| Abstract: | In this paper, we investigate in more detail some useful theorems related to conformable fractional derivative (CFD) and integral and introduce two classes of conformable fractional Sturm‐Liouville problems (CFSLPs): namely, regular and singular CFSLPs. For both classes, we study some of the basic properties of the Sturm‐Liouville theory. In the class of r‐CFSLPs, we discuss two types of CFSLPs which include left‐ and right‐sided CFDs, each of order α∈(n,n+1], and prove properties of the eigenvalues and the eigenfunctions in a certain Hilbert space. Also, we apply a fixed‐point theorem for proving the existence and uniqueness of the eigenfunctions. As an operator for the class of s‐CFSLPs, we first derive two fractional types of the hypergeometric differential equations of order α∈(0,1] and obtain their analytical eigensolutions as Gauss hypergeometric functions. Afterwards, we define the conformable fractional Legendre polynomial/functions (CFLP/Fs) as Jacobi polynomial and investigate their basic properties. Moreover, the conformable fractional integral Legendre transforms (CFILTs) based on CFLP/Fs‐I and ‐II are introduced, and using these new transforms, an effective procedure for solving explicitly certain ordinary and partial conformable fractional differential equations (CFDEs) are given. Finally, as a theoretical application, some fractional diffusion equations are solved. |
