Numerical solution of distributed-order fractional diffusion equations using quadratic hat functions

Abbasi, Afrooz; Nemati, Somayeh

doi:10.22055/jamm.2026.49875.2378

Numerical solution of distributed-order fractional diffusion equations using quadratic hat functions

Document Type : Original Paper

Authors

Afrooz Abbasi ¹

Somayeh Nemati ²

¹ Department of Applied Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎University of Mazandaran‎, ‎Babolsar‎, ‎Iran

² Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

10.22055/jamm.2026.49875.2378

Abstract

‎In this research‎, ‎quadratic hat functions and their properties are used to solve a class of distributed-order fractional diffusion equations‎. ‎The fractional derivative is considered in the Caputo sense‎. ‎To obtain the numerical solution‎, ‎after applying the second-order spatial integral to both sides of the equation and employing the Gauss-Legendre quadrature rule‎, ‎the problem is transformed into a multi-term time-fractional integro-differential equation‎. ‎Subsequently‎, ‎by introducing an approximation of the first-order time derivative of the unknown function‎, ‎derivatives of various orders of the function are approximated using the Riemann-Liouville operational matrix and the properties of quadratic hat functions‎. ‎By substituting these approximations and applying the initial and boundary conditions‎, ‎a system of algebraic equations is obtained‎. Then, the approximation error is ‎examined. ‎‎Finally‎, ‎numerical examples are provided to evaluate the efficiency and accuracy of the proposed method‎.

Keywords

Distributed-order diffusion equation

Quadratic hat functions

Caputo derivative

Riemann-Liouville integral

Operational matrix