A numerical method using Haar wavelets for solving fractional optimal control problems (FOCPs) is studied. The fractional derivative in these problems is in the Caputo sense. The operational matrix of fractional Riemann–Liouville integration and the direct collocation method are considered. The proposed technique is applied to transform the state and control variables into non-linear programming (NLP) parameters at collocation points. An NLP solver can then be used to solve FOCPs. Illustrative examples are included to demonstrate the validity and applicability of the proposed method.

In this paper, we apply the concept of fractional calculus to study three-dimensional Lotka-Volterra differential equations. We incorporate the Caputo-Fabrizio fractional derivative into this model and investigate the existence of a solution. We discuss the uniqueness of the solution and determine under what conditions the model offers a unique solution. We prove the stability of the nonlinear model and analyse the properties, considering the non-singular kernel of the Caputo-Fabrizio operator. We compare the stability conditions of this system with respect to the Caputo-Fabrizio operator and the Caputo fractional derivative. In addition, we derive a new numerical method based on the Adams-Bashforth scheme. We show that the type of differential operators and the value of orders significantly influence the stability of the Lotka-Volterra system and numerical results demonstrate that different fractional operator derivatives of the nonlinear population model lead to different dynamical behaviors.