Spectral Methods Using a Finite Class of Orthogonal Polynomials Related to Inverse Gamma Distribution

Subject Areas : International Journal of Mathematical Modelling & Computations

Amir Hossein Salehi Shayegan ^{1 *}

1 - Mathematics Department, Faculty of Basic Science, Khatam-ol-Anbia (PBU) University, Tehran, Iran

Received: 2023-10-05 Accepted : 2023-12-30 Published : 2024-11-15

Keywords: Spectral methods, Orthogonal polynomials, Finite approximation, Inverse Gamma distribution,

Abstract :

Classical orthogonal polynomials of Jacobi‎, ‎Laguerre and Hermite are characterized as the infinite sequences of orthogonal polynomials‎. ‎In this paper‎, ‎we present a sequence of orthogonal polynomials which is finitely orthogonal with respect to inverse Gamma distribution on infinite interval‎. ‎General properties of this sequence such as orthogonality relation‎, ‎Rodrigues type formula‎, ‎recurrence relations and also some of its applications such as Gauss quadrature‎, ‎Gauss-Radau quadrature formulas and so on are indicated‎. ‎In addition‎, ‎it is well-known that spectral methods for unbounded domains can be essentially classified into four categories; domain truncation‎, ‎approximation by classical orthogonal systems on unbounded domains‎, ‎approximation by other non-classical orthogonal systems and mapping‎. ‎In this paper based on the second category‎, ‎we propose a spectral method using the finite class of orthogonal polynomial $ {N_n^{(p)}(x)} $ related to inverse Gamma distribution‎. ‎Error analysis and convergence of the method are thoroughly investigated‎. ‎At the end‎, ‎two numerical examples are given for the efficiency and accuracy of the proposed method‎.

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