Construction of Second- and Fourth-Order RK-Type Methods for Solving Fractional Differential Equations of the First Order

In the present work, two distinct Runge–Kutta methods based on the conformable derivative were introduced for solving fractional differential equations of the first order. Both proposed methods were defined as second- and fourth-order and derived from Taylor series expansions and integral quadrature. Illustrative examples were discussed for the validity and efficiency of the methods. The absolute error graphs demonstrated the exceptional efficiency of the proposed methods and their greater accuracy than that of the conformable derivative. Furthermore, some approaches that provide more precise approximations than the fractional Runge–Kutta method were discovered.