The Search for Differential Equations for Orthogonal Polynomials by Using Computers
نویسنده
چکیده
We look for differential equations of the form ∞ ∑ i=0 ci(x)y (x) = λny(x), where the coefficients {ci(x)} ∞ i=0 do not depend on n, for the generalized Jacobi polynomials { P n (x) } ∞ n=0 found by T.H. Koornwinder in 1984 and for generalized Laguerre polynomials { L n (x) } ∞ n=0 which are orthogonal with respect to an inner product of Sobolev type. We introduce a method which makes use of computeralgebra packages like Maple and Mathematica and we will give some preliminary results.
منابع مشابه
Solving singular integral equations by using orthogonal polynomials
In this paper, a special technique is studied by using the orthogonal Chebyshev polynomials to get approximate solutions for singular and hyper-singular integral equations of the first kind. A singular integral equation is converted to a system of algebraic equations based on using special properties of Chebyshev series. The error bounds are also stated for the regular part of approximate solut...
متن کاملNumerical solution of delay differential equations via operational matrices of hybrid of block-pulse functions and Bernstein polynomials
In this paper, we introduce hybrid of block-pulse functions and Bernstein polynomials and derive operational matrices of integration, dual, differentiation, product and delay of these hybrid functions by a general procedure that can be used for other polynomials or orthogonal functions. Then, we utilize them to solve delay differential equations and time-delay system. The method is based upon e...
متن کاملA fractional type of the Chebyshev polynomials for approximation of solution of linear fractional differential equations
In this paper we introduce a type of fractional-order polynomials based on the classical Chebyshev polynomials of the second kind (FCSs). Also we construct the operational matrix of fractional derivative of order $ gamma $ in the Caputo for FCSs and show that this matrix with the Tau method are utilized to reduce the solution of some fractional-order differential equations.
متن کاملThe operational matrix of fractional derivative of the fractional-order Chebyshev functions and its applications
In this paper, we introduce a family of fractional-order Chebyshev functions based on the classical Chebyshev polynomials. We calculate and derive the operational matrix of derivative of fractional order $gamma$ in the Caputo sense using the fractional-order Chebyshev functions. This matrix yields to low computational cost of numerical solution of fractional order differential equations to the ...
متن کاملA spectral method based on Hahn polynomials for solving weakly singular fractional order integro-differential equations
In this paper, we consider the discrete Hahn polynomials and investigate their application for numerical solutions of the fractional order integro-differential equations with weakly singular kernel .This paper presented the operational matrix of the fractional integration of Hahn polynomials for the first time. The main advantage of approximating a continuous function by Hahn polynomials is tha...
متن کامل