Sobolev orthogonal polynomials in two variables and second order partial differential equations

We consider polynomials in two variables which satisfy an admissible second order partial differential equation of the form

(∗)     

Second order partial differential equations for gradients of orthogonal polynomials in two variables

In this work, we introduce the classical orthogonal polynomials in two variables as the solutions of a matrix second order partial differential equation involving matrix polynomial coefficients, the usual gradient operator, and the divergence operator. Here we show that the successive gradients of these polynomials also satisfy a matrix second order partial differential equation closely related to the first one.     

Orthogonal polynomials in two variables

Summary Some properties of orthogonal (and generalized orthogonal) polynomial sets in two variables are obtained, in particular a characterization of such sets based on generating functions. Then those linear homogeneous partial differential eqnations of the form L[w]+λw=0, having a set of polynomials as solution, are characterized; and a detailed study is made of all such equations of second order whose polynomial solutions form an orthogonal (or generalized orthogonal) set. Supported byN.S.F. Grant GP-5311.     

Matrix differential equations and scalar polynomials satisfying higher order recursions

We show that any scalar differential operator with a family of polynomials as its common eigenfunctions leads canonically to a matrix differential operator with the same property. The construction of the corresponding family of matrix valued polynomials has been studied in [A. Durán, A generalization of Favard's theorem for polynomials satisfying a recurrence relation, J. Approx. Theory 74 (1993) 83-109; A. Durán, On orthogonal polynomials with respect to a positive definite matrix of measures, Canad. J. Math. 47 (1995) 88-112; A. Durán, W. van Assche, Orthogonal matrix polynomials and higher order recurrence relations, Linear Algebra Appl. 219 (1995) 261-280] but the existence of a differential operator having them as common eigenfunctions had not been considered. This correspondence goes only one way and most matrix valued situations do not arise in this fashion. We illustrate this general construction with a few examples. In the case of some families of scalar valued polynomials introduced in [F.A. Grünbaum, L. Haine, Bispectral Darboux transformations: An extension of the Krall polynomials, Int. Math. Res. Not. 8 (1997) 359-392] we take a first look at the algebra of all matrix differential operators that share these common eigenfunctions and uncover a number of phenomena that are new to the matrix valued case.     

Linear partial difference equations of hypergeometric type: Orthogonal polynomial solutions in two discrete variables

In this paper a systematic study of the orthogonal polynomial solutions of a second order partial difference equation of hypergeometric type of two variables is done. The Pearson's systems for the orthogonality weight of the solutions and also for the difference derivatives of the solutions are presented. The orthogonality property in subspaces is treated in detail, which leads to an analog of the Rodrigues-type formula for orthogonal polynomials of two discrete variables. A classification of the admissible equations as well as some examples related with bivariate Hahn, Kravchuk, Meixner, and Charlier families, and their algebraic and difference properties are explicitly given.     

高阶非线性中立型微分方程的周期解

利用k-集压缩延拓理论 ,研究了一类高阶非线性中立型微分方程周期解的存在性 ,推广了文 [1 ],[2 ]的相应结果 .     

Even periodic solutions of higher order duffing differential equations

By using Mawhin’s continuation theorem, the existence of even solutions with minimum positive period for a class of higher order nonlinear Duffing differential equations is studied.     

关于二维光滑线性方程无解的注记

Abstract. A smooth linear complex partial differential equation in two variables which is without solutions is found.     

Kinetic decomposition for singularly perturbed higher order partial differential equations

In this paper, we consider singularly perturbed higher order partial differential equations. We establish the condition under which the approximate solutions converge in a strong topology to the entropy solution of a scalar conservation laws using methodology developed in Hwang and Tzavaras (Comm. Partial Differential Equations 27 (2002) 1229). First, we obtain the approximate transport equation for the given dispersive equations. Then using the averaging lemma, we obtain the convergence.     

Numbers of subnormal solutions for higher order periodic differential equations

In this paper, we estimate the number of subnormal solutions for higher order linear periodic differential equations, and estimate the growth of subnormal solutions and all other solutions. We also give a representation of subnormal solutions of a class of higher order linear periodic differential equations.     

On Kneser solutions of higher order nonlinear ordinary differential equations

The equationx ⁽ⁿ⁾(t)=(−1) ⁿ │x(t)│ ^k withk>1 is considered. In the casen≦4 it is proved that solutions defined in a neighbourhood of infinity coincide withC(t−t₀)^−n/(k−1), whereC is a constant depending only onn andk. In the general case such solutions are Kneser solutions and can be estimated from above and below by a constant times (t−t ₀)^−n/(k−1). It is shown that they do not necessarily coincide withC(t−t₀)^−n/(k−1). This gives a negative answer to two conjectures posed by Kiguradze that Kneser solutions are determined by their value in a point and that blow-up solutions have prescribed asymptotics. Dedicated to Professor Vladimir Maz'ya on the occasion of his 60th birthday. The author was supported by the Swedish Natural Science Research Council (NFR) grant M-AA/MA 10879-304.     

On global regular solutions of third order partial differential equations

Diffusion in the presence of high-diffusivity paths is an important issue of current technology. In metals high-diffusivity paths are identified with dislocations, grain boundaries, free surfaces and internal microcracks. Diffusion in a media with two distinct families of diffusion paths is modelled by two coupled linear partial differential equations of parabolic type with diffusivities D₁ and D₂. Physically the situation D₂ ? D₁ is of some considerable interest and previously established results, for D₂ non-zero, for the solution of boundary value problems, are not applicable to the idealized theory characterized by D₂ vanishing. An integral equation, which arises in the solution of boundary value problems for this idealized theory, is formally solved.     

Periodic solutions for higher order differential equations with deviating argument

By using the coincidence degree theory of Mawhin, we study the existence of periodic solutions for higher order differential equations with deviating argument . Some new results on the existence of periodic solutions of the equations are obtained. Meanwhile, an example is given to illustrate our results.     

Orthogonal polynomials and higher order singular Sturm-Liouville systems

In addition to the classic orthogonal polynomials which satisfy second order differential equations, there are a number of orthogonal polynomials which satisfy differential equations of orders four or six. Like the classic sets, they have distributional weight functions, are the eigenfunctions for certain self-adjoint boundary-value problems, and sometimes are involved with indefinite boundary-value problems.The purpose of this survey is to summarize the work of the last decade and to exhibit the state of the art as it now stands. Of particular interest is the development of the theory of singular Sturm-Liouville systems, which is so necessary in order to describe the boundary-value problems associated with these polynomials.     

On the oscillation of solutions of partial differential equations in three variables

We obtain conditions for oscillation and nonoscillation of solutions of the equation Lu + pu=0 in Euclidean, spherical, and hyperbolic spaces with three variables, and also direct and converse theorems on mean values over spheres.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 45–48.     

Existence of meromorphic solutions of some higher order linear differential equations

This article discusses the problems on the existence of meromorphic solutions of some higher order linear differential equations with meromorphic coefficients. Some nice results are obtained. And these results perfect the complex oscillation theory of meromorphic solutions of linear differential equations.     

Wavelet Galerkin schemes for higher order time dependent partial differential equations

Fourth‐order derivatives appearing in different linear and nonlinear transient higher order multidimensional equations modeling several physical problems have always posed computational challenges to widely prevailing numerical approaches such as FDM, FEM, and so forth. In this study, we address the issue effectively using the special features of Daubechies wavelets such as orthogonality, compact support, arbitrary regularity, high‐order vanishing moments, and good localization. An efficient compression strategy is proposed to reduce the computational cost significantly. Implicitly stable backward Euler scheme is used for time marching the evolving solution. A priori error estimates have been derived to prove the convergence of the numerical scheme. The proposed approach is successfully tested on few linear and nonlinear multidimensional PDEs.