<p>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.</p>