# Iranian Journal of Numerical Analysis and Optimization Volume:14 Issue: 1, Winter 2024

• تاریخ انتشار: 1402/10/11
• تعداد عناوین: 8
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• M. Yarahmadi *, S. Yaghobipour Pages 1-19
In this paper, a computational method based on parameterizing state and control variables is presented for solving Stochastic Optimal Control (SOC) problems. By using Chebyshev wavelets with unknown coefficients, state and control variables are parameterized, and then a stochastic optimal control problem is converted to a stochastic optimization problem. The expected cost functional of the resulting stochastic optimization problem is approximated by sample average approximation thereby the problem can be solved by optimization methods more easily. For facilitating and guar-anteeing convergence of the presented method, a new theorem is proved. Finally, the proposed method is implemented based on a newly designed algorithm for solving one of the well-known problems in mathematical fi-nance, the Merton portfolio allocation problem in finite horizon. The simu-lation results illustrate the improvement of the constructed portfolio return.
Keywords: Stochastic optimal control, Chebyshev wavelets, expansion, Optimal asset allocation
• D. Priyadarsini, M. Routaray, P.K. Sahu * Pages 20-43
This article presents a novel numerical approach to the solution of the nonlinear Kawahara equation. The desired approximations are obtained from the combination of Dickson polynomials and Taylor’s expansion. The combined approach is based on Taylor’s expansion for discretizing the time derivative and Dickson polynomials for space derivatives. The problem will be converted into a system of linear algebraic equations for each time step via some suitable collocation points. Error estimation is presented after obtaining the approximate solution. The newly proposed technique is compared with some existing numerical methods to show the method’s applicability, accuracy, and efficacy. Two problems are solved to demon-strate the method’s power and effect, and the results are presented as a table and graphics.
Keywords: Taylor’s expansion, Dickson polynomials, Kawahara equation, Discretization
• M. Amiri, A. Ashrafi * Pages 44-76
Ranking decision making units (DMUs) is an important topic in data en-velopment analysis (DEA). When efficient DMUs or inefficient DMUs have the same efficiency score, the traditional DEA model usually fails to rank all DMUs. For the sake of comparing and improving the discrimination power of DMUs, some proposed approaches use cooperative game theory for rank-ing DMUs. In this paper, communication game theory, which includes a transferable utility cooperative game and an undirected graph describing limited cooperation between players, can be used to rank DMUs. The idea is that the ranking of DMUs can be done by measuring the effect of remov-ing a subset of DMUs on the total share of the remaining DMUs obtained by the reference frontier share model. In the proposed approach, the play-ers are the DMUs, and the characteristic function measures the increase and decrease in the total share of each DMU. The current paper considers the total share for efficient and inefficient DMUs to rank all DMUs. The proposed approach has been tested on several datasets and compared with the results of the previous ranking methods, which sometimes coincide. In the empirical study, a complete ranking of DMUs is useful and reasonable.
Keywords: Data Envelopment Analysis, Communication game, Myerson value, Reference frontier share model
• A. Kumar, S. Gowrisankar * Pages 77-106
In this article, we develop an efficient numerical method for one-dimensional time-delayed singularly perturbed parabolic problems. The proposed nu-merical approach comprises an upwind difference scheme with modified graded mesh in the spatial direction and a backward Euler scheme on uni-form mesh in the temporal direction. In order to capture the local behavior of the solutions, stability and error estimations are obtained with respect to the maximum norm. The proposed numerical method converges uniformly with first-order up to logarithm in the spatial variable and also first-order in the temporal variable. Finally, the outcomes of the numerical experiments are included for two test problems to validate the theoretical findings.
Keywords: Singular perturbation problem, Finite difference methods, Mod-ified graded mesh, Boundary layers, Uniform convergence
• S. Shahbeyk * Pages 107-135
In this paper, we study approximate proper efficient (nondominated and minimal) solutions of vector optimization problems with variable ordering structures (VOSs). In vector optimization with VOS, the partial order-ing cone depends on the elements of the image set. Approximate proper efficient/nondominated/ minimal solutions are defined in different senses (Henig, Benson, and Borwein) for problems with VOSs from new stand-points. The relationships among the introduced notions are studied, and some scalarization approaches are developed to characterize these solutions. These scalarization results based on new functionals defined by elements from the dual cones are given. Moreover, some existing results are ad-dressed.
Keywords: Approximate proper solutions, Variable ordering structure, Scalar-ization, Vector optimization
• M. E.-B. Keddar, O. Belhamiti * Pages 136-171
This study presents a novel approach to understanding the global dynam-ics of COVID-19 transmission with vaccination based on a discrete-time model. We establish biologically meaningful constraints on the model parameters and prove the existence of a disease-free equilibrium and an endemic equilibrium under these constraints, along with their theoretical stabilities. Furthermore, we identify the most sensitive operational param-eters that have a substantial impact on the transmission of the epidemic. Through numerical simulations, we demonstrate the local stability of the two equilibria, depending on the parameter values. Our findings reveal that the discrete-time model is not only dynamically robust but also more realistic than its continuous counterpart under biologically meaningful con-straints. These results provide a foundation for future research in this area and contribute to our understanding of the global dynamics of COVID-19 transmission.
Keywords: Discrete-time epidemic model, Euler Scheme, Local stability, Vaccination reproduction number
• Y.H. Youssri *, A.G. Atta Pages 172-199
Herein, we construct an explicit modal numerical solver based on the spec-tral Petrov–Galerkin method via a specific combination of shifted Cheby-shev polynomial basis for handling the nonlinear time-fractional Burger-type partial differential equation in the Caputo sense. The process reduces the problem to a nonlinear system of algebraic equations. Solving this alge-braic equation system will yield the approximate solution’s unknown coef-ficients. Many relevant properties of Chebyshev polynomials are reported, some connection and linearization formulas are reported and proved, and all elements of the obtained matrices are evaluated neatly. Also, conver-gence and error analyses are established. Various illustrative examples demonstrate the applicability and accuracy of the proposed method and depict the absolute and estimated error figures. Besides, the current ap-proach’s high efficiency is proved by comparing it with other techniques in the literature.
Keywords: Time-fractional Burgers’ equation, Chebyshev polynomials, Petrov–Galerkin method, Convergence Analysis