Adaptive Interpolation Algorithm Based on a kd-Tree for Numerical Integration of Systems of Ordinary Differential Equations with Interval Initial Conditions

We consider issues related to the numerical solution of interval systems of ordinary differential equations. We suggest an algorithm that permits finding interval estimates of solutions with prescribed accuracy in reasonable time. The algorithm constructs an adaptive partition (a dynamic structured grid) based on a kd-tree over the space formed by interval initial conditions for the ordinary differential equations. In the operation of the algorithm, a piecewise polynomial function interpolating the dependence of the solution on the specific values of interval parameters is constructed at each step of solution of the original problem. We prove that the global error estimate linearly depends on the height of the kd-tree. The algorithm is tested on several examples; the test results show its efficiency when solving problems of the class under study.     

Time Optimal Control for Some Ordinary Differential Equations with Multiple Solutions

This paper studies a time optimal control problem for a class of ordinary differential equations. The control systems may have multiple solutions. Based on the properties fulfilled by the solutions of the concerned equations, we get both the existence and the Pontryagin maximum principle for optimal controls.     

与薛丁谔型谱问题相联系的孤子族的分解

The soliton hierarchy associated with a Schrodinger type spectral problem with four potentials is decomposed into a class of new finite-dimensional Hamiltonian systems by using the nonlinearized approach.It is worth to point that the solutions for the soliton hierarchy are reduced to solving the compatible Hamiltonian systems of ordinary differential equations.     

Numerical Solution of Systems of Loaded Ordinary Differential Equations with Multipoint Conditions

A system of loaded ordinary differential equations with multipoint conditions is considered. The problem under study is reduced to an equivalent boundary value problem for a system of ordinary differential equations with parameters. A system of linear algebraic equations for the parameters is constructed using the matrices of the loaded terms and the multipoint condition. The conditions for the unique solvability and well-posedness of the original problem are established in terms of the matrix made up of the coefficients of the system of linear algebraic equations. The coefficients and the righthand side of the constructed system are determined by solving Cauchy problems for linear ordinary differential equations. The solutions of the system are found in terms of the values of the desired function at the initial points of subintervals. The parametrization method is numerically implemented using the fourth-order accurate Runge–Kutta method as applied to the Cauchy problems for ordinary differential equations. The performance of the constructed numerical algorithms is illustrated by examples.     

Spaces of Quasi-Periodic Functions and Oscillations in Differential Equations

We build spaces of q.p. (quasi-periodic) functions and we establish some of their properties. They are motivated by the Percival approach to q.p. solutions of Hamiltonian systems. The periodic solutions of an adequatez partial differential equation are related to the q.p. solutions of an ordinary differential equation. We use this approach to obtain some regularization theorems of weak q.p. solutions of differential equations. For a large class of differential equations, the first theorem gives a result of density: a particular form of perturbated equations have strong solutions. The second theorem gives a condition which ensures that any essentially bounded weak solution is a strong one.     

Stochastic Retarded Evolution Equations: Green Operators,Convolutions, and Solutions

Abstract

In this work, we shall investigate solution (strong, weak and mild) processes and relevant properties of stochastic convolutions for a class of stochastic retarded differential equations in Hilbert spaces. We introduce a strongly continuous one-parameter family of bounded linear operators which will completely describe the corresponding deterministic systematical dynamics with time delays. This family, which constitutes the fundamental solutions (Green's operators) of our stochastic retarded systems, is applied subsequently to define mild solutions of the stochastic retarded differential equations considered. The relations among strong, weak and mild solutions are explored. By virtue of a strong solution approximation method, Burkholder–Davis–Gundy's type of inequalities for stochastic convolutions are established.     

非线性二阶常微分方程组边值问题的正解

本文利用拓扑方法研究了下列非线性二阶常微分方程组边值问题在某些条件下,我们证明了上述边值问题正解的存在性和个数.     

Global Solvability and Estimates of Solutions to the Cauchy Problem for the Retarded Functional Differential Equations That Are Used to Model Living Systems

We study the Cauchy problem for the retarded functional differential equations that model the dynamics of some living systems. We find certain conditions ensuring the existence, uniqueness, and nonnegativity of solutions on finite and infinite time intervals. We obtain upper bounds for solutions and prove the continuous dependence of solutions on the initial data on finite time intervals.     

Intelligent Front-End for Solving Nonlinear Systems of Differential and Algebraic Equations

INTENSE (INTElligent Nonlinear Systems of Equations solver) is a problem-solving environment that provides necessary computational facilities for finding numerical solutions of ordinary differential equations and systems of nonlinear algebraic equations. INTENSE offers facilities such as a user-friendly interface for specification of problems, an automatic detection of the properties of a problem, a user-friendly interface for specification of a method, an automatic detection of the properties of methods, a mechanism that matches the properties of problems with the properties of methods, the application of a numerical method and supervision of the errors, some lists or graphics of the values of approximate solutions, and solving the problem without specification of a particular method. Bibliography: 35 titles.     

一类交替型脉冲微分系统的解

主要讨论一类超前型与滞后型交替的脉冲微分系统.首先给出具常系数的脉冲微分系统解存在的充分条件以及解唯一的表达形式;对于变系数的微分系统也作了相应的讨论.     

Preservation of stability under discretization of systems of ordinary differential equations

We study the stability preservation problem while passing from ordinary differential to difference equations. Using the method of Lyapunov functions, we determine the conditions under which the asymptotic stability of the zero solutions to systems of differential equations implies that the zero solutions to the corresponding difference systems are asymptotically stable as well. We prove a theorem on the stability of perturbed systems, estimate the duration of transition processes for some class of systems of nonlinear difference equations, and study the conditions of the stability of complex systems in nonlinear approximation.     

Degasperis-Procesi方程的类孤子新解

利用辅助方程与函数变换相结合的方法,构造了Degasperis-Procesi(D-P)方程的无穷序列类孤子新解.首先,通过两种函数变换,把D-P方程化为常微分方程组.然后,利用常微分方程组的首次积分,把D-P方程的求解问题化为几种常微分方程的求解问题.最后,利用几种常微分方程的Bcklund变换等相关结论,构造了D-P方程的无穷序列类孤子新解.这里包括由Riemannθ函数、Jacobi椭圆函数、双曲函数、三角函数和有理函数组成的无穷序列光滑孤立子解、尖峰孤立子解和紧孤立子解.     

A Modified Regularization Method

We consider the Cauchy problem for the systems of the first-order ordinary differential equations with a small parameter E of degree q at the derivative. We study the possibility of solving this problem by means of the regularization method of the Lomov theory of singular perturbations. We show that, for q>1, an application of the procedure by Lomov leads only to the trivial solution to the problem in the class of resonance-free solutions. We suggest and describe a modification of the procedure which enables us to construct a nontrivial solution to the problem in the space of resonance-free solutions.     

Invariant subspaces and exact solutions of a class of dispersive evolution equations

The invariant subspace method is used to classify a class of systems of nonlinear dispersive evolution equations and determine their invariant subspaces and exact solutions. A crucial step is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that systems of evolution equations admit. A few examples of presenting exact solutions with generalized separated variables illustrate the effectiveness of the invariant subspace method in solving systems of nonlinear evolution equations.     

On Differential Equations with Retarded Argument

We study the limit properties of solutions to one class of systems of differential equations as the number of equations and some parameters tend to infinity. We establish a close connection between solutions to these systems and differential equations with retarded argument. Our convergence theorems provide a new method for approximation of solutions to nonlinear differential equations with retarded argument.Original Russian Text Copyright © 2005 Demidenko G. V. and Likhoshvai V. A.The authors were supported by the Russian Foundation for Basic Research (Grants 03-01-00328, 03-04-48506, 03-04-48829, 03-07-96833, 04-01-00458, 05-04-49068, and 05-07-90274), the Integration Grant of the Siberian Branch of the Russian Academy of Sciences (No. 119 and 148), the Program Development of the Scientific Potential of Higher School of the Ministry for Education of the Russian Federation (Grant 8273), and NSF:FIBR (Grant EF-0330786).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 538–552, May–June, 2005.     

Positive Radial Solutions of Some Nonlinear Partial Differential Equations

We consider Dirichlet boundary value problems for a class of nonlinear ordinary differential equations motivated by the study of radial solutions of equations which are perturbations of the p-Laplacian.     

On One Problem of the Investigation of Global Solutions of Linear Differential Equations with Deviating Argument

We present conditions under which global solutions of linear systems of differential equations with deviating argument are solutions of ordinary differential equations.     

非线性二阶常微分方程的正周期解

讨论一类非线性二阶常微分方程的周期解问题 ,利用Banach空间锥上的不动点定理得到了正周期解的存在性和多重性结果 ,大大改进了文献 [1 ]的结果     

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.     

Equivalence Groups for First-Order Balance Equations and Applications to Electromagnetism

We investigate the groups of equivalence transformations for first-order balance equations involving an arbitrary number of dependent and independent variables. We obtain the determining equations and find their explicit solutions. The approach to this problem is based on a geometric method that depends on Cartan's exterior differential forms. The general solutions of the determining equations for equivalence transformations for first-order systems are applied to a class of the Maxwell equations of electrodynamics.