This research investigated an optimal control problem formulated using a Hilfer fractional-order differential model to describe the dynamics between tumor cells and the immune system during immuno-chemotherapy. The model emphasized the activation timing of effector cells and their capacity to generate a potent immune response against the tumor. The primary objective was to minimize the costs associated with immuno-chemotherapy while simultaneously reducing the tumor cell population through optimal control strategies. By solving both the state and adjoint equations, the optimal control problem was addressed numerically. Simulation results demonstrated that the proposed immuno-chemotherapy protocol significantly reduced the tumor burden.

In this paper, we used the nonstandard compact finite difference method to numerically solve one-dimensional truncated Bratu-Picard equations and discussed the convergence analysis of the proposed method. Depending on the parameters in the mentioned equation, it may have no solution, one solution, or two solutions; also, it may have infinitely many solutions. The numerical results show that our method covers all mentioned aspects depending on the parameters in the equation.