On conformable fractional Legendre polynomials and their convergence properties with applications

TL;DR

This paper introduces conformable fractional Legendre polynomials, explores their properties and convergence, and applies them to solve fractional differential equations using collocation methods, enriching fractional special function theory.

Contribution

It presents a new class of conformable fractional Legendre polynomials with detailed properties, recurrence relations, and applications to fractional differential equations.

Findings

Established orthogonality and convergence properties of CFLPs

Derived recurrence and differential relations for CFLPs

Applied CFLPs to solve fractional differential equations successfully

Abstract

The main objective of this paper is to give a wide study on the conformable fractional Legendre polynomials (CFLPs). This study is assumed to be a generalization and refinement, in an easy way, of the scalar case into the context of the conformable fractional differentiation. We introduce the CFLPs via different generating functions and provide some of their main properties and convergence results. Subsequently, some pure recurrence and differential recurrence relations, Laplace's first integral formula and orthogonal properties are then developed for CFLPs. We append our study with presenting shifted CFLPs and describing applicable scheme using the collocation method to solve some fractional differential equations (FDEs) in the sense of conformable derivative. Some useful examples of FDEs are treated to support our theoretical results and examining their exact solutions. To the best of our knowledge, the obtained results are newly presented and could enrich the fractional theory of special functions.