Global solvability of one degenerate semilinear differential operator equation

We consider the Cauchy problem for the abstract semilinear differential equation where A and B are linear closed, generally speaking, degenerate operators acting from a Banach space X into a Banach space Y and f(t, u) is a continuously differentiable function. We assume that the resolvent (A + B)^–1 has a pole of at most second order at the point = 0. Global conditions for the existence and uniqueness of a solution of the Cauchy problem are obtained. The results are applied to a nonlinear degenerate initial boundary-value problem with partial derivatives and to a system of differential-algebraic equations of a nonlinear electric circuit.Translated from Neliniini Kolyvannya, Vol. 7, No. 3, pp. 414–429, July–September, 2004.     

Solutions of the generaln-th order variable coefficients linear difference equation

In this paper.variable operator and its product with shifting operator are studied.The product of power series of shifting operator with variable coefficient is defined andits convergence is proved under Mikusinski’s sequence convergence.After turning ageneral variable coefficient linear difference equation of order n into a set of operatorequations.we can obtain the solutions of the general n-th order variable coefficientlinear difference equation.     

Mikusinski’s operators solution of three-order linear difference equation with variable coefficients

This paper proceeds the papers [3] [4], we make use of the idea of the variable number operators and some concepts and conclusions of the shifting operators series with variable coefficients in the operational field of Mikusinski, it is devoted to the solution of the general three-order linear difference equation with variable coefficients, and it is also devoted to the better solution formula for the some special three-order linear difference equations with variable coefficients: in addition, we try to provide the idea and method for realizing solution of the more than three-order linear difference equation with variable coefficients. Project Supported by the Science Foundation of Anhui Province     

A high convergent precision exact analytic method for differential equation with variable coefficients

The exact analytic method was given by[1].It can be used for arbitrary variable coefficient differential equations and the solution obtained can have the second order convergent precision.In this paper,a new high precision algorithm is given based on[1],through a bending problem of variable cross-section beams.It can have the fourth convergent precision without increasing computation work.The present computation method is not only simple but also fast.The numerical examples are given at the end of this paper which indicate that the high convergent precision can be obtained using only a few elements.The correctness of the theory in this paper is confirmed.     

Exact solutions of a KdV equation with variable coefficients via Exp-function method

In this paper, the Exp-function method is used to obtain generalized solitonary solutions and periodic solutions of a KdV equation with five arbitrary functions. The results show that the Exp-function method with the help of symbolic computation provides a very effective and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.     

Boundedness of solutions of linear differential systems with periodic coefficients

Archive for Rational Mechanics and Analysis -     

On the similarity solutions of the KdV equation with variable coefficients

In this paper we apply the similarity methods to a modified Korteweg-deVries equation containing an arbitrary function of time which describes the inhomogeneity effects. Some classes of functional forms for this arbitrary function are characterized and the related similarity solutions are determined.     

On the solvability of one class of problems of optimal control by coefficients in the principal part of a nonlinear elliptic operator

We consider the problem of the existence of solutions of an optimal-control problem for a nonlinear elliptic equation with Dirichlet conditions on the boundary in the case where the control functions are the coefficients in the principal part of the differential operator. It is shown that this problem has an optimal solutions in the class of generalized solenoidal matrices. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 59–72, January–March, 2009.     

On the decomposition of a degenerate singularly perturbed linear system of differential equations

We study the problem of decomposition of degenerate singularly perturbed systems of differential equations. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 3, pp. 401–415, July–September, 2006.     

Constructions of the general solution for a system of partial differential equations with variable coefficients

Solving partial differential equations has not only theoretical significance,but alsopractical value.In this paper.by the property of conjugate operator.we give a method toconstruct the general solutions of a system of partial differential equations.     

The explicit solution of homogeneous linear oridinary differential equations with constant coefficients

In this paper by using tensor analysis we give the explicit expressions of the solution of the initial-value problem of homogeneous linear differential equations with constant coefficients and the n th-order homogeneous linear differential equation with constant coefficients. In fact, we give the general formula for calculating the elements of the matrix exp [At]. We also give the results when the characteristic equation has the repeated roots. The present method is simpler and better than, the other methods.     

A semi-analysis method of differential equations with variable coefficients under complicated boundary conditions

IntroductionInthemostcases,asfarasthesolutiontothedifferentialequationwithvariablecoefficientsisconcerned ,therehasbeennoanysatisfactoryanswer.Forthisreason ,thescholarsbothathomeandabroadhavedonesomeresearchjustasindicatedinRefs.[1 ] ,[2 ]and [3 ] ,etc.Theauthorsinthispaperhaveusedthefiniteelementmodelandthemetricfunctiontheoryincombinationwithadjustableparametermodelwithvariablecoefficientstosimplifythesolutiontothedifferentialequationwithvariablecoefficientsintothefeaturevalueaswellasthefea…