The paper presents a method for simultaneous computation of eigenfunction and eigenvalue of the stationary Schrödinger equation on a grid, without imposing boundary-value condition. The method is based on the filter operator, which selects the eigenfunction from wave packet at the rate comparable to δ function. The efficacy and reliability of the method are demonstrated by comparing the simulation results with analytical or numerical solutions obtained by using other methods for various boundary-value conditions. It is found that the method is robust, accurate, and reliable. Further prospect of filter method for simulation of the Schrödinger equation in higher-dimensional space will also be highlighted.
In this paper, optimal control of tumor growth model with dendritic cells as immunotherapy is provided. The model equation can be expressed into a nonlinear differential equation system consisting of four compartments namely, tumor cells, CTL cells, helper T cells, and dendritic cells. Dendritic cells as immunotherapy are injected to the body at time t. The aim of this optimal control is to minimize the tumor cells density as well as the cost of dendritic cells to be administered to the body. Optimal control problem is carried out based on Pontryagin's maximum principle and numerical simulation is solved by using Forward-Backward Sweep methods. Simulation results show that control strategy shrinks tumor cells density which is shown by tumor cells density graph that monotonically decreases after applying dendritic cells as immunotherapy.
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