Ferdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 320240901Nonpolynomial B-spline collocation method for solving singularly perturbed quasilinear Sobolev equation6386614505410.22067/ijnao.2024.85929.1363ENF. Edosa MergaDepartment of Mathematics, Jimma University, Jimma, Oromia, Ethiopia.0000-0002-8737-8684G. File DuressaDepartment of Mathematics, Jimma University, Jimma, Oromia, Ethiopia.0000-0003-1889-4690Journal Article20231217In 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’s linearization method. Time derivatives are discretized by implicit Euler’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.https://ijnao.um.ac.ir/article_45054_0a097eb4b8155b6a27a80789c0ea9231.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 320240901Differential-integral Euler–Lagrange equations6626804504610.22067/ijnao.2024.86104.1367ENMohammedd ShehataDepartment of Basic Science, Bilbeis Higher Institute for Engineering, Sharqia, Egypt.Journal Article20231228We 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–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–Lagrange equations. To illustrate our results, we provide simple examples and applications such as obtaining the minimum<br />power for an RLC circuit.https://ijnao.um.ac.ir/article_45046_cad7d6a95b26125cd6e798e8a0c566e3.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 320240901An improved imperialist competitive algorithm for solving an inverse form of the Huxley equation6817074516010.22067/ijnao.2024.86692.1384ENH. Dana MazraehDepartment of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, G.C. Tehran, Iran.K. ParandDepartment of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, G.C. Tehran, Iran.Department of Cognitive Modeling, Institute for Cognitive and Brain Sciences, Shahid Beheshti University, G.C. Tehran, Iran.H. FarahaniDepartment of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, G.C. Tehran, Iran.S.R. KheradpishehDepartment of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, G.C. Tehran, Iran.Journal Article20240204In 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.https://ijnao.um.ac.ir/article_45160_3412d03ed8e5684be0c07b2fa139fa51.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 320240901Stability analysis and optimal strategies for controlling a boycotting behavior of a commercial product7087354513410.22067/ijnao.2024.86892.1394ENO. AarabateLaboratory of Fundamental Mathematics and their Applications, Department of Mathematics, Faculty of Sciences, University of Chouaib Doukkali, El jadida, Morocco.0009-0005-3645-0307S. BelhdidLaboratory of Fundamental Mathematics and their Applications, Department of Mathematics, Faculty of Sciences, University of Chouaib Doukkali, El jadida, Morocco.0009-0002-3532-9028O. BalatifLaboratory of Fundamental Mathematics and their Applications, Department of Mathematics, Faculty of Sciences, University of Chouaib Doukkali, El jadida, Morocco.0000-0003-1887-5350Journal Article20240216In this work, we propose a mathematical model that describes citizens’ 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’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.https://ijnao.um.ac.ir/article_45134_cba6cf01762c588d2c25ea651df33fc4.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 320240901Highly accurate collocation methodology for solving the generalized Burgers–Fisher’s equation7367614509610.22067/ijnao.2024.86994.1398ENS. ShalluDepartment of Mathematics, Punjab Engineering College (Deemed to be University), Chandigarh, 160012, India.0000-0002-9797-3372V.K. KukrejaDepartment of Mathematics, SLIET Longowal 148106 (Punjab) India.Journal Article20240222An improvised collocation scheme is applied for the numerical treatment of the nonlinear generalized Burgers–Fisher’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.https://ijnao.um.ac.ir/article_45096_4897bc30a27e8f8294d593fda7004627.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 320240901Uniformly convergent numerical solution for caputo fractional order singularly perturbed delay differential equation using extended cubic B-spline collocation scheme7627954526410.22067/ijnao.2024.86894.1393ENN.A. EndrieDepartment of Mathematics,College of Natural and Computational Science, Arba Minch University, Arba Minch, Ethiopia.0009-0003-1158-0485G.F. DuressaDepartment of Mathematics, College of Natural and Computational Science, Jimma University, Jimma, Ethiopia.0000-0003-1889-4690Journal Article20240216This 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’s asymp-totic analysis. The Euler’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 μ (free param-eter, when the free parameter μ 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.https://ijnao.um.ac.ir/article_45264_1afaefb8fb8f14264e9c06ff1566c9e4.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 320240901Finite element analysis for microscale heat equation with Neumann boundary conditions7968324536410.22067/ijnao.2024.87084.1403ENM.H. HashimDepartment of Mathematics, College of Sciences, University of Basrah, Basrah, Iraq.0009-0002-8900-2987A.J. HarfashDepartment of Mathematics, College of Sciences, University of Basrah, Basrah, Iraq.Journal Article20240229In 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.https://ijnao.um.ac.ir/article_45364_8a6d2ea6328851c555f7eb35e121d9c4.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 320240901Numerical method for the solution of high order Fredholm integro-differential difference equations using Legendre polynomials8338744546110.22067/ijnao.2024.87599.1425ENP.T. PantuvoDepartment of Mathematics and Statistics, Federal University Wukari, Taraba State, Nigeria.G. AjileyeDepartment of Mathematics and Statistics, Federal University Wukari, Taraba State, Nigeria.0000-0002-4161-686xR. TaparkiDepartment of Mathematical Sciences, Taraba State University, Jalingo, Taraba State, Nigeria.O.O. AdurojaDepartment of Mathematics, University of Ilesa, Ilesa, Osun State, Nigeria.0000-0001-6767-3953Journal Article20240413This 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’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.https://ijnao.um.ac.ir/article_45461_c4d38138721dbb79021dbd7ef60dde9a.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 320240901A pseudo−operational collocation method for optimal control problems of fractal−fractional nonlinear Ginzburg−Landau equation8758994518010.22067/ijnao.2024.86362.1375ENT. ShojaeizadehDepartment of Mathematics, Qom Branch, Islamic Azad University, Qom, Iran.E. Golpar-RabokyDepartment of Mathematics, University of Qom, Qom, Iran.Parisa RahimkhaniFaculty of Science, Mahallat Institute of Higher Education, Mahallat, Iran.Journal Article20240112The 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 ap-proximate 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 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.https://ijnao.um.ac.ir/article_45180_33ed19b82c9cb25190682917ab534342.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 320240901A numerical computation for solving delay and neutral differential equations based on a new modification to the Legendre wavelet method9009374527810.22067/ijnao.2024.87373.1412ENN.M. El-ShazlyMathematics and Computer Science Department, Faculty of Science, Menoufia University, Menoufia, Egypt.M.A RamadanMathematics and Computer Science Department, Faculty of Science, Menoufia University, Menoufia, Egypt.Journal Article20240323The 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’s spread, population dynamics, epidemiology, physiology, immunology, and neural networks. The use of 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’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.https://ijnao.um.ac.ir/article_45278_1e11f7507115107a4e2d9bd90a258b47.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 320240901Extending quasi-GMRES method to solve generalized Sylvester tensor equations via the Einstein product9389694537710.22067/ijnao.2024.87481.1418ENM.M. IzadkhahDepartment of Computer Science, Faculty of Computer and Industrial Engineering, Birjand University of Technology, Birjand, Iran.0000-0002-6728-5528Journal Article20240404This 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.https://ijnao.um.ac.ir/article_45377_3c2791321f03dfc950f6c6ddebb9cf9a.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 320240901A stabilized simulated annealing-based Barzilai–Borwein method for the solution of unconstrained optimization problems9709904518410.22067/ijnao.2024.86481.1379ENH. SharmaDepartment of Mathematics, International Institute of Information Technology, Bhubaneswar, Odisha, India, 751029.R.K. NayakDepartment of Mathematics, International Institute of Information Technology, Bhubaneswar, Odisha, India, 751029.0000-0002-4163-4275Journal Article20240119The Barzilai–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–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.https://ijnao.um.ac.ir/article_45184_c0e62196daf3c45270f4f0dba1b54cf0.pdf