Spectral Methods Using a Finite Class of Orthogonal Polynomials Related to Inverse Gamma Distribution
محورهای موضوعی : فصلنامه ریاضی
Amir Hossein Salehi Shayegan
1
1 - Mathematics Department, Faculty of Basic Science, Khatam-ol-Anbia (PBU) University, Tehran, Iran
کلید واژه: Spectral methods, Orthogonal polynomials, Finite approximation, Inverse Gamma distribution,
چکیده مقاله :
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.
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.
