Regular-Singular Crossings on Nonlinear Systems with Turning Point

We consider in this paper the boundary value problems of nonlinear systems the form εY″=F(t,Y,Y′,ε), -1<t<1, Y(-1,ε)=A(ε), Y(1,ε)=B(ε). Supoosing some or all of the components of F , that is, f i satisfy 2 f y′ 2 i t =0 =0, we say that F possesses a generalized turning point at t =0. Our goal is to give sufficient conditions for the existence of solution of the problems and to study the asymptotic behavior of the solution when F possesses a generalized turning point at t =0. We mainly discuss regular singular crossings.

The Well-Posed Operators with Their Spectra in Lpw-Spaces

In this paper, we have considered the general ordinary quasi-differential operators generated by a general quasi-differential expression τp,q in Lpw-spaces of order n with complex coefficients and its formal adjoint τ+q',p' in Lpw-spaces for arbitrary p,q∈[1,∞). We have proved in the case of one singular end-point that all well-posed extensions of the minimal operator T0 (τp,q) generated by such expression τp,q and their formal adjoint on the interval [a,b) with maximal deficiency indices have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all the standard essential spectra to be empty. Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions can be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while others are new.

Frobenius Method for Solving Second-Order Ordinary Differential Equations

As we know that the power series method is a very effective method for solving the Ordinary differential equations (ODEs) which have variable coefficient, so in this paper we have studied how to solve second-order ordinary differential equation with variable coefficient at a singular point t = 0 and determined the form of second linearly independent solution. Based on the roots of initial equation there are real and complex cases. When the roots of initial equation are real then there are three kinds of second linearly independent solutions. If the roots of the initial equation are distinct complex numbers, then the solution is complex-valued.

On the Domains of General Ordinary Differential Operators in the Direct Sum Spaces

Given general quasi-differential expressions , each of order n with complex coefficients and their formal adjoint are on the interval [a,b) respectively, we give a characterization of all regularly solvable operators and their adjoints generated by a general ordinary quasi-differential expression in the direct sum Hilbert spaces . The domains of these operators are described in terms of boundary conditions involving -solutions of the equations and their adjoint on the intervals [ap,bp). This characterization is an extension of those obtained in the case of one interval with one and two singular end-points of the interval (a,b), and is a generalization of those proved in the case of self-adjoint and J-self-adjoint differential operators as a special case, where J denotes complex conjugation.

On the Spectra of General Ordinary Quasi-Differential Operators and Their L2w-Solutions

In this paper, we consider the general ordinary quasi-differential expression τ of order n with complex coefficients and its formal adjoint τ+ on the interval [a,b). We shall show in the case of one singular end-point and under suitable conditions that all solutions of a general ordinary quasi-differential equation are in the weighted Hilbert space provided that all solutions of the equations and its adjoint are in . Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions may be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while the others are new.

INTERFACE PROBLEMS FOR ELLIPTIC DIFFERENTIAL EQUATIONS

A new approach is given to analyse the regularity of solutions near singular points for the interface problems of second order elliptic partial differential equations. For general equations with nonsymmetric dominant terms and discontinuous piecewise smooth coefficients, it is proved that solutions in H 1 can be docomposed into two parts, one of which is a finite sum of particular solutions to the corresponding homogeneous equations with piecewise constant coefficients, and the other one of which is the regular part. Moreover a priori estimations are proven.