Mathematical modeling of the ice–water phase transition during liquid flow inside a pipe with a small ice buildup on the wall at high Reynolds numbers is considered. As a mathematical model describing the dynamics of the phase transition, a double-deck boundary layer model and a phase field system are used. Results of numerical simulation are presented.
This article is considering the stability property of the solution with inner layer for singularly perturbed stationary equation with Neumann boundary conditions. The right-hand side is assumed to have discontinuity on some arbitrary curve (h(t)). Stability analysis is performed by obtaining the first non-zero coefficient of the series for eigenvalue and eigenfunction from the Sturm–Liouville problem. Theory of the asymptotic approximations is used in order to construct them.
The design of fault-tolerant production and delivery systems is one of the priority areas in modern operations research. The traditional approach to modeling such systems is based on the use of stochastic models that describe the choice of a possible scenario of actions in the event of problems in a production or transportation network. Along with a number of advantages, this approach has a known drawback. The occurrence of problems of an unknown nature that can jeopardize the performance of the entire simulated system significantly complicates its use. This paper introduces the minimax problem of constructing fault-tolerant production plans (reliable production process design problem, RPPDP), the purpose of which is to ensure the uninterrupted operation of a distributed production system with minimal guaranteed cost. It is shown that the RPPDP is NP-hard in the strong sense and remains intractable under quite specific conditions. To find exact and approximate solutions with accuracy guarantees for this problem, branch-and-bound methods are developed based on the proposed compact model of the mixed integer linear program (MILP) and novel heuristic of adaptive search in large neighborhoods (adaptive large neighborhood search, ALNS) as part of extensions of the well-known Gurobi MIP solver. The high performance and complementarity of the proposed algorithms is confirmed by the results of numerical experiments carried out on a public library of benchmarking instances developed by the authors based on instances from the PCGTSPLIB library.
We examine the existence of solutions to the Sturm–Liouville problem with a non-self-adjoint differential operator and discontinuous nonlinearity in the phase variable. For positive values of the spectral parameter, theorems on the existence of nontrivial (positive and negative) solutions of the problem are proved. Examples illustrating the theorems are given.
The functional of eigenvalues on the manifold of periodic potentials is described analytically and topologically.
Initial-boundary value problems are considered for homogeneous parabolic systems with Dini-continuous coefficients and zero initial conditions in a semibounded plane domain with a nonsmooth lateral boundary admitting cusps, on which general boundary conditions with variable coefficients are given. A theorem on unique classical solvability of these problems in the space of functions that are continuous and bounded together with their first spatial derivatives in the closure of the domain is proved by applying the boundary integral equation method. A representation of the resulting solutions in the form of vector single-layer potentials is given.
The problem of controllability for problems of optimal control and optimization of distributed parameter systems governed by partial differential equations is considered. The concept of controllability understood as Tikhonov correctness for solving optimization problems is introduced. A theorem formulating controllability conditions for directly solving optimization problems (direct minimization of the objective functional) is presented. A test example of the numerical solution of the optimization problem for a nonlinear hyperbolic system describing the unsteady flow of water in an open channel is considered. The analysis of controllability is demonstrated that ensures the correctness of the problem solution and high accuracy of optimization of the distributed friction coefficient in the flow equations.
The influence of certain parameters on the stability conditions of the reaction front in a porous medium is studied in this article. The mathematical model includes the heat equation, the concentration equation and the equations of motion under the Boussinesq–Darcy approximation. An asymptotic analysis was carried out using the method of Zeldovich and Frank-Kamentskii to obtain the interface problem. A stability analysis was performed to determine a linearized problem that will be solved numerically using the finite difference method with an implicit scheme. This will allow to conclude the effect of each parameter on the stability of the front, in particular the amplitude and the frequency of the vibrations.
The main aim of this paper is to introduce generalized quaternions with hyper-number coefficients. For this, firstly, a new number system is defined, which is the generalization of bicomplex numbers, hyper-double numbers and hyper-dual numbers. And any element of this generalization is called a hyper-number. Then, real matrix representation and vector representation of a hyper-number are given. Secondly, hyper-number generalized quaternions and their algebraic properties are introduced. For a hyper-number generalized quaternion, (4 times 4) real generalized quaternion matrix representation is presented. Next, because of lack of commutativity, for a hyper-number generalized quaternion, two different hyper-number matrix representations are calculated. Moreover, real matrix representations of a hyper-number generalized quaternion is expressed by matrix representation of a hyper-number. Finally, vector representations of a hyper-number generalized quaternion are given and properties of this representations are investigated.
We consider an optimal distributed control problem in a strictly convex planar domain with a smooth boundary and a small parameter multiplying a highest derivative of an elliptic operator. A zero Dirichlet condition is set on the boundary of the domain, and control is additively involved in the inhomogeneity. The set of admissible controls is the unit ball in the corresponding space of square integrable functions. The solutions of the obtained boundary value problems are considered in the generalized sense as elements of a Hilbert space. The optimality criterion is the sum of the squared norm of the deviation of the state from a given state and the squared norm of the control with some coefficient. Due to this structure of the optimality criterion, the role of the first or second term of the criterion can be strengthen, if necessary. It is more important to achieve a given state in the first case and to minimize the resource cost in the second case. The asymptotics of the problem generated by the sum of a second-order differential operator with a small coefficient at a highest derivative and a zero-order differential operator is studied in detail.
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