# Laguerre Functions and Their Applications to Tempered Fractional Differential Equaitons on Infinite Intervals

## TL;DR

This paper develops Laguerre spectral methods for solving tempered fractional differential equations on infinite intervals, demonstrating their effectiveness through numerical experiments and theoretical properties.

## Contribution

It introduces two types of generalized Laguerre functions tailored for TFDEs and establishes their properties for spectral approximation on infinite domains.

## Key findings

- Spectral scheme efficiently solves TFDEs.
- Properties of Laguerre functions facilitate numerical analysis.
- Numerical experiments confirm scheme's accuracy and efficiency.

## Abstract

Tempered fractional derivatives originated from the tempered fractional diffusion equations (TFDEs) modeled on the whole space R (see [23]). For numerically solving TFDEs, two kinds of generalized Laguerre functions were defined and some important properties were proposed to establish the approximate theory. The related prototype tempered fractional differ- ential problems was proposed and solved as the guidance. TFDEs are numerically solved by two domains Laguerre spectral method and the numerical experiments show some properties of the TFDEs and verify the efficiency of the spectral scheme.