Numerical Solution of Variable-Order Fractional Differential Equations Using Bernoulli Polynomials

We introduce a new numerical method, based on Bernoulli polynomials, for solving multiterm variable-order fractional differential equations. The derivative was considered in the Caputo sense, while Riemann–Liouville integral operator used to give approximations unknown function and its derivatives. An operational matrix of integration introduced functions. By assuming that solution problem is sufficiently smooth, we approximated given order using polynomials. Then, find some Using these collocation points, reduced system nonlinear algebraic error estimate approximate obtained by proposed method. Finally, five illustrative examples were demonstrate applicability high accuracy technique, comparing our results with ones existing methods literature making clear novelty work. showed method efficient, giving high-accuracy solutions even small number basis functions when not infinitely differentiable, providing better smaller compared state-of-the-art methods.