Optimal solution of the fractional-order smoking model and its public health implications

This paper proposes a nonlinear smoking model (SM) by means of a system of fractional-order differential equations. The SM is formulated in the sense of the fractional Caputo derivative. The method consists of an optimization based on a new class of basis functions, namely the generalized shifted Legendre polynomials (GSLP), to solve the fractional SM (FSM). The solution is first approximated by the GSLP with unknown coefficients and parameters in the matrix form; afterward, the operational matrices for the fractional derivatives are calculated. This means that after combining the operational matrices and the Lagrange multipliers technique, an optimization method for solving the nonlinear FSM is obtained. The convergence analysis is also proved, while several examples illustrate the applicability the proposed method.