Abstract
A solution of the Cauchy problem is obtained for one degenerate equation with the Dzhrbashyan–Nersesyan fractional derivative, particular solutions of which are represented using the Kilbas–Saigo function.
A solution of the Cauchy problem is obtained for one degenerate equation with the Dzhrbashyan–Nersesyan fractional derivative, particular solutions of which are represented using the Kilbas–Saigo function.
A necessary and sufficient condition is established for the closedness of the range or surjectivity of a differential operator acting on smooth sections of vector bundles. For connected noncompact manifolds it is shown that these conditions are derived from the regularity conditions and the unique continuation property of solutions. An application of these results to elliptic operators (more precisely, to operators with a surjective principal symbol) with analytic coefficients, to second-order elliptic operators on line bundles with a real leading part, and to the Hodge–Laplace–de Rham operator is given. It is shown that the top de Rham (respectively, Dolbeault) cohomology group on a connected noncompact smooth (respectively, complex-analytic) manifold vanishes. For elliptic operators, we prove that solvability in smooth sections implies solvability in generalized sections.
For a nonlinear differential matrix equation, we study a multipoint boundary value problem by a constructive method of regularization over the linear part of the equation using the corresponding fundamental matrices. Based on the initial data of the problem, sufficient conditions for its unique solvability are obtained. Iterative algorithms containing relatively simple computational procedures are proposed for constructing a solution. Effective estimates are given that characterize the rate of convergence of the iteration sequence to the solution, as well as estimates of the solution localization domain.
The paper discusses the features of constructing numerical schemes for solving coefficient inverse problems for nonlinear partial differential equations of the reaction–diffusion–advection type with data of various types. As input data for the inverse problem, we consider (1) data at the final moment of time, (2) data at the spatial boundary of a domain, (3) data at the position of the reaction front. To solve the inverse problem in all formulations, the gradient method of minimizing the target functional is used. In this case, when constructing numerical minimization schemes, both an approach based on discretization of the analytical expression for the gradient of the functional and an approach based on differentiating the discrete approximation of the functional to be minimized are considered. Features of the practical implementation of these approaches are demonstrated by the example of solving the inverse problem of reconstructing the linear gain coefficient in a nonlinear Burgers-type equation.
We consider initial–boundary value problems for homogeneous parabolic systems with coefficients satisfying the double Dini condition with zero initial conditions in a semibounded plane domain with nonsmooth lateral boundary. The method of boundary integral equations is used to prove a theorem on the unique classical solvability of such problems in the space of functions that are continuous together with their first spatial derivative in the closure of the domain. An integral representation of the obtained solutions is given. It is shown that the condition for the solvability of the posed problems considered in the paper is equivalent to the well-known complementarity condition.
We study the feedback control problem for a mathematical model that describes the motion of a viscoelastic fluid with memory along the trajectories of the velocity field. We prove the existence of an optimal control that delivers a minimum to a given bounded and lower semicontinuous cost functional.
In physics, the singular heat equation with the Bessel operator is used to explain the basic process of heat transport in a substance with spherical or cylinder symmetry. This paper examines the solution of the Cauchy problem for the heat equation with the Bessel operator acting in the space variable. We obtain some properties of the solution and consider the normalized modified Bessel function of the first kind.
A new methodology for solving inverse dynamics problem is developed. The methodology is based on using a mathematical model of a dynamical system and robust stabilization methods for a system under uncertainty.
Most exhaustively the theory is described for linear finite-dimensional time-invariant scalar systems and multiple-input multiple-output systems.
The study shows that with this approach, the zero dynamics of the original system is of crucial significance. This dynamics, if exists, is assumed to be exponentially stable.
It is established that zero-dynamics, relative order, and the corresponding equations of motion cannot be defined correctly in multiple-input multiple-output systems. For correct inverse transformation of the solution of the problem, additional assumptions have to be introduced, which generally limits the inverse system category.
Special attention is given to the synthesis of elementary (minimal) inverters, i.e., least-order dynamical systems that solve the transformation problem.
It is also established that the inversion methods sustain the efficiency with finite parameter variations in the initial system as well as with uncontrolled exogenous impulses having no impact on the system’s internal dynamics.
We consider a linear-convex control system defined by a set of differential equations with continuous matrix coefficients. The system may have control parameters, as well as uncertainties (interference) the possible values of which are subject to strict pointwise constraints. For this system, over a finite period of time, taking into account the constraints, we study the problem of guaranteed hitting the target set from a given initial position despite the effect of uncertainty. The main stage of solving the problem is the construction of an alternating integral and a solvability set. To construct the latter, the greatest computational complexity is the calculation of the geometric difference between the target set and the set determined by the uncertainty. A two-dimensional example of this problem is considered for which a method is proposed for finding the solvability set without the need to calculate the convex hull of the difference of the support functions of the sets.
We study the stability of a modified (with variation in the nonlinearity parameter) “super-twisting” algorithm. The analysis is based on majorizing the trajectories of the system with an arbitrary nonlinearity parameter by the trajectories of systems of the classical “super-twisting” algorithm. Stability conditions for the modified systems are obtained, as well as estimates for the size of the stability domain depending on system parameters.
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