<p>The primary objective of this paper is to introduce operational matrices for the fractional integration of fractional-order hat functions and to apply them to the solution of certain fractional dynamic systems. First, reviewing the definitions of the classic hat functions, we consider fractional-order linear and quadratic hat functions and explore the key properties of these two classes of basis functions. Next, we examine how these basis functions can be used to approximate a given function and evaluate the accuracy of this approximation through a rigorous error analysis. We then propose numerical methods for solving some problems involving fractional derivative operators, which rely on the use of operational matrices for fractional integration associated with both types of basis functions. These matrices, derived through direct computations, simplify the main problems by converting them into some systems of equations with low complexity. Finally, the proposed methods are applied to test problems to demonstrate their effectiveness.</p>