In this paper, we developed a novel numerical method for solving general nonlinear fractional ordinary differential equations (FODEs). First, we transformed the nonlinear FODEs into the equivalent Volterra integral equations. We then developed a time-stepping algorithm for the numerical solution of the Volterra integral equations based on the third-order Taylor expansion for approximating the integrands in the Volterra integral equations on a chosen mesh with the mesh parameter $h$. This approximation led to implicit nonlinear algebraic equations in the unknowns at each given mesh point, and an iterative algorithm based on Newton's method was developed to solve the resulting implicit equations. A convergence analysis of this numerical scheme showed that the error between the exact solution and numerical solution at each mesh point is $\mathcal{O}(h^3)$, independent of the fractional order. Finally, four numerical examples were solved to verify the theoretical results and demonstrate the effectiveness of the proposed method.

In this paper, we proposed a dynamic optimization problem involving a two-stage fractional system subjected to both a terminal state inequality constraint and continuous state inequality constraints in a microbial batch process. The objective function was the productivity of 1,3-propanediol at the terminal time, while the decision variables were the initial concentrations of biomass and glycerol, and the terminal time of the batch process. We first equivalently transformed the problem with free terminal time into one with fixed terminal time in a new time horizon by applying a proposed time-scaling transformation. We then converted the equivalent problem into an optimization problem with only box constraints by using an exact penalty function method. A novel third-order numerical scheme was presented for solving the two-stage fractional system. On this basis, we developed an improved particle swarm optimization algorithm to solve the resulting optimization problem. Finally, numerical results showed that a significant increase in the productivity of 1,3-propanediol at the terminal time was obtained compared with the previously reported results.