Fourth-order differential equation for the co-modified of semi-classical orthogonal polynomials

Periodic solutions of fourth-order delay differential equation

In this paper the periodic solutions of fourth order delay differential equation of the form $ddddot{x}(t)+adddot{x}(t)+f(ddot{x}(t-tau(t)))+g(dot{x}(t-tau(t)))+h({x}(t-tau(t)))=p(t)$  is investigated. Some new positive periodic criteria are given.  

Approximate solution of fourth order differential equation in Neumann problem

Generalized solution on Neumann problem of the fourth order ordinary differential equation in space $W^2_alpha(0,b)$ has been discussed, we obtain the condition on B.V.P when the solution is in classical form. Formulation of Quintic Spline Function has been derived and the consistency relations are given.Numerical method,based on Quintic spline approximation has been developed. Spline solution ...

The relationship between semi-classical Laguerre polynomials and the fourth Painlevé equation

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semiclassical Laguerre weight and classical solutions of the fourth Painlevé equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions which arise in the description of special function solutions of the fo...

Classical 2-orthogonal polynomials and differential equations

We construct the linear differential equations of third order satisfied by the classical 2orthogonal polynomials. We show that these differential equations have the following form: R4,n(x)P (3) n+3(x)+R3,n(x)P ′′ n+3(x)+R2,n(x)P ′ n+3(x)+R1,n(x)Pn+3(x)=0, where the coefficients {Rk,n(x)}k=1,4 are polynomials whose degrees are, respectively, less than or equal to 4, 3, 2, and 1. We also show tha...

Differential and Difference Equations for Products of Classical Orthogonal Polynomials