Unified predictor-corrector method for fractional differential equations with general kernel functions

Recently, fractional derivatives with respect to general kernel functions have been investigated extensively. Since the classical equidistant partition makes numerical analysis much more complicated, numerical methods for such general fractional differential equations become challenging. In order to address this problem, a nonequidistant partition is adopted for numerical discretization and a predictor-corrector scheme is proposed in space AC delta[a, b]. The coefficients of the rectangle and trapezoidal formulae are derived, respectively. Numerical examples are illustrated which show the efficiency in comparative study of fractional differential equations with different memory effects. This study also provides a numerical tool to determine which fractional derivative is better in real-world applications.