# M\"untz Sturm-Liouville Problems: Theory and Numerical Experiments

## TL;DR

This paper introduces new classes of M"untz functions with spectral properties, orthogonal projections, quadrature rules, and applies them to solve fractional differential equations with numerical validation.

## Contribution

It develops Jacobi-M"untz functions of two types, establishing their spectral properties and deriving new quadrature rules and projections for fractional differential equations.

## Key findings

- New Jacobi-M"untz functions with orthogonality and recurrence relations
- Derived error bounds for orthogonal projections
- Numerical solutions for fractional differential equations using these functions

## Abstract

This paper presents two new classes of M\"untz functions which are called Jacobi-M\"untz functions of the first and second types. These newly generated functions satisfy in two self-adjoint fractional Sturm-Liouville problems and thus they have some spectral properties such as: orthogonality, completeness, three-term recurrence relations and so on. With respect to these functions two new orthogonal projections and their error bounds are derived. Also, two new M\"untz type quadrature rules are introduced. As two applications of these basis functions some fractional ordinary and partial differential equations are considered and numerical results are given.