An approximate solution for variable-order fractional optimal control problem via Müntz-Legendre wavelets with an application in epidemiology

A class of variable-order fractional optimal control problems (VO-FOCPs) is solved by applying Müntz-Legendre wavelets. Different from classical wavelets (such as Legendre and Chebyshev), the Müntz-Legendre wavelets have an extra parameter representing the fractional order; therefore, they provide more reliable results for certain fractional calculus problems. Using the regularized beta function, the Riemann-Liouville fractional integral operator (RLFIO) of these wavelets is precisely determined. We then transform the given VO-FOCPs into parameter optimization problems that can be easily solved. The convergence analysis and error estimation of the proposed method are provided. Four examples are solved to illustrate the high accuracy of the approach. The method is also applied to a cancer-obesity interaction model to analyze the interactions between the tumor, immune, normal, and fat cells when the chemotherapeutic drugs are injected into a body.