Transformation of equations with retarded argument to equations with the best argument

The problem of choosing the best argument in the Cauchy problem for a system of ordinary differential equations with retarded argument is studied from the viewpoint of the method of continuation of the solution with respect to a parameter. It is proved that the arc length counted along the integral curve of the problem is the best argument for the system of continuation equations to be well-posed in the best possible way. A transformation of the Cauchy problem to the best argument is obtained. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 62–68, January, 1998.     

Parametrization of the Cauchy problem for systems of ordinary differential equations with limiting singular points

The method of solution continuation with respect to a parameter is used to solve an initial value problem for a system of ordinary differential equations with several limiting singular points. The solution is continued using an argument (called the best) measured along the integral curve of the problem. Additionally, a modified argument is introduced that is locally equivalent to the best one in the considered domain. Theoretical results are obtained concerning the conditioning of the Cauchy problem parametrized by the modified argument in a neighborhood of each point of its integral curve.     

Numerical integration of systems of delay differential-algebraic equations

The numerical solution of the initial value problem for a system of delay differential-algebraic equations is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are obtained for transforming this problem to the best argument, which ensures the best condition for the corresponding system of continuation equations. The best argument is the arc length along the integral curve of the problem. Algorithms and programs based on the continuous and discrete continuation methods are developed for the numerical integration of this problem. The efficiency of the suggested transformation is demonstrated using test examples.     

Numerical solution to systems of delay integrodifferential algebraic equations

The numerical solution of the initial value problem for a system of delay integrodifferential algebraic equations is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are obtained for transforming this problem to the best argument, which is the arc length along the integral curve of the problem. The efficiency of the transformation is demonstrated using test examples.     

The cauchy problem for mechanical systems with a finite number of degrees of freedom as a problem of continuation on the best parameter

The Cauchy problem for a system of ordinary differential equations is formulated as a problem of continuation on the best parameter. It is proved that the length of an integral curve of the problem is such a parameter. The merits of the proposed transformation are demonstrated by a test example in which a stiff system of equations describing the perturbed motion of an aircraft is solved numerically.     

On the multisummability of divergent solutions of linear partial differential equations with constant coefficients

We consider the Cauchy problem for general linear partial differential equations in two complex variables with constant coefficients. We obtain necessary and sufficient conditions for the multisummability of formal solutions in terms of analytic continuation properties and growth estimates of the Cauchy data.     

Stabilization of unstable stationary points in equations with delayed argument

We solve the problem of localization and stabilization of an unstable stationary point of a nonlinear system of ordinary differential equations (ODE) with a delayed argument for parameter values when the ODE system has chaotic dynamics. Translated from Nelineinaya Dinamika i Upravlenie, pp. 133–141, 1999.     

On the Cauchy problem for linear hyperbolic functional-differential equations

We study the question of the existence, uniqueness, and continuous dependence on parameters of the Carathéodory solutions to the Cauchy problem for linear partial functional-differential equations of hyperbolic type. A theorem on the Fredholm alternative is also proved. The results obtained are new even in the case of equations without argument deviations, because we do not suppose absolute continuity of the function the Cauchy problem is prescribed on, which is rather usual assumption in the existing literature.     

Integral representation of a solution of the Cauchy problem for a degenerating hyperbolic equation with retarded argument

By using the method of integral equations, we prove the existence and uniqueness of a regular solution of the Cauchy problem for a degenerating hyperbolic equation with retarded argument. Orel Pedagogic Institute, Russia. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10, pp. 1332–1336, October, 1997.     

A criterion of solvability of the elliptic Cauchy problem in a multi-dimensional cylindrical domain

In this paper, we consider the Cauchy problem for multidimensional elliptic equations in a cylindrical domain. The method of spectral expansion in eigenfunctions of the Cauchy problem for equations with deviating argument establishes a criterion of the strong solvability of the considered elliptic Cauchy problem. It is shown that the ill-posedness of the elliptic Cauchy problem is equivalent to the existence of an isolated point of the continuous spectrum for a self-adjoint operator with deviating argument.     

Inverse problem of the variational calculus for differential equations of second order with deviating argument

We obtain conditions for the solvability of the inverse problem of the variational calculus for differential equations of second order with deviating argument of special form as well as the formula for the functional of the inverse problem defined by the integral that differs from the standard one by that the required function has a retarded argument.     

Differential equations with state-dependent piecewise constant argument

A new class of differential equations with state-dependent piecewise constant argument is introduced. It is an extension of systems with piecewise constant argument. Fundamental theoretical results for the equations—the existence and uniqueness of solutions, the existence of periodic solutions, and the stability of the zero solution—are obtained. Appropriate examples are constructed.     

Numerical-analytic method for the investigation of multipoint boundary-value problems for systems of differential equations with transformed argument

The problem of existence and approximate construction is studied for the solution of a nonlinear system of differential equations with a transformed argument and linear multipoint boundary conditions. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1581–1584, November, 1998.     

Sturm's theorem for equations with delayed argument

Sturm's type theorems on separation of zeros of solutions are proved for second order linear differential equations with delayed argument.     

On a mixed problem for systems of definite quasilinear partial differential equations with deviating argument

For a mixed problem for a system of definite quasilinear pseudoparabolic equations with deviating argument, we prove a theorem on differential inequalities and existence of a unique regular solution and a comparison theorem and give sufficient conditions of existence of solutions with constant sign.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 11, pp. 1581–1585, November, 1994.     

Energy-transport and drift-diffusion limits of nonisentropic Euler–Poisson equations

Two relaxation limits in critical spaces for the scaled nonisentropic Euler–Poisson equations with the momentum relaxation time and energy relaxation time are considered. As the first step of this justification, the uniform (global) classical solutions to the Cauchy problem in Chemin–Lerner?s spaces with critical regularity are constructed. Furthermore, by the compactness argument, it is rigorously justified that the scaled classical solutions converge to the solutions of energy-transport equations and drift-diffusion equations, respectively, with respect to different time scales.     

Optimal waiting time bounds for some flux-saturated diffusion equations

We consider the Cauchy problem for two prototypes of flux-saturated diffusion equations. In arbitrary space dimension, we give an optimal condition on the growth of the initial datum which discriminates between occurrence or nonoccurrence of a waiting time phenomenon. We also prove optimal upper bounds on the waiting time. Our argument is based on the introduction of suitable families of subsolutions and on a comparison result for a general class of flux-saturated diffusion equations.     

The Cauchy problem for the system of thermoelasticity equations in space

We consider the problem of analytic continuation of the solution of the system of thermoelasticity equations in a bounded three-dimensional domain on the basis of known values of the solution and the corresponding stress on a part of the boundary, i.e., the Cauchy problem. We construct an approximate solution of the problem based on the method of Carleman's function.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 212–217, August, 1998.In conclusion, the authors wish to thank Professor M. M. Lavrent'ev and Professor Sh. Ya. Yarmukhamedov for setting the problem and for discussions in the course of the solution.     

On periodic solutions of difference equations with continuous argument

We establish conditions for the existence of periodic solutions of systems of nonlinear difference equations with continuous argument.     

Local analytic continuation of holomorphic solutions of partial differential equations

Summary We ask. When is it possible to continue analytically holomorphic solutions of partial differential equations. Using objects called cones of analytic continuation we get sufficient conditions generalizing results by J.-M. Bony and P. Schapira and by Y. Tsuno. The results are counterparts of earlier results by the author on local uniqueness in the Cauchy problem. We also give a necessary condition by constructing solutions with singularities. We think that the technique used here should also have other applications. Anyhow this result is a generalization of a result by Y. Tsuno related to simple characteristic hypersurfaces. It corresponds to the existence of local null solution when the initial hypersurface has constant multiplicity in the Cauchy problem. Entrata in Redazione il 24 dicembre 1975.