Some solvability problems for the Boltzmann equation in the framework of the Shakhov model

We consider the nonlinear Boltzmann equation in the framework of the Shakhov model for the classical problem of gas flow in a plane layer. The problem reduces to a system of nonlinear integral equations. The nonlinearity of the studied system can be partially simplified by passing to a new argument depending on the solution of the problem itself. We prove the existence theorem for a unique solution of the linear system and the existence theorem for a positive solution of the nonlinear Urysohn equation. We determine the temperature jumps on the lower and upper walls in the linear and nonlinear cases, and it turns out that the difference between them is rather small.     

A nonlinear boundary eigenvalue problem for TM-polarized electromagnetic waves in a nonlinear layer

We consider the propagation of TM-polarized electromagnetic waves in a nonlinear dielectric layer located between two linear media. The nonlinearity in the layer is described by the Kerr law. We reduce the problem to a nonlinear boundary eigenvalue problem for a system of ordinary differential equations. We obtain a dispersion relation and a first approximation for eigenvalues of the problem. We compare the results with those obtained for the case of a linear medium in the layer.     

Optimization for nonlinear hyperbolic equations without the uniqueness theorem for a solution of the boundary-value problem

We consider the optimal control problem for a system governed by a nonlinear hyperbolic equation without any constraints on the parameter of nonlinearity. No uniqueness theorem is established for a solution to this problem. The control-state mapping of this system is not Gateaux differentiable. We study an approximate solution of the optimal control problem by means of the penalty method.     

A class of optimal state-delay control problems

We consider a general nonlinear time-delay system with state-delays as control variables. The problem of determining optimal values for the state-delays to minimize overall system cost is a non-standard optimal control problem–called an optimal state-delay control problem–that cannot be solved using existing optimal control techniques. We show that this optimal control problem can be formulated as a nonlinear programming problem in which the cost function is an implicit function of the decision variables. We then develop an efficient numerical method for determining the cost function’s gradient. This method, which involves integrating an auxiliary impulsive system backwards in time, can be combined with any standard gradient-based optimization method to solve the optimal state-delay control problem effectively. We conclude the paper by discussing applications of our approach to parameter identification and delayed feedback control.     

The Inverse Problem Method on the Half-Line and a Nested System of Riccati Equations

A mixed problem for the nonlinear Bogoyavlenskii system on the half-line is studied by the inverse problem method. The solution of the mixed problem is reduced to the solution of the inverse spectral problem of recovering a forth-order differential operator on the half-line from the Weyl matrix. We derive evolution equations for the elements of the Weyl matrix and give an algorithm for the solution of the mixed problem. Evolution equations of the elements of the Weyl matrix are nonlinear. It is shown that they can be reduced to a nested system of three successively solvable matrix Riccati equations.     

Global existence of solutions to a coupled coagulation system

We consider a model equations describing the coagulation process of a gas on a surface. The problem is modeled by two coupled equations. The first one is a nonlinear transport equation with bilinear coagulation operator while the second one is a nonlinear ordinary differential equation. The velocity and the boundary condition of the transport equation depend on the supersaturation function satisfying the nonlinear ode. We first prove global existence and uniqueness of solution to the nonlinear transport equation then, we consider the coupled problem and prove existence in the large of solutions to the full coagulation system.     

微分变换法求解二维非线性Volterra积分微分方程

为了解决二维非线性Volterra积分微分方程的求解问题,本文给出微分变换法.利用该方法将方程中的微分部分和积分部分进行变换,这样简化了原方程,进而得到非线性代数方程组,从而将原问题转换为求解非线性代数方程组的解,使得计算更简便.文中最后数值算例说明了该方法的可行性和有效性.     

Differential variational–hemivariational inequalities with application to contact mechanics

In this paper we study a dynamical system which consists of the Cauchy problem for a nonlinear evolution equation of first order coupled with a nonlinear time-dependent variational–hemivariational inequality with constraint in Banach spaces. The evolution equation is considered in the framework of evolution triple of spaces, and the inequality which involves both the convex and nonconvex potentials. We prove existence of solution by the Kakutani–Ky Fan fixed point theorem combined with the Minty formulation and the theory of hemivariational inequalities. We illustrate our findings by examining a nonlinear quasistatic elastic frictional contact problem for which we provide a result on existence of weak solution.     

Well-posedness of a kind of nonlinear coupled system of fractional differential equations

We study the boundary value problem of a coupled differential system of fractional order, and prove the existence and uniqueness of solutions to the considered problem. The underlying differential system is featured by a fractional differential operator, which is defined in the Riemann-Liouville sense, and a nonlinear term in which different solution components are coupled. The analysis is based on the reduction of the given system to an equivalent system of integral equations. By means of the nonlinear alternative of Leray-Schauder, the existence of solutions of the factional differential system is obtained. The uniqueness is established by using the Banach contraction principle.     

Erratum to: Integration of the Continual System of Nonlinear Interaction Waves

We derive the continual system of nonlinear interaction waves from the compatibility of the transport equation on the whole line and the equation governing the time-evolution of the eigenfunctions of the transport operator. The transport equation represents the continual generalization from the n-component system of first-order ordinary differential equations. The continual system describes a nonlinear interaction of waves. We prove that the continual system can be integrated by the inverse scattering method. The method is based on applying the results of the inverse scattering problem for the transport equation to finding the solution of the Cauchy initial-value problem for the continual system. Indeed, the transition operator for the scattering problem admits right and left Volterra factorizations. The intermediate operator for this problem determines the one-to-one correspondence between the preimages of a solution of the transport equation. This operator is related to the transition operator and admits not only right and left Volterra factorizations but also the analytic factorization. By virtue of this fact the potential in the transport equation is uniquely reconstructed in terms of the solutions of the fundamental equations in inverse problem.     

Decoupling problems around a trajectory: from linearity to nonlinearity

Given a nonlinear control system for which an admissible statetrajectory is specified, we solve approximately the input outputdecoupling problem around this nominal trajectory. An approximatesolution for this problem is obtained by dealing with the linearizedsystem along this trajectory. An exact solution to the inputoutput decoupling problem for the linearization is shown tobe an approximate solution to the input output decoupling problemaround the nominal trajectory for the original nonlinear system.In a similar way, we provide an approximate solution to thedisturbance decoupling problem around a specified trajectoryof the nonlinear system. The nonlinear model of a two link robotmanipulator is used to illustrate the results on input outputdecoupling.     

Finite-Dimensional Reductions of Conservative Dynamical Systems and Numerical Analysis. I

We study infinite-dimensional Liouville–Lax integrable nonlinear dynamical systems. For these systems, we consider the problem of finding an appropriate set of initial conditions leading to typical solutions such as solitons and traveling waves. We develop an approach to the solution of this problem based on the exact reduction of a given nonlinear dynamical system to its finite-dimensional invariant submanifolds and the subsequent investigation of the system of ordinary differential equations obtained by qualitative analysis. The efficiency of the approach proposed is demonstrated by the examples of the Korteweg–de Vries equation, the modified nonlinear Schrödinger equation, and a hydrodynamic model.     

Numerical analysis to discontinuous Galerkin methods for the age structured population model of marine invertebrates

In this article we consider the age structured population growth model of marine invertebrates. The problem is a nonlinear coupled system of the age‐density distribution of sessile adults and the abundance of larvae. We propose the semidiscrete and fully‐discrete discontinuous Galerkin schemes to the nonlinear problem. The DG method is well suited to approximate the local behavior of the problem and to easily take the locally refined meshes with hanging nodes adaptively. The simple communication pattern between elements makes the DG method ideal for parallel computation. The global existence of the approximation solution is proved for the nonlinear approximation system by using the broken Sobolev spaces and the Schauder's fixed point theorem, and error estimates are obtained for both the semidiscrete scheme and the fully‐discrete scheme. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Nonlinear spectral problem for a Hamiltonian system of differential equations with redundant conditions

We consider a nonlinear spectral problem for a self-adjoint Hamiltonian system of differential equations. The boundary conditions correspond to a self-adjoint problem. It is assumed that the input data (the matrix of the system and the matrices of the boundary conditions) satisfy certain conditions of monotonicity with respect to the spectral parameter. In addition to the main boundary conditions, a redundant nonlocal condition given by a Stieltjes integral is imposed on the solution. For the nontrivial solvability of the over-determined problem thus obtained, the original problem is replaced by an auxiliary problem that is consistent with the entire set of conditions. This auxiliary problem is obtained from the original one by allowing a discrepancy of a specific form. We study the resulting problem and give a numerical method for its solution.     

Coupling nonlinear Stokes and Darcy flow using mortar finite elements

We study a system composed of a nonlinear Stokes flow in one subdomain coupled with a nonlinear porous medium flow in another subdomain. Special attention is paid to the mathematical consequence of the shear-dependent fluid viscosity for the Stokes flow and the velocity-dependent effective viscosity for the Darcy flow. Motivated by the physical setting, we consider the case where only flow rates are specified on the inflow and outflow boundaries in both subdomains. We recast the coupled Stokes–Darcy system as a reduced matching problem on the interface using a mortar space approach. We prove a number of properties of the nonlinear interface operator associated with the reduced problem, which directly yield the existence, uniqueness and regularity of a variational solution to the system. We further propose and analyze a numerical algorithm based on mortar finite elements for the interface problem and conforming finite elements for the subdomain problems. Optimal a priori error estimates are established for the interface and subdomain problems, and a number of compatibility conditions for the finite element spaces used are discussed. Numerical simulations are presented to illustrate the algorithm and to compare two treatments of the defective boundary conditions.     

A numerical study of parabolic problems with nonlinear boundary conditions

In this paper, we consider an initial‐boundary value problem for a parabolic equation with nonlinear boundary conditions. The solution to the problem can be expressed as a convolution integral of a Green's function and two unknown functions. We change the problem to a system of two nonlinear Volterra integral equations of convolution type. By using an explicit procedure on the basis of Sinc‐function properties, the resulting integral equations are replaced by a system of nonlinear algebraic equations, whose solution yields an accurate approximate solution to the parabolic problem. Some examples are considered to illustrate the ability of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Asymptotic solution method for a quasilinear minimum force control problem

For a transient process in a quasilinear system, we consider an optimization problem of finding a (multi-dimensional) control with minimum intensity. We suggest an algorithm for constructing asymptotic approximations to the solution of this problem. The main advantage of the algorithm is that an optimal control problem for a linear system is solved instead of the original essentially nonlinear problem.     

Representation of systems disturbed by wide band noise

In this paper, a method of handling and working with wide band noise is developed. We represent wide band noise as a distributed delay of white noise and use it to reduce a nonlinear system disturbed by wide band noise to a nonlinear system disturbed by white noise. An application of this reduction to a nonlinear filtering problem under a wide band noise disturbance is discussed.     

Finding multiple roots of a box-constrained system of nonlinear equations with a biased random-key genetic algorithm

Several numerical methods for solving nonlinear systems of equations assume that derivative information is available. Furthermore, these approaches usually do not consider the problem of finding all solutions to a nonlinear system. Rather, most methods output a single solution. In this paper, we address the problem of finding all roots of a system of equations. Our method makes use of a biased random-key genetic algorithm (BRKGA). Given a nonlinear system, we construct a corresponding optimization problem, which we solve multiple times, making use of a BRKGA, with areas of repulsion around roots that have already been found. The heuristic makes no use of derivative information. We illustrate the approach on seven nonlinear equations systems with multiple roots from the literature.     

Existence and uniqueness of solutions to singular Cahn–Hilliard equations with nonlinear viscosity terms and dynamic boundary conditions

We prove global existence and uniqueness of solutions to a Cahn–Hilliard system with nonlinear viscosity terms and nonlinear dynamic boundary conditions. The problem is highly nonlinear, characterized by four nonlinearities and two separate diffusive terms, all acting in the interior of the domain or on its boundary. Through a suitable approximation of the problem based on abstract theory of doubly nonlinear evolution equations, existence and uniqueness of solutions are proved using compactness and monotonicity arguments. The asymptotic behaviour of the solutions as the diffusion operator on the boundary vanishes is also shown.