An approach for approximate solution of fractional-order smoking model with relapse class

In this paper, we develop a fractional-order smoking model by considering relapse class. First, we formulate the model and find the unique positive solution for the proposed model. Then we apply the Grünwald-Letnikov approximation in the place of maintaining a general quadrature formula approach to the Riemann-Liouville integral definition of the fractional derivative. Building on this foundation avoids the need for domain transformations, contour integration or involved theory to compute accurate approximate solutions of fractional-order giving up smoking model. A comparative study between Grünwald-Letnikov method and Runge-Kutta method is presented in the case of integer-order derivative. Finally, we present the obtained results graphically.