The Numerical Solution of the L-Fractional Abel Differential Equations

In this paper, approximate solutions of the L (Leibniz)-fractional Abel differential equations of the first and second kind are computed by using the shifted Morgan-Voyce (SMV) polynomials. Fractional derivatives are defined in the Leibniz sense. Firstly, the unknown function and its L-fractional derivative are represented by truncated series of SMV polynomials. After applying the collocation procedure, matrix relations are obtained for the unknown function and its derivatives. These relations transform the L-fractional Abel differential equation into a nonlinear system of algebraic equations, which is efficiently solved by using the Newton's method. Finally, five numerical examples are considered to demonstrate validity of the method. The results show excellent agreement with exact solutions, with maximum absolute errors of $10^{-16}$ for $\alpha=1$.Comparisons with existing methods show that the proposed scheme is both accurate and computationally efficient.