Numerical method of solution to loaded nonlocal boundary value problems for ordinary differential equations

A numerical method is suggested for solving systems of nonautonomous loaded linear ordinary differential equations with nonseparated multipoint and integral conditions. The method is based on the convolution of integral conditions into local ones. As a result, the original problem is reduced to an initial value (Cauchy) problem for systems of ordinary differential equations and linear algebraic equations. The approach proposed is used in combination with the linearization method to solve systems of loaded nonlinear ordinary differential equations with nonlocal conditions. An example of a loaded parabolic equation with nonlocal initial and boundary conditions is used to show that the approach can be applied to partial differential equations. Numerous numerical experiments on test problems were performed with the use of the numerical formulas and schemes proposed.     

Differential variational inequalities

This paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash games, and hybrid engineering systems with variable structures. The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic complementarity systems, and evolutionary variational inequalities. Conditions are presented under which the DVI can be converted, either locally or globally, to an equivalent ODE with a Lipschitz continuous right-hand function. For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensional variational inequalities. Borrowing results from differential inclusions (DIs) with upper semicontinuous, closed and convex valued multifunctions, we establish the convergence of such a procedure for solving initial-value DVIs. We also present a class of DVIs for which the theory of DIs is not directly applicable, and yet similar convergence can be established. Finally, we extend the method to a boundary-value DVI and provide conditions for the convergence of the method. The results in this paper pertain exclusively to systems with “index” not exceeding two and which have absolutely continuous solutions. The work of J.-S. Pang is supported by the National Science Foundation under grants CCR-0098013 CCR-0353074, and DMS-0508986, by a Focused Research Group Grant DMS-0139715 to the Johns Hopkins University and DMS-0353016 to Rensselaer Polytechnic Institute, and by the Office of Naval Research under grant N00014-02-1-0286. The work of D. E. Stewart is supported by the National Science Foundation under a Focused Research Group grant DMS-0138708.     

Nontrivial solutions for impulsive fractional differential systems through variational methods

Fractional differential equations (FDEs) as a generalization of ordinary differential equations and integration to arbitrary noninteger orders have gained importance due to their numerous applications in many fields of science and engineering. Indeed, there are a large number of phenomena, including fluid flow, diffusive transport akin to diffusion, rheology, probability, and electrical networks, that are modeled by different equations involving fractional order derivatives. This paper deals with multiplicity results of solutions for a class of impulsive fractional differential systems. The approach is based on variational methods and critical point theory. Indeed, we establish existence results for our system under some algebraic conditions on the nonlinear part with the classical Ambrosetti–Rabinowitz (AR) condition on the nonlinear and the impulsive terms. Moreover by combining two algebraic conditions on the nonlinear term, which guarantee the existence of two weak solutions, applying the mountain pass theorem, we establish the existence of third weak solution for our system.     

Complete group classification of systems of two linear second‐order ordinary differential equations: the algebraic approach

We give a complete group classification of the general case of linear systems of two second‐order ordinary differential equations. The algebraic approach is used to solve the group classification problem for this class of equations. This completes the results in the literature on the group classification of two linear second‐order ordinary differential equations including recent results which give a complete group classification treatment of such systems. We show that using the algebraic approach leads to the study of a variety of cases in addition to those already obtained in the literature. We illustrate that this approach can be used as a useful tool in the group classification of this class of equations. A discussion of the subsequent cases and results is given. Copyright © 2014 John Wiley & Sons, Ltd.     

On the numerical solution of loaded systems of ordinary differential equations with nonseparated multipoint and integral conditions

We propose a numerical method of solving systems of loaded linear nonautonomous ordinary differential equations with nonseparated multipoint and integral conditions. This method is based on the convolution of integral conditions to obtain local conditions. This approach allows one to reduce solving the original problem to solving a Cauchy problem for a system of ordinary differential equations and linear algebraic equations. Numerous computational experiments on several test problems with the formulas and schemes proposed for the numerical solution have been carried out. The results of the experiments show that the approach is reasonably efficient.     

On the solution of boundary value problems with nonseparated multipoint and integral conditions

We suggest a numerical method for solving systems of linear nonautonomous ordinary differential equations with nonseparated multipoint and integral conditions. By using this method, which is based on the operation of convolution of integral conditions into local ones, one can reduce the solution of the original problem to the solution of a Cauchy problem for systems of ordinary differential equations and linear algebraic equations. We establish bounded linear growth of the error of the suggested numerical schemes. Numerical experiments were carried out for specially constructed test problems.     

Solving some problems in the theory of thin shells of revolution with complex boundary conditions

An approach is proposed to solving linear boundary-value problems for shells of revolution that are closed in the circumferential direction, with complex boundary conditions in which the coefficients of the solving functions depend on the circumferential coordinate. The approach relies on reduction of the boundary-value problem to a number of boundary-value problems for systems of ordinary differential equations and systems of algebraic equations. We solve a specific problem for the stressed state of a conical shell with one of its ends supported by an elastic foundation with a variable modulus.Institute of Mechanics, Ukrainian Academy of Sciences. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 85–93, 1989.     

A dynamical systems approach to constrained minimization

We present an ordinary differential equations approach to solve general smooth minimization problems including a convergence analysis. Generically often the procedure ends up at a point which fulfills sufficient conditions for a local minimum. This procedure will then be rewritten in the concept of differential algebraic equations which opens the route to an efficient implementation. Furthermore, we link this approach with the classical SQP-approach and apply both techniques onto two examples relevant in applications.     

Dynamical systems and variational inequalities

The variational inequality problem has been utilized to formulate and study a plethora of competitive equilibrium problems in different disciplines, ranging from oligopolistic market equilibrium problems to traffic network equilibrium problems. In this paper we consider for a given variational inequality a naturally related ordinary differential equation. The ordinary differential equations that arise are nonstandard because of discontinuities that appear in the dynamics. These discontinuities are due to the constraints associated with the feasible region of the variational inequality problem. The goals of the paper are two-fold. The first goal is to demonstrate that although non-standard, many of the important quantitative and qualitative properties of ordinary differential equations that hold under the standard conditions, such as Lipschitz continuity type conditions, apply here as well. This is important from the point of view of modeling, since it suggests (at least under some appropriate conditions) that these ordinary differential equations may serve as dynamical models. The second goal is to prove convergence for a class of numerical schemes designed to approximate solutions to a given variational inequality. This is done by exploiting the equivalence between the stationary points of the associated ordinary differential equation and the solutions of the variational inequality problem. It can be expected that the techniques described in this paper will be useful for more elaborate dynamical models, such as stochastic models, and that the connection between such dynamical models and the solutions to the variational inequalities will provide a deeper understanding of equilibrium problems.     

关于具有转向点的一类常微分方程的边值问题

本文应用多重尺度法研究具有转向点的一类常微分方程的边值问题.避免了[1]中出现的悖理,以及[2]中关于确定任意常数的变分运算;构造出边值问题的解的一致有效渐近近似式.并研究了非共振的情形.     

A unified theory of necessary conditions for nonlinear nonconvex control systems

We consider optimal problems for a general nonlinear nonconvex input-output relation for Banach space valued functions. A maximum principle is obtained using Ekeland's variational principle. The formulation applies to systems described by ordinary differential equations, functional differential equations, and partial differential equations (both for distributed and boundary control systems).This work was supported in part by the National Science Foundation under Grant No. DMS-8200645.     

MIP-based instantaneous control of mixed-integer PDE-constrained gas transport problems

We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.     

Infection Time in Multistable Gene Networks. A Backward Stochastic Variational Inequality with Nonconvex Switch-Dependent Reflection Approach

We investigate a mathematical model associated to the infection time in multistable gene networks. The mathematical processes are of hybrid switch type. The switch is governed by pure jump modes and linked to DNA bindings. The differential component follows backward stochastic dynamics reflected in some mode-dependent nonconvex domains. First, we study the existence of solutions to the resulting stochastic variational inclusions, by reducing the model to a family of ordinary variational inclusions with generalized reflection in semiconvex domains. Second, by considering control-dependent drivers, we hint to some model-selection approach by embedding the controlled backward stochastic variational inclusion in a family of regular measures. Regularity and structural properties of these sets are given.     

Lepage forms,closed 2-forms and second-order ordinary differential equations

Lepage 2-forms appear in the variational sequence as representatives of the classes of 2-forms. In the theory of ordinary differential equations on jet bundles they are used to construct exterior differential systems associated with the equations and to study solutions, and help to solve the inverse problem of the calculus of variations: since variational equations are characterized by Lepage 2-forms that are closed. In this paper, a general setting for Lepage forms in the variational sequence is presented, and Lepage 2-forms in the theory of second-order differential equations in general and of variational equations in particular, are investigated in detail. The text was submitted by the authors in English.     

On the boundedness of outer polyhedral estimates for reachable sets of linear systems

New properties of outer polyhedral (parallelepipedal) estimates for reachable sets of linear differential systems are studied. For systems with a stable matrix, it is determined what the orientation matrices are for which the estimates possessing the generalized semigroup property are bounded/unbounded on an infinite time interval. In particular, criteria are found (formulated in terms of the eigenvalues of the system’s matrix and the properties of bounding sets) that guarantee for previously mentioned tangent estimates and estimates with a constant orientation matrix that either there are initial orientation matrices for which the corresponding estimate tubes are bounded or all these tubes are unbounded. For linear stationary systems, a system of ordinary differential equations and algebraic relations is derived that determines estimates with constant orientation matrices for reachable sets that have no generalized semigroup property but are tangent and also bounded if the matrix of the system is stable.     

Oldroyd B流体依时性管内流动的变分解析方法

在本文中，研究上随体Ｏｌｄｒｏｙｄ Ｂ流体在水平管内依时性流动，该问题可归结为无量纲速度分量三阶偏微分方程的初边值问题，采用改进的Ｋａｎｔｏｒｏｖｉｃｈ方法，将该方程化为各级近似的二阶常微分方程组的初值问题，通过Ｌａｐｌａｃｅ变换，求得其二阶常微分方程的解析解。在本文中，提出了变分解析的新概念，获得了二级近似变分解析解，其中包括常压力力梯度和周期性压力梯度两种情形，应用计算机符呈处理和Ｌａｐｌａｃ     

Numerical detection and study of singularities in solutions of differential equations

New simple and robust methods are proposed for detecting singularities, such as poles, logarithmic poles, and mixed singularities, in systems of ordinary differential equations. The methods produce characteristics of these singularities with an a posteriori asymptotically precise error estimate. They are applicable in the case of an arbitrary parametrization of integral curves, including one in terms of the arc length, which is optimal for stiff and ill-conditioned problems. Following this approach, blowup solutions can be detected for a broad class of important nonlinear partial differential equations, since they are reducible by the method of lines to systems of ordinary differential equations of huge orders. The simplicity and reliability of the approach are superior to those of previously known methods.