Orthogonal Polynomials Based Operational Matrices with Applications to Bagley-Torvik Fractional Derivative Differential Equations

Abstract

Orthogonal polynomials are the natural way to express the elements of the inner product spaces as an infinite sum of orthonormal basis sets. The construction and development of the many important numerical algorithms are based on the operational matrices of orthogonal polynomials including spectral tau, spectral collocation, and operational matrices approach are few of them. The widely used orthogonal polynomials are Legendre, Jacobi, and Chebyshev. However, only a few papers are available where the Hermite polynomials (HPs) were exploited to solve numerically the differential equations. The notable characteristic of the HPs is its ability to approximate the square-integrable functions on the entire real line. The prime objective of this chapter is to introduce the two new generalized operational matrices of HPs which are developed in the sense of the Riemann-Liouville fractional-order integral operator and Hilfer fractional-order derivative operator. The newly derived operational matrices are further used to construct a numerical algorithm for solving the Bagley--Trovik types fractional derivative differential equations (FDDE). Moreover, the results obtained by using the proposed algorithm are compared with the results obtained otherwise to demonstrate the efficiency and accuracy of the proposed numerical algorithm. Some examples are solved for application purposes.