Implicit Numerical Schemes Based on the Lower Incomplete Gamma Function for Solving a Class of Nonlinear Fractional-Ordinary Differential Equation Problems Arising from a Stochastic Process

In this paper, we have developed new implicit numerical schemes based on the incomplete gamma function for a class of nonlinear fractional-order compartmental models arising from a stochastic process. These nonlinear integro-differential equation models have fractional derivatives on the right-hand side of the systems with order \(1-\alpha \) , where \(0< \alpha <1\) , in contrast to conventional fractional order compartmental models, which have fractional derivatives on the left-hand side of the systems. Based on the convergence analysis, it was determined that the implicit schemes’ rates of convergence are \(O(h^{2+\alpha })\) and \(O(h^{1+\alpha })\) , respectively, where h represents the time step size. We provide multiple numerical examples to support our theoretical claims and show the effectiveness of the methods.