Iranian Journal of Numerical Analysis and Optimization
https://ijnao.um.ac.ir/
Iranian Journal of Numerical Analysis and Optimizationendaily1Sat, 01 Jun 2024 00:00:00 +0330Sat, 01 Jun 2024 00:00:00 +0330Quantum solutions of a nonlinear Schrödinger equation
https://ijnao.um.ac.ir/article_44712.html
In the present paper, we precisely conduct a quantum calculus method for the numerical solutions of PDEs. A nonlinear Schr&ouml;dinger equation is considered. Instead of the known classical discretization methods based on the finite difference scheme, Adomian method, and third modified ver-sions, we consider a discretization scheme leading to subdomains according to q-calculus and provide an approximate solution due to a specific value of the parameter q. Error estimates show that q-calculus may produce effi-cient numerical solutions for PDEs. The q-discretization leads effectively to higher orders of convergence provided with faster algorithms. The numer-ical tests are applied to both propagation and interaction of soliton-type solutions.Discontinuous Galerkin approach for two-parametric convection-diffusion equation with discontinuous source term
https://ijnao.um.ac.ir/article_44733.html
In this article, we explore the discontinuous Galerkin finite element method for two-parametric singularly perturbed convection-diffusion problems with a discontinuous source term. Due to the discontinuity in the source term, the problem typically shows a weak interior layer. Also, the presence of multiple perturbation parameters in the problem causes boundary layers on both sides of the boundary. In this work, we develop the nonsymmetric discontinuous Galerkin finite element method with interior penalties to handle the layer phenomenon. With the use of a typical Shishkin mesh, the domain is discretized, and a uniform error estimate is obtained. Numerical experiments are conducted to validate the theoretical conclusions.High order second derivative multistep collocation methods for ordinary differential equations
https://ijnao.um.ac.ir/article_44807.html
In this paper, we introduce second derivative multistep collocation meth-ods for the numerical integration of ordinary differential equations (ODEs). These methods combine the concepts of both multistep methods and col-location methods, using second derivative of the solution in the collocation points, to achieve an accurate and efficient solution with strong stability properties, that is, A-stability for ODEs. Using the second-order deriva-tives leads to high order of convergency in the proposed methods. These methods approximate the ODE solution by using the numerical solution in some points in the r previous steps and by matching the function values and its derivatives at a set of collocation methods. Also, these methods utilize information from the second derivative of the solution in the colloca-tion methods. We present the construction of the technique and discuss the analysis of the order of accuracy and linear stability properties. Finally, some numerical results are provided to confirm the theoretical expecta-tions. A stiff system of ODEs, the Robertson chemical kinetics problem, and the two-body Pleiades problem are the case studies for comparing the efficiency of the proposed methods with existing methods.Jarratt and Jarratt-variant families of iterative schemes for scalar and system of nonlinear equations
https://ijnao.um.ac.ir/article_44931.html
This manuscript puts forward two new generalized families of Jarratt&rsquo;s iterative schemes for deciding the solution of scalar and systems of non-linear equations. The schemes involve weight functions that are based on bi-variate rational approximation polynomial of degree two in both its numerator and denominator. The convergence study conducted on the schemes, indicated that they have convergence order (CO) four in scalar space and retain the same number of CO in vector space. The numerical experiments conducted on the schemes when used to decide the solutions of some real-life nonlinear models show that they are good challengers of some well-known and robust existing iterative schemes.An innovative particle physics optimization algorithm for efficient test case minimization in software testing
https://ijnao.um.ac.ir/article_44767.html
Software testing is a crucial step in the development of software that guar-antees the dependability and quality of software products. A crucial step in software testing is test case minimization, which seeks to minimize the number of test cases while ensuring maximum coverage of the system being tested. It is observed that the existing algorithms for test case minimization still suffer in efficiency and precision. This paper proposes a new optimiza-tion algorithm for efficient test case minimization in software testing. The proposed algorithm is designed on the base parameters of the metaheuristic algorithms, inspired by scientific principles. We evaluate the performance of the proposed algorithm on a benchmark suite of test cases from the literature. Our experimental results show that the proposed algorithm is highly effective in reducing the number of test cases while maintaining high coverage of the system under test. The algorithm outperforms the existing optimization algorithms in terms of efficiency and accuracy. We also con-duct a sensitivity analysis to investigate the effect of different parameters on the performance of the proposed algorithm. The sensitivity analysis results show that the performance of the algorithm is robust to changes in the parameter values. The proposed algorithm can help software testers reduce the time and effort required for testing while ensuring maximum coverage of the system under test.Dynamics of Cholera Pathogen Carriers and Effect of Hygiene Consciousness in Cholera Outbreaks
https://ijnao.um.ac.ir/article_44923.html
We derive a deterministic mathematical model that scrutinizes the dy-namics of cholera pathogen carriers and the hygiene consciousness of in-dividuals, before the illness, during its prevalence, and after the disease&rsquo;s outbreaks. The dynamics can effectively help in curtailing the disease, but its effects had less coverage in the literature. Boundedness of the solu-tion of the model, its existence, and uniqueness are ascertained. Effects of cholera pathogen carriers and hygiene consciousness of individuals in controlling the disease or allowing its further spread are analyzed. The differential transformation method is used to obtain series solutions of the differential equations that make the system of the model. Simulations of the series solutions of the model are carried out and displayed in graphs. The dynamics of the concerned state variables and parameters in the model are interpreted via the obtained graphs. It is observed that higher hygiene consciousness of individuals can drastically reduce catching cholera disease at onset and further spread of its infections in the population, this in turn, shortens the period of cholera epidemic.Differential-integral Euler-Lagrange equations
https://ijnao.um.ac.ir/article_45046.html
In this paper, we will study the calculus of variations problem in the presence of a system of differential-integral equations. In order to identify the necessary optimality conditions for this problem, we will derive the so-called differential-integral(D-I) Euler-Lagrange equations. We will also generalize this problem to other cases, such as the case of higher orders, and the problem of optimal control, and we will derive the so-called (D-I) Pontryagin equations. In special cases, these formulations lead to classical Euler-Lagrange equations. To illustrate our results, we will provide simple examples and applications such as obtaining the minimum power for an RLC circuit.Drug release from a two-layer stent coating considering the viscoelastic property of the arterial wall: A mathematical and numerical study
https://ijnao.um.ac.ir/article_44963.html
Atherosclerosis is one of the most common diseases in the world. Med-ication with metal stents plays an important role in treating this disease. There are many models for delivering drugs from stents to the arterial wall. This paper presents a model that describes drug delivery from the stent coating layers to the arterial wall tissue. This model complements the previous models by considering the mec hanical properties of the arte-rial wall tissue, which changes due to atherosclerosis and improves results for designing stents. The stability behavior of the model is analyzed, and a number of numerical results are provided with explanations. A compar-ison between numerical and experimental results, which examine a more accurate match between the in vivo and in vitro, is shown.Non-polynomial B-spline collocation method for solving singularly perturbed quasilinear Sobolev equation
https://ijnao.um.ac.ir/article_45054.html
In this paper, singularly perturbed one-dimensional initial boundary value problem of quasilinear Sobolev type equation is presented. The non-linear term of the problem is linearized by Newton's linearization method. Time derivatives is discretized by implicit Euler's method on non-uniform step size. A uniform trigonometric B-spline collocation method is used to treat the spatial variable. The convergence analysis of the scheme proved and the accuracy of the method is of order two in space and order one in time direction respectively. To test the efficiency of the method a model example is demonstrated. The results of the scheme is presented in tabular and figure indicates the scheme is uniformly convergent and has initial layer at $t=0$.Mathematical analysis and forecasting of controlled Spatio-temporal dynamics of the EG.5 Virus
https://ijnao.um.ac.ir/article_44943.html
In this article, we propose a mathematical approach that connects an innovative spatio-temporal model to the problem of the EG.5 variant of COVID-19 in a human population. We demonstrate the existence and uniqueness of the global positive solution for our suggested system. The implementation and analysis of an applicable optimal control issue are as follows. The methods of optimal control theory are applied in this work to demonstrate the existence of optimal control, and with necessary op-timality conditions, we discover the explicit expression of optimal control that minimizes the negative impacts of this infectious disease on countries. We provide numerical simulations at the conclusion to demonstrate the efficacy of our chosen strategy.Highly accurate collocation methodology for solving the generalized Burgers-Fisher’s equation
https://ijnao.um.ac.ir/article_45096.html
An improvised collocation scheme is applied for the numerical treatment of the nonlinear generalized Burgers-Fisher&rsquo;s (gBF) equation using splines of degree three. In the proposed methodology, some subsequent rectifications are done in the spline interpolant, which resulted in the magnification of the order of convergence along the space direction. A finite difference approach is followed to integrate the time direction. Von Neumann methodology is opted to discuss the stability of the method. The error bounds and convergence study show that the technique has (s4+∆t2) order of convergence. The correspondence between the approximate and analytical solutions is shown by graphs, plotted using MATLAB, and by evaluating absolute error.Optimizing natural gas liquids (NGL) production process: A multi-objective approach for energy-efficient operations using genetic algorithm and artificial neural networks
https://ijnao.um.ac.ir/article_45030.html
There are various techniques for separating natural gas liquid (NGL) from natural gas, one of which is refrigeration. In this method, the temper-ature is reduced in the dew point adjustment stage to condense the NGLs. The purpose of this paper is to introduce a methodology for optimizing the NGLs production process by determining the optimal values for specific set-points such as temperature and pressure in various vessels and equip-ment. The methodology also focuses on minimizing energy consumption during the NGL production process. To do this, this research defines a multi-objective problem and presents a hybrid algorithm, including a ge-netic algorithm (NSGA II) and artificial neural network (ANN) system. We solve the defined multi-objective problem using NSGA II. In order to de-sign a tool that is a decision-helper for selecting the appropriate set-points, the ability of the ANN algorithm along with multi-objective optimization is evaluated. We implement our proposed algorithm in an Iranian chemical factory, specifically the NGL plant, which separates NGL from natural gas, as a case study for this article. Finally, we demonstrate the effectiveness of our proposed algorithm using the nonparametric statistical Kruskal&ndash;Wallis test.Stability analysis and optimal strategies for controlling a boycotting behavior of a commercial product.
https://ijnao.um.ac.ir/article_45134.html
In this work, we propose a mathematical model that describes citizens' behavior toward a product, where individuals are generally divided into three main categories: potential consumers, boycotters who abstain from it for various reasons, and actual consumers. Therefore, our work contributes to understanding product boycott behavior and the factors influencing this phenomenon. Additionally, it proposes optimal strategies to control boycott behavior and limit its spread, thus protecting product marketing and encouraging consumer reuse. We use mathematical theoretical analysis to study the local and global stability, as well as sensitivity analysis to identify parameters with a high impact on the reproduction number $R_{0}$. Subsequently, we formulate an optimal control problem aimed at minimizing the number of boycotters and maximizing consumer participation. Pontryagin&rsquo;s maximum principle is employed to characterize the optimal controls. Finally, numerical simulations conducted using Matlab confirm our theoretical results, with a specific application to the case of the boycott of Centrale Danone by several Moroccan citizens in April 2018.A new bi-level data envelopment analysis model to evaluate the Human Development Index
https://ijnao.um.ac.ir/article_44952.html
In the 1990s, the united nations development programme (UNDP) in-troduced the human development index (HDI) to determine the develop-ment degrees of countries. One deficiency in the HDI calculation is the use of equal weights for its sub-indicators. Many scholars have tried to solve this problem using a data envelopment analysis (DEA) method, particu-larly the one enhanced by weight restrictions. Indeed no specific methodhas been yet suggested to determine the parameters of the weight restric-tions. In this paper, we use four DEA/benefit of the doubt (BoD) models enriched by the assurance regions type I (AR-I) constraints to assess human development; we aim to objectively determine the AR-I bounds. Therefore, we consider a basis as the accepted human development values and propose a bi-level optimization problem to extract the AR-I bounds in such a way that the efficiency scores are almost the same as the basic values. On the other hand, the HDI is a globally accepted index that shows small changes year by year. So, if the UNDP decides to apply a BoD model for calculat-ing the HDI instead of the traditional method, then it is better than the scores obtained by the BoD model, showing small changes in comparison with the HDI, at least in the first few years. Therefore, the HDI values are considered as the basis. Moreover, the objectively achieved AR-I bounds provide us with an insight into the way the sub-indicators affect the de-velopment scores. The bounds can be modified by the experts opinions, in the future.An improved imperialist competitive algorithm for solving an inverse form of the Huxley equation
https://ijnao.um.ac.ir/article_45160.html
In this paper, we present an improved imperialist competitive algorithm for solving an inverse form of the Huxley equation, which is a nonlinear partial differential equation. To show the effectiveness of our proposed algorithm, we conducted a comparative analysis with the original imperialist competitive algorithm and a genetic algorithm. The improvement suggested in this study made the original imperialist competitive algorithm a more powerful method for function approximation. The numerical results show that the improved imperialist competitive algorithm is an efficient algorithm for determining the unknown boundary conditions of the Huxley equation and solving the inverse form of nonlinear partial differential equations.Modeling individual mobility’s impact on COVID-19 transmission: Insights from a two-patch SEIR-V approach
https://ijnao.um.ac.ir/article_45031.html
This research explores the influence of individual mobility on COVID-19 transmission, utilizing a temporal mathematical model to clarify disease spread and vaccination dynamics across diverse regions. Employing a com-putationally efficient two-patch configuration that emphasizes regional in-teractions, our study aims to guide optimal disease control strategies. The introduced SEIR-V model with a two-patch setup estimates the vaccination reproduction number, Rv, while equilibrium points and system stability are identified. Visualizations from numerical simulations and sensitivity analyses illustrate key parameters affecting the vaccination reproduction number and COVID-19 control measures. Our findings underscore system responsiveness, emphasizing the intricate relationship between Rv , migra-tion rates, and disease prevalence.Global convergence of new conjugate gradient methods with application in conditional model regression function
https://ijnao.um.ac.ir/article_45163.html
The conjugate gradient method is one of the most important ideas in scientific computing, it is applied to solving linear systems of equations and nonlinear optimization problems. In this paper, based on a variant of the Hestenes-Stiefel (HS) method and the Polak-Ribiere-Polyak (PRP) method, two modified CG methods ( named ` MHS&lowast; and MPRP&lowast; ) are presented and analyzed. The search direction of the presented methods fulfills the sufficient descent condition at each iteration. We establish the global convergence of the proposed algorithms under normal assumptions and strong Wolfe line search. Preliminary elementary numerical experiment results are presented, demonstrating the promise and the effectiveness of the proposed methods. Finally, the proposed methods were further extended to solve the problem of the conditional model regression function.Goursat problem in Hyperbolic partial differential equations with variable coefficients solved by Taylor collocation method
https://ijnao.um.ac.ir/article_45057.html
The hyperbolic partial differential equation (PDE) has important practical uses in science and engineering. This article provides an estimate for solving the Goursat problem in hyperbolic linear PDEs with variable coefficients. The Goursat PDE is transformed into a second kind of linear Volterra in-tegral equation. A convergent algorithm that employs Taylor polynomials is created to generate a collocation solution, and the error using the maxi-mum norm is estimated. The paper includes numerical examples to prove the method&rsquo;s effectiveness and precision.A pseudo−operational collocation method for optimal control problems of fractal−fractional nonlinear Ginzburg−Landau equation
https://ijnao.um.ac.ir/article_45180.html
The presented work introduces a new class of nonlinear optimal control problems in two dimensions whose constraints are nonlinear Ginzburg$-$Landau equations with fractal$-$fractional (FF) derivatives. To acquire their approximate solutions, a computational strategy is expressed using the FF derivative in the Atangana$-$Riemann$-$Liouville (A-R-L) concept with the Mittage-Leffler kernel. The mentioned scheme utilizes the shifted Jacobi polynomials (SJPs) and their operational matrices of fractional and FF derivatives. A method based on the derivative operational matrices of SJR and collocation scheme Is suggested and employed to reduce the problem into solving a system of algebraic equations. We approximate state and control functions of the variables derived from &nbsp;SJPs with unknown coefficients into the objective function, the dynamic system, and the initial and Dirichlet boundary conditions. The effectiveness and efficiency of the suggested approach are investigated through the different types of test problems.A stabilized simulated annealing based Barzilai-Borwein method for the solution of unconstrained optimization problems
https://ijnao.um.ac.ir/article_45184.html
The Barzilai-Borwein (BB) method offers efficient step sizes for large-scale unconstrained optimization problems. However, it may not guarantee global convergence for non-quadratic objective functions. The simulated annealing-based Barzilai-Borwein (SABB) method addresses this issue by incorporating a simulated annealing rule. This work proposes a novel step-size strategy for the SABB method, referred to as the SABBm method. Furthermore, we introduce two stabilized variants: SABBstab and SABBmstab. SABBstab combines a simulated annealing rule with a stabilization step to ensure convergence. SABBmstab builds upon SABBstab, incorporating the modified step size derived from the SABBm method. The effectiveness and competitiveness of the proposed methods are demonstrated through numerical experiments on CUTEr benchmark problems.A numerical solution of parabolic quasi variational inquality non-linear using Newton-Multigrid method
https://ijnao.um.ac.ir/article_45195.html
In this article, we apply three numerical methods to study the uniform convergence of the Newton-Multigrid method for parabolic quasi-variational inequalities with a non-linear right-hand side. To discretize the problem, we utilize a finite element method for the operator and Euler scheme for the time. To obtain the system discretization of the problem, we reformulate the parabolic quasi-variational inequality as a Hamilton-Jacobi-Bellman equation. For linearizing the problem on the coarse grid, we employ Newton's method as an external iteration to obtain the Jacobian system. On the smooth grid, we apply the multi-grid method as an interior iteration of the Jacobian system. Finally, we provide proof of the uniform convergence of the Newton-Multigrid method for parabolic quasi-variational inequalities with a nonlinear right hand, by giving a numerical example of this problem.Designing the sinc neural networks to solve the fractional optimal control problem
https://ijnao.um.ac.ir/article_45196.html
Sinc numerical methods are essential approaches to solving nonlinear problems. In this work, based on this method, the sinc neural networks (SNNs) are designed and applied to solve the fractional optimal control problem (FOCP) in the sense of the Riemann&ndash;Liouville (RL) derivative. To solve the FOCP, we first approximate the RL derivative using Grunwald&ndash;Letnikov (GL) operators. Then, according to Pontryagin&rsquo;s minimum principle (PMP) for FOCP and using an error function, we construct an unconstrained minimization problem. We approximate the solution of the ordinary differential equation obtained from the Hamiltonian condition using the sinc neural network. Simulation results show the efficiencies of the proposed approach.Smith chart-based particle swarm optimization algorithm for multi-objective engineering problems
https://ijnao.um.ac.ir/article_45253.html
Particle swarm optimization (PSO) is a widely recognized bio-inspired algorithm for systematically exploring solution spaces and iteratively identifying optimal points. Through updating local and global best solutions, PSO effectively explores the search process, enabling the discovery of the most advantageous outcomes. This study proposes a novel Smith chart-based particle swarm optimization (SC-PSO) to solve convex and non-convex multi-objective engineering problems by representing complex plane values in a polar coordinate system. The main contribution of this paper lies in the utilization of the Smith chart&rsquo;s impedance and admittance circles to dynamically update the location of each particle, thereby effectively determining the local best particle. The proposed method is applied to three test functions with different behaviors, namely concave, convex, non-continuous, and non-convex, and performance parameters are examined. The simulation results show that the proposed strategy offers successful convergence performance for multi-objective optimization applications and meets performance expectations with a well-distributed solution set.