Short abstract: In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian differential-algebraic boundary value problem for a matrix Lyapunov equation.
Extended abstract: The study of differential-algebraic boundary value problems was initiated in the works of K. Weierstrass, N. N. Luzin and F. R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the work of S. Campbell, Yu. E. Boyarintsev, V. F. Chistyakov, A. M. Samoilenko, M. O. Perestyuk, V. P. Yakovets, O. A. Boichuk, A. Ilchmann and T. Reis. The study of the differential-algebraic boundary value problems is associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, theory of control, theory of motion stability. At the same time, the study of differential algebraic boundary value problems is closely related to the study of boundary value problems for difference equations, initiated in A. A. Markov, S. N. Bernstein, Ya. S. Besikovich, A. O. Gelfond, S. L. Sobolev, V. S. Ryaben'kii, V. B. Demidovich, A. Halanay, G. I. Marchuk, A. A. Samarskii, Yu. A. Mitropolsky, D. I. Martynyuk, G. M. Vayniko, A. M. Samoilenko, O. A. Boichuk and O. M. Standzhitsky. Study of nonlinear singularly perturbed boundary value problems for difference equations in partial differences is devoted to the work of V. P. Anosov, L. S. Frank, P. E. Sobolevskii, A. L. Skubachevskii and A. Asheraliev. Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A. M. Samoilenko and O. A. Boichuk on linear boundary value problems for difference-algebraic equations, in particular finding the necessary and sufficient conditions for the existence of the desired solutions, and also the construction of the Green's operator of the Cauchy problem and the generalized Green operator of a linear boundary value problem for a difference-algebraic equation. Thus, the actual problem is the transfer of the results obtained in the articles and monographs of S. Campbell, A. M. Samoilenko and O. A. Boichuk on the linear boundary value problems for the differential-algebraic boundary value problem for a matrix Lyapunov equation, in particular, finding the necessary and sufficient conditions of the existence of the desired solutions of the linear differential-algebraic boundary value problem for a matrix Lyapunov equation. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian differential-algebraic boundary value problem for a matrix Lyapunov equation. The proposed scheme of the research of the linear differential-algebraic boundary value problem for a matrix Lyapunov equation in the critical case in this article can be transferred to the seminonlinear differential-algebraic boundary value problem for a matrix Lyapunov equation.
Keywords: differential-algebraic equation; boundary value problem; matrix Lyapunov equation.
2010 Mathematics Subject Classification: 15A24; 34В15; 34C25. [ Full-text available (PDF) ] Top of the page.
Short abstract: A numerical-analytical algorithm for investigation of the electrostatic potential of a sphere with a circular hole and а charge surrounded by ribbon spheres is constructed. The method of inversion of the integral operator and semi-inversion of the matrix operator of the Dirichlet problem for the Laplace equation is used. A system of the second kind with a compact operator in space $\ell_2$ is obtained. Particular variants of the problem are considered.
Extended abstract: A numerical-analytical algorithm for investigation of the potential of a sphere with a circular hole, surrounded by external and internal closed ribbon spheres, is constructed. The number of ribbons on the spheres is arbitrary. The ribbons on the spheres are separated by non-conductive, infinitely thin partitions. The partitions are located in planes parallel to the shear plane of the sphere with a hole. Each ribbon has its own independent potential. An electrostatic charge is placed between the outer sphere and the sphere with a hole in the axis of the structure. The full potential must satisfy, in particular, Maxwell's equations, taking into account the absence of magnetic fields, satisfy the boundary conditions, have the required singularity at the point where the charge is placed. To solve this problem, we first used the method of partial domains and the method of separating variables in a spherical coordinate system. In this case, for the Fourier series, we use power functions and Legendre polynomials of integer orders. From the boundary conditions, using an auxiliary system of 3 equations with 4 unknowns, a pairwise system of functional equations of the first kind with respect to the coefficients of the Fourier series is obtained. The system is not effective for solving by direct methods. The method of inversion of the Volterra integral operator and semi-inversion of the matrix operators of the Dirichlet problem for the Laplace equation are applied. The method is based on the ideas of the analytical method of the Riemann - Hilbert problem. In this case, integral representations for the Legendre polynomials are used. A system of linear algebraic equations of the second kind with a compact matrix operator in the Hilbert space $\ell_2$ is obtained. The system is effectively solvable numerically for arbitrary parameters of the problem and analytically for the limiting parameters of the problem. Particular variants of the problem are considered.
Keywords: spheres; hole; electrostatics; linear system of the second kind; compact.
2010 Mathematics Subject Classification: 65N12; 35A25; 78A45. [ Full-text available (PDF) ] Top of the page.
Short abstract: It is proposed the generalized numerical based approaches to optimal control the heat conduction processes based on solving the especially built ordinary differential equations. The example of using the proposed approaches is discussed and it is shown that using these approaches can allow to decrease almost twice the transient time required for heating-up structures.
Extended abstract: It is proposed the generalized mathematical formulation of the problem about the optimal control for the heat conduction processes representing by the partial differential equation. The proposed formulation not includes the necessary clarifications about the conditions which must be satisfied by the current and required temperature fields. But, during the generalized solving of the formulated problem, it is established that the current and required temperature fields must be agreed with the mathematical model of the heat conduction so that to have possibilities to provide uniquely these temperature fields by means the control vector. To solve the problem about the optimal control for the heat conduction processes it is developed the numerical approaches based on reducing to the especially built ordinary differential equations and minimization problem. This reducing is based on discretisation the heat conduction by using the grid method and on defining the unknown control vector as the numerical solution of the especially built Cauchy problem. To satisfy the all limitations it is proposed to build the permissible velocity of the unknown control vector considering with the requirements of necessary switching in some moments of the time. The particular example of using the proposed generalized approaches is considered to illustrate their application technique. It is shown that the proposed generalized mathematical formulation is fully corresponded with the considered particular example. In this considered particular example, the resolving Cauchy problem can be built and the switching time can be found in the depending on the grid node choosing. It is shown that the transient time can be decrease almost twice due to optimizing the control in the particular example at least. All these results will allow giving the clear representation of the proposed approaches and the technique of their using to solve the engineering problems about the optimal control of the heat conduction processes in different industrial systems.
Keywords: control; modeling; optimization; methods; heat conduction.
2010 Mathematics Subject Classification: 49Mxx; 80Axx. [ Full-text available (PDF) ] Top of the page.
Short abstract: A quasilinear system of three differential equations of hyperbolic type, which describes the settling of aggregating particles of a suspension placed in a thin long tube in a field of external forces inhomogeneous along the tube, is studied. The system of equations for mass and volume concentrations and the average size of aggregates in a one-dimensional formulation allows discontinuous solutions. The characteristics of the system correspond to the surfaces of discontinuities, of which the outer one describes the subsidence rate, and the inner one can have a different structure from a simple concentration jump to a jump accompanied by a rarefaction wave and a fan of characteristics at the moving boundary. A detailed study of the conditions for the existence of different types of solutions is carried out. The application of the results for different applied problems is discussed.
Extended abstract: The mathematical model of the sedimentation process of suspension particles is usually a quasilinear hyperbolic system of partial differential equations, supplemented by initial and boundary conditions. In this work, we study a complex model that takes into account the aggregation of particles and the inhomogeneity of the field of external mass forces. The case of homogeneous initial conditions is considered, when all the parameters of the arising motion depend on only one spatial Cartesian coordinate $x$ and on time $t$. In contrast to the known formulations for quasilinear systems of equations (for example, as in gas dynamics), the solutions of which contain discontinuities, in the studied formulation the basic system of equations occurs only on one side of the discontinuity line in the plane of variables $(t; x)$. On the opposite side of the discontinuity surface, the equations have a different form in general. We will restrict ourselves to considering the case when there is no motion in a compact zone occupied by settled particles, i.e. all velocities are equal to zero and the volumetric contents of all phases do not change over time. The problem of erythrocyte sedimentation in the field of centrifugal forces in a centrifuge, with its uniform rotation with angular velocity $\omega=const$ is considered. We have studied the conditions for the existence of various types of solutions. One of the main problems is the evolution (stability) problem of the emerging discontinuities. The solution of this problem is related to the analysis of the relationships for the characteristic velocities and the velocity of the discontinuity surface. The answer depends on the number of characteristics that come to the jump, and the number of additional conditions set on the interface. The discontinuity at the lower boundary of the area occupied by pure plasma is always stable. But for the surface separating the zones of settled and of moving particles, the condition of evolution may be violated. In this case, it is necessary to adjust the original mathematical model.
Keywords: differential equations; hyperbolic systems; characteristics; sedimentation; aggregation.
2010 Mathematics Subject Classification: 35L57; 76T20. [ Full-text available (PDF) ] Top of the page.
2010 Mathematics Subject Classification: 01A70. [ Full-text available (PDF) ] Top of the page.
2010 Mathematics Subject Classification: 01A70. [ Full-text available (PDF) ] Top of the page.