<p>Fractional-order basis functions have gained increasing attention for solving fractional differential equations due to their flexibility and effectiveness in capturing nonlocal behaviors. This work presents an explicit expression for the Riemann–Liouville integral of fractional-order Bernoulli wavelet functions and applies it to numerically solve distributed-order fractional differential equations. Traditional basis functions—such as polynomials, piecewise polynomials, sine–cosine, or exponential functions—often fail to approximate solutions accurately when the solutions lack smoothness, which is common in such problems. To address this, we approximate the highest-order derivative of the unknown solution and apply Gauss–Legendre quadrature along with the derived Riemann–Liouville integrals of Bernoulli wavelets at collocation points, thereby reducing the problem to a system of nonlinear algebraic equations. Due to the high computational cost arising from the nested nonlinear structure and integral terms over distributed-order derivatives, this method benefits from high-performance computing resources. Although the final system must be solved sequentially, the small size and repeated structure of each subsystem make the method ideal for accelerated computation, especially for large-scale or real-time simulations. Furthermore, a comprehensive error analysis is provided for the expansion of a given function using fractional-order Bernoulli wavelet functions. A criterion is established for determining the number of basis functions required to achieve a desired error bound in the numerical solution of the problem under study. To validate the effectiveness and practicality of the proposed technique, five numerical examples are presented along with their simulations.</p>