Iranian Journal of Numerical Analysis and Optimization
https://ijnao.um.ac.ir/
Iranian Journal of Numerical Analysis and Optimizationendaily1Sun, 01 Sep 2024 00:00:00 +0330Sun, 01 Sep 2024 00:00:00 +0330Nonpolynomial B-spline collocation method for solving singularly perturbed quasilinear Sobolev equation
https://ijnao.um.ac.ir/article_45054.html
In this paper, a singularly perturbed one-dimensional initial boundary value problem of a quasilinear Sobolev-type equation is presented. The nonlinear term of the problem is linearized by Newton&rsquo;s linearization method. Time derivatives are discretized by implicit Euler&rsquo;s method on nonuniform step size. A uniform trigonometric B-spline collocation method is used to treat the spatial variable. The convergence analysis of the scheme is 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. Results of the scheme are presented in tabular, and the figure indicates the scheme is uniformly convergent and has an initial layer at t = 0.Differential-integral Euler–Lagrange equations
https://ijnao.um.ac.ir/article_45046.html
We study the calculus of variations problem in the presence of a system of differential-integral (D-I) equations. In order to identify the necessary optimality conditions for this problem, we derive the so-called D-I Euler&ndash;Lagrange equations. We also generalize this problem to other cases, such as the case of higher orders, the problem of optimal control, and we derive the so-called D-I Pontryagin equations. In special cases, these formulations lead to classical Euler&ndash;Lagrange equations. To illustrate our results, we provide simple examples and applications such as obtaining the minimumpower for an RLC circuit.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 conduct a comparative analysis with the original imperialist competitive algorithm and a genetic algorithm. The improvement suggested in this study makes 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.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&rsquo; be-havior 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 boy-cott 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 R0. 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 sim-ulations 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.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&ndash;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 conver-gence 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.Uniformly convergent numerical solution for caputo fractional order singularly perturbed delay differential equation using extended cubic B-spline collocation scheme
https://ijnao.um.ac.ir/article_45264.html
This article presents a parameter uniform convergence numerical scheme for solving time fractional order singularly perturbed parabolic convection-diffusion differential equations with a delay. We give a priori bounds on the exact solution and its derivatives obtained through the problem&rsquo;s asymp-totic analysis. The Euler&rsquo;s method on a uniform mesh in the time direction and the extended cubic B-spline method with a fitted operator on a uniform mesh in the spatial direction is used to discretize the problem. The fitting factor is introduced for the term containing the singular perturbation pa-rameter, and it is obtained from the zeroth-order asymptotic expansion of the exact solution. The ordinary B-splines are extended into the extended B-splines. Utilizing the optimization technique, the value of &mu; (free param-eter, when the free parameter &mu; tends to zero the extended cubic B-spline reduced to convectional cubic B-spline functions) is determined. It is also demonstrated that this method is better than some existing methods in the literature.Finite element analysis for microscale heat equation with Neumann boundary conditions
https://ijnao.um.ac.ir/article_45364.html
In this paper, we explore the numerical analysis of the microscale heat equation. We present the characteristics of numerical solutions obtained through both semi- and fully-discrete linear finite element methods. We establish a priori estimates and error bounds for both semi-discrete and fully-discrete finite element approximations. Additionally, the existence and uniqueness of the semi-discrete and fully-discrete finite element ap-proximations have been confirmed. The study explores error bounds in various spaces, comparing the semi-discrete to the exact solutions, the semi-discrete against the fully-discrete solutions, and the fully-discrete solutions with the exact ones. A practical algorithm is introduced to address the sys-tem emerging from the fully-discrete finite element approximation at every time step. Additionally, the paper presents numerical error calculations to further demonstrate and validate the results.Numerical method for the solution of high order Fredholm integro-differential difference equations using Legendre polynomials
https://ijnao.um.ac.ir/article_45461.html
This research paper deals with the numerical method for the solution of high-order Fredholm integro-differential difference equations using Legen-dre polynomials. We obtain the integral form of the problem, which is transformed into a system of algebraic equations using the collocation method. We then solve the algebraic equation using Newton&rsquo;s method. We establish the uniqueness and convergence of the solution. Numerical problems are considered to test the efficiency of the method, which shows that the method competes favorably with the existing methods and, in some cases, approximates the exact solution.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&minus;Landau equations with fractal&minus;fractional (FF) derivatives. To acquire their ap-proximate solutions, a computational strategy is expressed using the FF derivative in the Atangana&minus;Riemann&minus;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 SJPs with unknown coef-ficients 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 numerical computation for solving delay and neutral differential equations based on a new modification to the Legendre wavelet method
https://ijnao.um.ac.ir/article_45278.html
The goal of this study is to use our suggested generalized Legendre wavelet method to solve delay and equations of neutral differential form with pro-portionate delays of different orders. Delay differential equations have some application in the mathematical and physical modelling of real-world prob-lems such as human body control and multibody control systems, electric circuits, dynamical behavior of a system in fluid mechanics, chemical en-gineering, infectious diseases, bacteriophage infection&rsquo;s spread, population dynamics, epidemiology, physiology, immunology, and neural networks. The use of &nbsp;orthonormal polynomials is the key advantage of this method because it reduces computational cost and runtime. Some examples are provided to demonstrate the effectiveness and accuracy of the suggested strategy. The method&rsquo;s accuracy is reported in terms of absolute errors. The numerical findings are compared to other numerical approaches in the literature, particularly the regular Legendre wavelets method, and show that the current method is quite effective in order to solve such sorts of differential equations.Extending quasi-GMRES method to solve generalized Sylvester tensor equations via the Einstein product
https://ijnao.um.ac.ir/article_45377.html
This paper aims to extend a Krylov subspace technique based on an in-complete orthogonalization of Krylov tensors (as a multidimensional exten-sion of the common Krylov vectors) to solve generalized Sylvester tensor equations via the Einstein product. First, we obtain the tensor form of the quasi-GMRES method, and then we lead to the direct variant of the proposed algorithm. This approach has the great advantage that it uses previous data in each iteration and has a low computational cost. More-over, an upper bound for the residual norm of the approximate solution is found. Finally, several experimental problems are given to show the acceptable accuracy and efficiency of the presented method.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&ndash;Borwein method offers efficient step sizes for large-scale un-constrained optimization problems. However, it may not guarantee global convergence for nonquadratic objective functions. Simulated annealing-based on Barzilai&ndash;Borwein (SABB) method addresses this issue by in-corporating 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 sta-bilization 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 demon-strated through numerical experiments on CUTEr benchmark problems.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.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.A space-time least-squares support vector regression scheme for inverse source problem of the time-fractional wave equation
https://ijnao.um.ac.ir/article_45349.html
The inverse problems in various fields of applied sciences and industrial design are concerned with the estimation of parameters that cannot be directly measured. In this work, we present a novel numerical approach for addressing the fractional inverse source problem by a machine learning algorithm and considering the ideas behind the spectral methods. The introduced algorithm utilizes a space-time Galerkin type of least-squares support vector regression (LS-SVR) to approximate the unknown source in a finite-dimensional space, providing a stable and efficient solution. With the proposed machine learning method, we overcome the limitations of classical numerical methods and offer a promising alternative for tackling inverse source problems while avoiding overfitting by carefully selecting regularization parameters. To validate the effectiveness of our approach and illustrate an exponential convergence, we present some test problems along with the corresponding numerical results. The proposed method's superior accuracy compared to the existing methods is also illustrated.A semi-analytic and numerical approach to the fractional differential equations
https://ijnao.um.ac.ir/article_45383.html
A class of linear and nonlinear fractional differential equations (FDEs) in the Caputo sense are considered and studied through two novel techniques called the Homotopy analysis method (HAM) a reliable approach is proposed for solving fractional order nonlinear ordinary differential equations, and the Haar wavelet technique (HWT) is a numerical approach for both integer and non-integer order. Perturbation techniques are widely applied to gain analytic approximations of nonlinear equations. However, perturbation methods are essentially based on small physical parameters (called perturbation quantity), but unfortunately, many nonlinear problems have no such kind of small physical parameters at all but HAM overcomes this and HWM doesn&rsquo;t require any parameters. Due to this we opted HAM and HWM to study FDEs. We have drawn a semi-analytical solution in terms of a series of polynomials and numerical solutions for FDEs. First, we solved the models by HAM by choosing the preferred control parameter. Secondly, HWT is considered, through this technique, the operational matrix of integration is used to convert the given FDEs into a set of algebraic equation system. Four problems are discussed using both techniques. Obtained results are expressed in graphs and tables. Theorems on convergence have been discussed in terms of theorems.Designing a sliding mode controller for a class of multi-controller Covid-19 disease model
https://ijnao.um.ac.ir/article_45417.html
The recent outbreak of the Covid-19 disease has just appeared at the end of 2019&lrm; &lrm;and has now become a global pandemic&lrm;. &lrm;Analysis of mathematical models in the prediction and control of this pandemic helps to make the right decisions about vaccination&lrm;، &lrm;quarantine and other control measures&lrm;. In this article&lrm;, &lrm;the aim is to analyze and review the three control measures of educational campaigns&lrm;, &lrm;social distancing, and treatment control&lrm;, &lrm;that these control measures can reduce the spread of this disease&lrm;. For this purpose&lrm;, &lrm;due to the uncertainty in the model parameters&lrm;, &lrm;a sliding mode control law is used&lrm;. &lrm;Further&lrm;, &lrm;because the model parameters are changing and the upper limit of the parameters that have uncertainty should be known&lrm;. &lrm;Then an adaptive control is used to estimate the switching gain online&lrm;. In addition&lrm;, &lrm;in order to prevent the chattering phenomenon&lrm;, &lrm;the sign function is used in the sliding control law&lrm;. Also&lrm;, &lrm;the obtained properties are expressed and proven analytically&lrm;. Therefore&lrm;, &lrm;initially&lrm;, &lrm;the controller is designed with uncertainty in mind&lrm;. &lrm;After that&lrm;, &lrm;the parameters that have uncertainty in the simulation are obtained by online estimation of the adaptive control&lrm;. The efficiency and performance of the controller in the absence of the certainty of the model parameters is investigated&lrm;, &lrm;and the results show the desired performance of this controller&lrm;. &lrm;Finally&lrm;, &lrm;the performance and efficiency of the controller are evaluated by simulation&lrm;.A fuzzy solution approach to multi-objective fully triangular fuzzy optimization problem
https://ijnao.um.ac.ir/article_45495.html
Numerous optimization problems comprise uncertain data in practical circumstances and such uncertainty can be suitably addressed using the concept of fuzzy logic. This paper proposes a computationally efficient solution methodology to generate a set of fuzzy non-dominated solutions of a fully fuzzy multi-objective linear programming problem which incorporates all its parameters and decision variables expressed in the form of triangular fuzzy numbers. The fuzzy parameters associated with the objective functions are transformed into interval forms by utilizing the fuzzy cuts which subsequently generate the equivalent interval-valued objective functions and the concept of the centroid of triangular fuzzy numbers derives the deterministic form of the constraints.Furthermore, the scalarization process of the weighting sum approach and certain concepts of interval analysis are used to generate the fuzzy non-dominated solutions from which the compromise solution can be determined based on the corresponding real-valued expressions of fuzzy optimal objective values resulting due to the ranking function. Three numerical and one practical problem are solved for illustration and validation of the proposed approach. The computational results are also discussed as compared to some existing methods.Adaptive mesh based Haar wavelet approximation for a singularly perturbed integral boundary problem
https://ijnao.um.ac.ir/article_45503.html
This research presents a non-uniform Haar wavelet approximation of a singularly perturbed convection-diffusion problem with an integral boundary. The problem is discretized by approximating the second derivative of the solution with the help of a non-uniform Haar wavelets basis on an arbitrary non-uniform mesh. To resolve the multiscale nature of the problem, adaptive mesh is generated using the equidistribution principle. This approach allows for the dynamical adjusting of the mesh based on the solution&rsquo;s behavior without requiring any information about the solution. The combination of non-uniform wavelet approximation and the use of adaptive mesh leads to improved accuracy, efficiency, and the ability to handle the multiscale behavior of the solution. On the adaptive mesh rigorous error analysis is performed showing that the proposed method is a second-order parameter uniformly convergent. Numerical stability and computational efficiency are validated in various tables and plots for numerical results obtained by the implementation of two test examples.A Petrov-Galerkin approach for the numerical analysis of soliton and multi-soliton solutions of the Kudryashov-Sinelshchikov equation
https://ijnao.um.ac.ir/article_45541.html
This study delves into the potential polynomial and rational wave solutions of the Kudryashov-Sinelshchikov equation. This equation has multiple applications including the modeling of propagation for nonlinear waves in various physical systems. Through detailed numerical simulations using the finite element approach, we present a set of accurate solitary and soliton solutions for this equation. To validate the effectiveness of our proposed method, we utilize a collocation finite element approach based on quintic B-spline functions. Error norms, including $L_2$ and $L_8$, are employed to assess the precision of our numerical solutions, ensuring their reliability and accuracy. Visual representations, such as graphs derived from tabulated data, offer valuable insights into the dynamic changes of the equation over time or in response to varying parameters. Furthermore, we compute conservation quantities of motion and investigate the stability of our numerical scheme using Von Neumann's theory, providing a comprehensive analysis of the Kudryashov-Sinelshchikov equation and the robustness of our computational approach. The strong alignment between our analytical and numerical results underscores the efficacy of our methodology, which can be extended to tackle more complex nonlinear models with direct relevance to various fields of science and engineering.Mathematical modeling and optimal control of customer's behavior toward e-commerce
https://ijnao.um.ac.ir/article_45542.html
The extensive influence of digital platforms has reshaped societal interactions and daily routines, integrating e-commerce into every aspect of modern life. This evolution not only redefines traditional business models but also fosters global connectivity and economic restructuring. However, despite its critical role in the global economy, e-commerce faces challenges, notably the hesitation of some consumers due to concerns about security and trust. To address this, we propose a novel mathematical model, to examine customer behavior dynamics toward e-commerce, particularly the impact of the refusal behavior. Our study comprehensively examines the characteristics of our mathematical model, conducts a thorough stability analysis, and investigates the parameters sensitivity. Furthermore, control theory has been adopted to optimize the e-commerce adoption using Pontryagin's maximum principle, with numerical simulations to evaluate the effectiveness of our proposed strategies.Modified hat functions: Application in space-time-fractional differential equations with Caputo derivative
https://ijnao.um.ac.ir/article_45556.html
The present article introduces an operational approach based on modified hat functions to solve the space-time-fractional differential equations in the Caputo sense. In this method, the derivative of the unknown function is considered as a linear combination of the modified hat functions. We use Caputo's fractional derivative and Riemann-Liouville integral to approximate the functions in the equation in order to reduce the problem to simple linear equations. Also, we use the operational matrix of the fractional integral for The integral approximation of hat functions is used. The error of the mentioned method is of the order of $O(h^3)$. In addition, we examine several numerical examples to confirm the ability of the proposed approach.A parallel hybrid variable neighborhood descent algorithm for non-linear optimal control problems
https://ijnao.um.ac.ir/article_45587.html
In this paper, a numerical method for solving bounded continuous-time nonlinear optimal control problems (NOCP) that is based on a variable neighborhood descent algorithm (VND) is proposed. First, an improved VND that uses efficient neighborhood interchange, is applied to the discrete form of NOCP. Then, to improve the efficiency of the algorithm for practical and large-scale problems, the parallel processing approach is implemented. It performs the required complex computations in parallel. The resulting parallel algorithm is applied to a benchmark of nine practical problems such as the Van Der Pol Problem (VDP) and Chemical Reactor Problem (CRP). For large-scale problems, the parallel hybrid variable neighborhood descent algorithm (PHVND) is capable of obtaining optimal control values effectively. Our experimentation shows that PHVND outperforms the best-known heuristics in terms of solution quality and computational effort. In addition, computational results indicate that PHVND produces superior results compared to Particle swarm optimization (PSO) or genetic algorithm (GA).A novel mid-point upwind scheme for fractional order singularly perturbed convection-diffusion delay differential equation
https://ijnao.um.ac.ir/article_45588.html
This study presents a numerical approach for solving temporal fractional order singularly perturbed parabolic convection-diffusion differential equations with a delay using a uniformly convergent scheme. We use the asymptotic analysis of the problem to offer a priori bounds on the exact solution and its derivatives. To discretize the problem, we use the implicit Euler technique on a uniform mesh in time and the midpoint upwind finite difference approach on a piece-wise uniform mesh in space. The proposed technique has a nearly first-order uniform convergence order in both spatial and temporal dimensions. To validate the theoretical analysis of the scheme, two numerical test situations for various values of $\epsilon$ are explored.An efficient collocation scheme for new type of variable-order fractional Lane-Emden equation
https://ijnao.um.ac.ir/article_45600.html
The fractional Lane-Emden model illustrates different phenomena in astrophysics and mathematical physics. This paper involves the Vieta-Lucas (Vt-L) bases to solve types of variable-order (V-O) fractional Lane-Emden equations (linear and nonlinear). The operational matrix of V-O fractional derivative is obtained for the Vt-L polynomials. In the established approach, these polynomials are applied to transform the main problem into an algebraic equations system. To indicate the performance and capability of the scheme, a number of examples are presented for various types of V-O fractional Lane-Emden equations. Also, for one example, a comparison is done between the calculated results by our technique with those obtained via the Bernoulli polynomials. Overall, this paper introduces a new methodology for solving variable-order fractional Lane-Emden equations using Vieta-Lucas bases. The derived operational matrix and the transformation to an algebraic equation system offer practical advantages in solving these equations efficiently. The presented examples and comparative analysis highlight the effectiveness and validity of the proposed technique, contributing to the understanding and advancement of fractional Lane-Emden models in astrophysics and mathematical physics.Advanced mathematical modeling and prognostication of regulated spatio-temporal dynamics of monkeypox
https://ijnao.um.ac.ir/article_45677.html
This study explores a continuous spatiotemporal mathematical model to illustrate the dynamics of monkeypox virus spread across various regions, considering both human and animal hosts. We propose a comprehensive strategy that includes awareness campaigns, security measures, and health interventions in areas where the virus is prevalent. The goal is to reduce transmission between humans and animals, thereby decreasing human infections and eradicating the virus in animal populations. Our model, which integrates spatial variables, accurately reflects the geographical spread of the virus and the impact of interventions, followed by the implementation and analysis of an applicable optimal control problem. Optimal control theory methods were applied in this work to demonstrate the existence of optimal control and the necessary conditions for optimality. We conduct numerical simulations using Matlab with the Forward-Backward Sweep Method, revealing the efficiency of strategies focused on protecting vulnerable populations, preventing contact with infected individuals and animals, and promoting the use of quarantine facilities as the most effective means to control the spread of the monkeypox virus. Additionally, the study examines the socio-economic impacts of the virus and the benefits of timely intervention. This approach provides valuable insights for policymakers and public health officials in managing and controlling the spread of monkeypox.Parameter-uniform numerical treatment of singularly perturbed parabolic delay differential equations with non-local boundary conditions
https://ijnao.um.ac.ir/article_45689.html
This paper focuses on solving singularly perturbed parabolic equations of the convection-diffusion type with a large negative spatial shift and an integral boundary condition. A higher-order uniformly convergent numerical approach is proposed that uses Crank-Nicolson and a hybrid finite difference approximation on a piece-wise uniform Shishkin mesh. Simpson's 1/3 integration rule is used to treat the integral boundary condition. The proposed method has been shown to achieve almost second-order uniform convergence. The computational results derived from the numerical experiment are consistent with the theoretical estimates. Furthermore, the method produces a more accurate result than certain other methods in the literature.Effect of demographic stochasticity in the persistence zone of a two patch model with nonlinear harvesting
https://ijnao.um.ac.ir/article_45704.html
In this study, Allee type, single-species (prey), two-patch model with non-linear harvesting rate and species migration across two patches have been developed and analysed. As we all know, the population of any species in an ecosystem is greatly dependent on the carrying capacity of the corresponding ecosystem; the main focus of our work is on how carrying capacity affects system dynamics in the presence and absence of randomness (deterministic and stochastic case, respectively). In deterministic case, we find that the carrying capacity of both patches increases the number of interior equilibrium points, and a maximum of eight interior equilibrium points can be observed. Also, we observe some interesting dynamics, including bi-stability, tri-stability, and catastrophic bifurcations. On the other hand, we use the continuous-time Markov chain (CTMC) modelling approach to construct an equivalent stochastic model of the corresponding deterministic model based on deterministic assumptions. Based on the extinction or persistence of the species, we compare the dynamics of deterministic and stochastic models in order to assess the impact of demographic stochasticity on the population of the species in two patches. The stochastic model shows the possibility of species extinction in a finite amount of time, whereas the deterministic model shows the persistence of the species at the same time, which is the major difference between these two models. We also derive the implicit equation for the expected time needed for species extinction. Finally, a graphic is used to illustrate how the patch's carrying capacity affects the expected time.Optimal control of water pollutant transmission by utilizing a combined Jacobi collocation method and mountain gazelle algorithm
https://ijnao.um.ac.ir/article_45708.html
Water pollution can have many adverse effects on the environment and human health. The study of the transmission of water pollutants over a finite lifespan is carried out using an optimal control problem (OCP), with the system governed by ordinary differential equations. By utilizing the collocation approach, the OCP is transmuted to a non-linear programming problem, and then the mountain Gazelle algorithm (MGA) is applied to determine the optimal control and state solutions.A practical study demonstrated the effect of treatment on reducing water pollutants during a finite time.An efficient hybrid conjugate gradient method for unconstrained optimization and image restoration problems
https://ijnao.um.ac.ir/article_45709.html
The conjugate gradient (CG) method is an optimization technique known for its rapid convergence; it has blossomed into significant developments and applications. Numerous variations of CG methods have emerged to enhance computational efficiency and address real-world challenges. In this work, a novel conjugate gradient method is introduced to solve nonlinear unconstrained optimization problems. Based on the combination of PRP (Polak-Ribi&egrave;re-Polyak), HRM (Hamoda-Rivaie-Mamat) and NMFR (New modified Fletcher-Reeves) algorithms, our method produces a descent direction without depending on any line search. Moreover, it enjoys global convergence under mild assumptions and is applied successfully on various standard test problems as well as image processing. The numerical results indicate that the proposed method outperforms several existing methods in terms of efficiency.