposed can be used in the study of nonstationary dynamical systems described by systems of ordinary nonlinear differential equations with time-dependent delays. The method can be used as the basis for studying the stability of Cohen–Grossberg neural networks with discontinuous coefficients and discontinuous activation functions. [ABSTRACT FROM AUTHOR]

The Ambartsumian equation, along with its generalizations, is one of the main integral equations of astrophysics, which have found wide application in many areas of physics and technology. The Ambartsumian equation plays an important role in the study of light scattering in media of infinite optical thickness. Nowadays the analytical solution of this equation is not known; therefore, the development of approximate methods for its solution is urgent. To solve the Ambartsumian equation, several iterative methods are proposed that are used in solving practical problems. Methods of collocations and mechanical quadratures have also been constructed, the substantiation of which has been carried out under rather severe conditions. In the previous work of the authors, a spline-collocation method for solving the Ambartsumian equation with zero-order splines was constructed and substantiated. The accuracy of this method is , where N is a number of collocation nodes. It is of considerable interest to construct an iterative method adapted to the smoothness of the coefficients and kernels of the equation. Light scattering in media of finite optical thickness is described by Ambartsumian equation systems, for an approximate solution of which it is necessary to construct and substantiate effective numerical methods. This study is devoted to the construction of such methods. The construction and substantiation of iterative methods for solving systems of Ambartsumian equations is based on a generalization of the continuous method for solving nonlinear operator equations. The method and its generalization are based on the Lyapunov stability theory and are stable against perturbation of the initial conditions, coefficients, and kernels of the equations being solved. An additional advantage of the continuous method for solving nonlinear operator equations is that its implementation does not require the reversibility of the Gateaux derivative of the nonlinear operator. In this work, spline-collocation methods with first-order splines are constructed for solving the Ambartsumian equation and systems of equations, and their justification is given. Model examples that illustrate the effectiveness of the methods are solved. Equations generalizing the classical Ambartsumian equations are considered. To solve them, the computational schemes of spline-collocation methods are constructed and substantiated. The results obtained can be used to solve a number of astrophysical issues. [ABSTRACT FROM AUTHOR]

It is shown that any kernel of the collision integral of the nonlinear Boltzmann equation isotropic in velocities can be expressed in terms of a universal protokernel independent of the model of interaction between particles by single integration with the interaction cross section. This makes it possible to calculate any kernels of an anisotropic Boltzmann equation and use for their constructing the recurrent relations obtained earlier at the protokernel level. We consider three different expressions for the protokernel, including that for an arbitrary ratio of the masses of colliding particles. [ABSTRACT FROM AUTHOR]