On the properties of solutions to a class of nonlinear systems of differential equations of large dimension

We consider the Cauchy problem for a class of nonlinear systems of differential equations of large dimension, establish some properties of solutions, and prove that for a sufficiently large number of differential equations the last component of the solution is an approximate solution to the initial value problem for a delay differential equation.     

On One Class of Systems of Differential Equations and on Retarded Equations

We establish a connection between solutions to a broad class of large systems of ordinary differential equations and solutions to retarded differential equations. We prove that solving the Cauchy problem for systems of ordinary differential equations reduces to solving the initial value problem for a retarded differential equation as the number of equations increases unboundedly. In particular, the class of systems under consideration contains a system of differential equations which arises in modeling of multiphase synthesis.     

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.     

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.     

Differential properties of a generalized solution to a hyperbolic system of first-order differential equations

We study some questions of the qualitative theory of differential equations. A Cauchy problem is considered for a hyperbolic system of two first-order differential equations whose right-hand sides contain some discontinuous functions. A generalized solution is defined as a continuous solution to the corresponding system of integral equations. We prove the existence and uniqueness of a generalized solution and study the differential properties of the obtained solution. In particular, its first-order partial derivatives are unbounded near certain parts of the characteristic lines. We observe that this property contradicts the common approach which uses the reduction of a system of two first-order equations to a single second-order equation.     

Time decay of the solution to the Cauchy problem for a three-dimensional model of nonsimple thermoelasticity

This paper is devoted to the investigation of the solution to the Cauchy problem for a system of partial differential equations describing thermoelasticity of nonsimple materials in a three-dimensional space. The model of linear dynamical thermoelasticity of nonsimple materials is considered as the system of partial differential equations of fourth order. In this paper, we proposed a convenient evolutionary method of approach to the system of equations of nonsimple thermoelasticity. We proved the L^p−L^q time decay estimates for the solution to the Cauchy problem for linear thermoelasticity of nonsimple materials.     

A new continuous dependence result for impulsive retarded functional differential equations

We consider a large class of impulsive retarded functional differential equations (IRFDEs) and prove a result concerning uniqueness of solutions of impulsive FDEs. Also, we present a new result on continuous dependence of solutions on parameters for this class of equations. More precisely, we consider a sequence of initial value problems for impulsive RFDEs in the above setting, with convergent right-hand sides, convergent impulse operators and uniformly convergent initial data. We assume that the limiting equation is an impulsive RFDE whose initial condition is the uniform limit of the sequence of the initial data and whose solution exists and is unique. Then, for sufficient large indexes, the elements of the sequence of impulsive retarded initial value problem admit a unique solution and such a sequence of solutions converges to the solution of the limiting Cauchy problem.     

Global well-posedness and large-time behavior of 1D compressible Navier-Stokes equations with density-depending viscosity and vacuum in unbounded domains

We consider the Cauchy problem for one-dimensional(1D) barotropic compressible Navier-Stokes equations with density-depending viscosity and large external forces. Under a general assumption on the densitydepending viscosity, we prove that the Cauchy problem admits a unique global strong(classical) solution for the large initial data with vacuum. Moreover, the density is proved to be bounded from above time-independently.As a consequence, we obtain the large time behavior of the solution without external forces.     

Sufficient tests for the existence and uniqueness of a solution of the cauchy problem for a differential equation with a hysteresis nonlinearity

For a differential equation with a hysteresis nonlinearity of general type, which can be time-varying, we obtain sufficient conditions for the existence and uniqueness of a solution of the Cauchy problem similar to the well-known Cauchy-Picard, Peano, Perron, and Rosenblatt theorems for ordinary differential equations. We consider examples in which a test for the solution uniqueness similar to the Perron-Rosenblatt theorem is applied to specific differential equations with hysteresis nonlinearities of the form of the Prandtl model of a viscoelastic fiber and a time-varying nonlinearity.     

On the properties of ambiguity and irreversibility of dynamical maps of the initial data space of Cauchy problem

We investigate the Cauchy problems for evolutionary differential equations which possess the following properties: the solutions of considered problems admit the arising on the bounded time interval of singularities such that destroying of existence or uniqueness of solution and unbounded growth of norm of solution in Cauchy problem Banach space. The opportunity of continuation (probably of many-valued continuation) of the dynamical maps of the space of initial data by the procedure of passage to the limit for the sequences of approximating Cauchy problems is studied.     

On bounded solutions of a nonlinear Schrödinger equation on the half-line

We consider the problem on nonzero solutions of the Schrödinger equation on the half-line with potential that implicitly depends on the wave function via a nonlinear ordinary differential equation of the second order under zero boundary conditions for the wave function and the condition that the potential is zero at the beginning of the interval and its derivative is zero at infinity. The problem is reduced to the analysis and investigation of solutions of the Cauchy problem for a system of two nonlinear second-order ordinary differential equations with initial conditions depending on two parameters. We show that if the solution of the Cauchy problem for some parameter values can be extended to the entire half-line, then there exists a nonzero solution of the original problem with finitely many zeros.     

Well-posedness of the analytic Cauchy problem in classes of entire functions of finite order

We study the Cauchy problem for a system of complex linear differential equations in scales of spaces of functions of exponential type with an integral metric. Conditions under which this problem is well posed are obtained. These sufficient conditions are shown to be also necessary for the well-posedness of the Cauchy problem in the case of systems of ordinary differential equations with a parameter.     

Transmutations and irregular singular points

We consider transumtations for a class of problems in partial differential equations where the underlying equation, involving two assignable parameters, is an associated ordinary differential equation with an irregular singular point. An integral formula for the solution of this associated problem, valid for negative values of a timelike variable t, permits relating the solution of the problems in partial differential equations to be bounded or slow groth solutions of generalized heat problems. Applications of the formulas are made to Cauchy and boundary type problems.     

Global Solutions and Their Large-time Behavior of Cauchy Problem for Equations of Deep Water Type

We consider the large time behavior of the global solution of the Cauchy problem for the equation of Benjamin-Ono type. A series of large-time global estimates for Cauchy problem arc constructed. By means of the obtained global estimates uniformly for 0 ≤ t < ∞, the attractors of the Cauchy problem for the mentioned nonlinear equations are considered. And also the dimensions of the global attractor are estimated.     

The Cauchy problem for a system of two partial differential equations of infinite order

Starting from the generalized scheme of separation of variables, we propose a new effective method of constructing the solution of the Cauchy problem for a system of two partial differential equations, in general of infinite order with respect to the spatial variable. We consider the example of the Cauchy problem for the system of Lamé equations in the case of a two-dimensional strain.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 204–210.     

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 concept and existence of solution for impulsive fractional differential equations

This paper is motivated from some recent papers treating the problem of the existence of a solution for impulsive differential equations with fractional derivative. We firstly show that the formula of solutions in cited papers are incorrect. Secondly, we reconsider a class of impulsive fractional differential equations and introduce a correct formula of solutions for a impulsive Cauchy problem with Caputo fractional derivative. Further, some sufficient conditions for existence of the solutions are established by applying fixed point methods. Some examples are given to illustrate the results.     

On the Majorant Method for the Cauchy--Kovalevskaya Problem

We present a theory which allows us to justify the existence theorems and derive majorant estimates for solutions of the Cauchy problem for a countable system of partial differential equations. Our results also allow to estimate the maximum existence interval for a solution.     

On the Cauchy problem for a class of iterated Fuchsian partial differential equations

In this paper, we consider the Cauchy problem with ramified data for a class of iterated Fuchsian partial differential equations. We give an explicit representation of the solution in terms of Gauss hypergeometric functions. Our results are illustrated through some examples.     

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.