Barenblatt solutions for the time-fractional porous medium equation: approach via integral equations

TL;DR

This paper investigates Barenblatt solutions for the time-fractional porous medium equation using integral equations, proving their existence, properties, and developing convergent numerical schemes for practical computation.

Contribution

It introduces a novel integral equation approach to establish existence and properties of solutions, along with new numerical methods for accurate approximation.

Findings

Proved existence of Barenblatt solutions with fractional time derivatives.

Established key properties like mass conservation and regularity.

Developed convergent numerical schemes validated by examples.

Abstract

This paper explores Barenblatt solutions of the time-fractional porous medium equation, characterized by a Caputo-type time derivative. Employing an integral equation approach, we rigorously prove the existence of these solutions and establish several fundamental properties, including upper and lower estimates, mass conservation, regularity, and monotonicity. To bridge theory and practice, we introduce a family of convergent numerical schemes specifically designed to compute the Barenblatt solutions, ensuring reliable and efficient approximations. The theoretical framework is enriched with various examples that illustrate the concepts and validate the effectiveness of the proposed numerical methods, enhancing the understanding and applicability of our results.