A high-order predictor-corrector method for initial value problems with
fractional derivative involving Mittag-Leffler kernel: epidemic model
case study
Abstract
In this paper, we propose a numerical scheme of the predictor-corrector
type for solving nonlinear fractional initial value problems, the chosen
fractional derivative is called the Atangana-Baleanu derivative defined
in Caputo sense (ABC). This proposed method is based on Lagrangian
quadratic polynomials to approximate the nonlinearity implied in the
Volterra integral which is obtained by reducing the given fractional
differential equation via the properties of the ABC-fractional
derivative. Through this technique, we get corrector formula with high
accuracy which is implicit as well as predictor formula which is
explicit and has the same precision order as the corrective formula. On
the other hand, the so-called memory term is computed only once for both
prediction and correction phases, which indicates the low cost of the
proposed method. Also, the error bound of the proposed numerical scheme
is offered. Furthermore, numerical experiments are presented in order to
assess the accuracy of the new method on two differential equations.
Moreover, a case study is considered where the proposed
predictor-corrector scheme is used to obtained approximate solutions of
ABC-fractional generalized SI (susceptible-infectious) epidemic model
for the purpose of analyzing dynamics of the suggested system as well as
demonstrating the effectiveness of the new method to solve systems
dealing with real-world problems.