Generalizing Averaging Techniques for Approximating Fractional Differential Equations With Caputo Derivative

This work investigates the concept of numerically approximating fractional differential equations (FDEs) by using function average in an interval. First, the equivalent integral equation is obtained. Then, averaging methods (first-order product integration methods), such as Euler and midpoint, consist in replacing the right-hand side of the FDE by its point evaluation within the interval of integration. In this work, we generalize this idea to leverage on any numerical quadrature rule from classical calculus. Our idea is based on viewing this point evaluation as an average of the function. By definition, this average is a definite integral which can be separately approximated by any classical quadrature rule. A more accurate quadrature rule leads to a more accurate method for the FDE. Here, we use the nonstandard fourth-order trapezoid rule, and this leads to a prediction–correction method. We formulate a modified version of the corrector for the predictor. The convergence of the method is proved. Several numerical experiments are presented to compare the new trapezoid rule with Euler’s and midpoint methods. The results reveal that the new method is more accurate than the Euler and midpoint methods. The conclusion from the work is that a more accurate classical quadrature rule for the averaging leads to a more accurate solution of the FDE.