Two-sided linear numerical methods for the Cauchy problem for normal equations

Two-sided approximating collocation polynomials of the same order are constructed. These polynomials are used to develop two-sided extrapolation and interpolation difference methods and Krylov-type methods for the one-dimensional Cauchy problem for normal equations. The required integrals are evaluated by matrix numerical integration algorithms.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 3–14, 1992.     

Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations

The paper considers the Cauchy problem for linear partial differential equations of non-Kowalevskian type in the complex domain. It is shown that if the Cauchy data are entire functions of a suitable order, the problem has a formal solution which is multisummable. The precise bound of the admissible order of entire functions is described in terms of the Newton polygon of the equation.     

Set-valued solutions to the Cauchy problem for hyperbolic systems of partial differential inclusions

We prove the existence of global set-valued solutions to the Cauchy problem for partial differential equations and inclusions, with either single-valued or set-valued initial conditions. The method is based on the equivalence between this problem and problem of finding viability tubes of the associated characteristic system of ordinary differential equations. As an application we construct the value function of the Mayer problem arising in control theory. Received August 25, 1995     

Asymptotic solutions and error estimates for linear systems of difference and differential equations

Classical results concerning the asymptotic behavior solutions of systems of linear differential or difference equations lead to formulas containing factors that are asymptotically constant, i.e., k+o(1) as t tends to infinity. Here we are interested in more precise information about the o(1) terms, specifically how they depend precisely on certain perturbation terms in the equation. Results along these lines were given by Gel'fond and Kubenskaya for scalar difference equations and we will both extend and generalize one of them as well as provide some corresponding results for differential equations.     

The maximal nonuniqueness classes of solutions to the Cauchy problem for linear equations

We study linear partial differential equations with increasing coefficients in a half-plane. We establish maximal nonuniqueness classes of solutions to the Cauchy problem for these equations. The proof is based on a new estimation method for a solution to the dual differential equation with a parameter.     

Solvability of the Cauchy problem for higher-order linear functional differential equations

We obtain necessary and sufficient conditions for the unique solvability of the Cauchy problem for higher-order linear non-Volterra functional differential equations.     

An approach to deducing approximate solutions to the Cauchy problem for nonlinear differential equations

A new technique for the construction of numerical methods based on continued fractions is proposed. A characteristic feature of these algorithms is the fact that for certain values of the parameters it is possible to obtain both novel and traditional (explicit and implicit) numerical methods for the solution of the Cauchy problem for ordinary differential equations. Two-sided formulas are proposed by means of which it is possible to obtain on each integration step not only upper and lower approximations to the exact solution, but also information concerning the magnitude of the leading term of the error without the need for additional calculations of the right-hand side of the initial differential equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 12, pp. 1695–1701, December, 1992.     

The Cauchy problem for a linear parabolic partial differential equation

Numerical approximation of the solution of the Cauchy problem for the linear parabolic partial differential equation is considered. The problem: (p(x)u_x)_x ? q(x)u = p(x)u_t, 0 < x < 1,0 < t? T; $\text{u(0, t) = ?}{}_1\text{(t), 0 < t ? T}$; $\text{u(1,t) = ?}{}_2\text{(t), 0 < t ? T}$; p(0) u_x(0, t) = g(t), 0 < t₀ ? t ? T, is ill-posed in the sense of Hadamard. Complex variable and Dirichlet series techniques are used to establish Hölder continuous dependence of the solution upon the data under the additional assumption of a known uniform bound for ¦ u(x, t)¦ when 0 ? x ? 1 and 0 ? t ? T. Numerical results are obtained for the problem where the data ?₁, ?₂ and g are known only approximately.     

On a weakened Cauchy problem for a linear differential inclusion

For a differential inclusion $\dot x(t) \in \mathcal{A}x(t)$ with linear relation A, we establish the connection between a weakened and the ordinary Cauchy problem. We prove the uniqueness of and the representation for the weakened solution.     

Estimates of the solutions of linear systems with delay

Kiev. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No 5, pp. 196–198, September–October, 1991.     

On acyclicity of the set of solutions to the Cauchy problem for differential equations

The acyclicity of sets of solutions to the Cauchy problem is considered from the viewpoint of the axiomatic theory of solutions spaces of ordinary differential equations. We prove that the class of acyclic solution spaces is closed with respect to passing to the limit space. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 836–842, December, 1997 Translated by M. A. Shishkova     

Analysis and solution of the Cauchy problem for linear Volterra integro-differential equations with functional delay

We study how to transform Cauchy problems for Volterra integro-differential equations with functional delays to resolving Volterra integral equations with conventional argument by using a modification of a function of flexible structure. We show that such a transformation is possible for all linear Volterra integro-differential equations of retarded type. There exists a unique solution of the resolving equation provided that the kernels and the right-hand side are bounded in the closed square. The presence of parameters in the expression for the function of flexible structure permits one to choose these parameters in an optimal way in the course of the solution of the problem so as to represent the solution in closed form or, if this is difficult, optimize an approximate solution method. The accuracy of the approximate solutions is estimated.     

On the Cauchy type problem for systems of functional differential equations

Using theorems on functional differential inequalities, we establish new efficient conditions for the solvability as well as unique solvability of the Cauchy type problem for systems of functional differential equations in both linear and nonlinear cases.     

Algebraically simple involutive differential systems and the Cauchy problem

We consider systems of polynomial nonlinear partial differential equations (PDEs) possessing certain properties. Such systems were studied by the American mathematician Thomas in the 1930s, and he called them (algebraically) simple. Thomas gave a constructive procedure to split an arbitrary system of PDEs into a finite number of simple subsystems. The class of simple involutive systems of PDEs includes normal, or Kovalevskaya-type, systems and Riquier orthonomic passive systems. Systems of this class admit well-posed Cauchy problems. We discuss the basic features of the splitting algorithm, completion of simple systems to involution, and the well-posedness of the Cauchy problem. Two illustrative examples are given. Bibliography: 17 titles.     

Breakdown of classical solutions to Cauchy problem for inhomogeneous quasilinear hyperbolic systems

This paper deals with the asymptotic behavior of the life-span of classical solutions to Cauchy problem for inhomogeneous quasilinear strictly hyperbolic systems with weaker decaying initial data. Under the assumption that the source term satisfies the corresponding matching condition, we obtains a blow-up result for C ¹ solution to the Cauchy problem.     

Multiple periodic solutions for non-canonical Hamiltonian systems with application to differential delay equations

We first establish Maslov index for non-canonical Hamiltonian system by using symplectic transformation for Hamiltonian system. Then the existence of multiple periodic solutions for the non-canonical Hamiltonian system is obtained by applying the Maslov index and Morse theory. As an application of the results, we study a class of non-autonomous differential delay equation which can be changed to non-canonical Hamiltonian system and obtain the existence of multiple periodic solutions for the equation by employing variational method.