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

2025/07/25 by J. Caballero, Hanna Okrasińska-Płociniczak, Caballero, Josefa +5
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Numerical Analysis (math.NA) #Numerical methods in engineering #Thermoelastic and Magnetoelastic Phenomena

paper · pdf · doi:10.48550/arxiv.2507.19217

openalex publication_date 2025/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

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• Barenblatt profiles for a nonlocal porous media equation
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Related

• Time-fractional porous medium equation: Erdélyi-Kober integral equations, compactly supported solutions, and numerical methods
• Compactly supported solution of the time-fractional porous medium equation on the half-line
• Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type
• On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density
• Non-Local Porous Media Equations with Fractional Time Derivative