Ferdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 420241201A numerical solution of parabolic quasi-variational inequality nonlinear using Newton-multigrid method99110154519510.22067/ijnao.2024.86954.1395ENM. BahiDepartment of Mathematics, Faculty of exact Sciences, University of EL-OUED, Algeria.0009-0008-1668-8803M. BeggasDepartment of Mathematics, Faculty of exact Sciences, University of EL-OUED, Algeria.N. NesbaDepartment of Mathematics, Faculty of exact Sciences, University of EL-OUED, Algeria.A. ImtiazInstitute of Informatics and Computing in Energy (IICE), Universiti Tenaga Nasional, Kajang, Selangor, Malaysia.Journal Article20240220In this article, we apply three numerical methods to study the L∞-convergence of the Newton-multigrid method for parabolic quasi-variational inequalities with a nonlinear 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 interior iteration of the Jacobian system. On the smooth grid, we apply the multigrid method as an interior iteration on the Jacobian system. Finally, we provide a proof for the L∞-convergence of the Newton-multigrid method for parabolic quasi-variational inequalities with a nonlinear right-hand, by giving a numerical example for this problem. https://ijnao.um.ac.ir/article_45195_55c07fc9e1fe5b124502eadae7defc89.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 420241201Designing the sinc neural networks to solve the fractional optimal control problem101610364519610.22067/ijnao.2024.86494.1380ENR. Heydari DastjerdiDepartment of Mathematics, Payame Noor University, Tehran, Iran.0000-0001-8996-5068G. AhmadiDepartment of Mathematics, Payame Noor University, Tehran, Iran.0000-0003-1331-7253Journal Article20240120Sinc numerical methods are essential approaches for 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–Liouville (RL) derivative. To solve the FOCP, we first approximate the RL derivative using Grunwald–Letnikov operators. Then, according to Pontryagin’s minimum principle 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 SNN. Simulation results show the efficiencies of the proposed approach. https://ijnao.um.ac.ir/article_45196_621d8d6516c4ea0aff9c8a92689768ed.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 420241201A space-time least-squares support vector regression scheme for inverse source problem of the time-fractional wave equation103710684534910.22067/ijnao.2024.86883.1392ENA. MohammadiDepartment of Mathematics, Shahed University, Tehran, Iran.A. Tari MarzabadDepartment of Mathematics, Shahed University, Tehran, Iran.Journal Article20240215The 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 to approximate the unknown source in a finitedimensional 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. https://ijnao.um.ac.ir/article_45349_2601aae85fb9232901031fc13bfae522.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 420241201A semi-analytic and numerical approach to the fractional differential equations106911054538310.22067/ijnao.2024.85120.1331ENK.N. SachinDepartment of Mathematics, Bangalore University, Bengaluru-560056, India.K. SugunthaDepartment of Mathematics, Bangalore University, Bengaluru-560056, India.S. KumbinarasaiahDepartment of Mathematics, Bangalore University, Bengaluru-560056, India.0000-0001-8942-7892Journal Article20231031A class of linear and nonlinear fractional differential equations (FDEs) in the Caputo sense is 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 noninteger orders. 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. HAM overcomes this, and HWT does not require any parameters. Due to this, we opt for HAM and HWT to study FDEs. We have drawn a semi-analytical solution in terms of a series of polynomials and numerical solutions for FDEs. First, we solve the models by HAM by choosing the preferred control parameter. Second, HWT is considered. Through this technique, the operational matrix of integration is used to convert the given FDEs into a set of algebraic equation systems. Four problems are discussed using both techniques. Obtained results are expressed in graphs and tables. Results on convergence have been discussed in terms of theorems. https://ijnao.um.ac.ir/article_45383_7ad12a85ead14ee04dcb1072361bedc6.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 420241201A fuzzy solution approach to multi-objective fully triangular fuzzy optimization problem110611394549510.22067/ijnao.2024.87836.1434ENN. SwainDepartment of Mathematics, School of Applied Sciences, KIIT Deemed to be University, Bhubaneswar, 751024, Odisha, India.S. MaharanaDepartment of Mathematics, School of Applied Sciences, KIIT Deemed to be University, Bhubaneswar, 751024, Odisha, India.S. NayakDepartment of Mathematics, School of Applied Sciences, KIIT Deemed to be University, Bhubaneswar, 751024, Odisha, India.0000-0002-7340-9574Journal Article20240429Numerous 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 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 generates the equivalent interval valued objective functions. The concept of centroid of triangular fuzzy numbers derives the deterministic form of the constraints. Furthermore, the scalarization process of 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 resulted due to the ranking function. Three numerical problems 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. https://ijnao.um.ac.ir/article_45495_bea34233ad53e99dc9c192dfafd5fdab.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 420241201Adaptive mesh based Haar wavelet approximation for a singularly perturbed integral boundary problem114011674550310.22067/ijnao.2024.87550.1421ENP. ShuklaDepartment of Engineering Sciences, Indian Institute of Information Technology
and Management Gwalior, Gwalior, Madhya Pradesh, 474015, India.S. SainiDepartment of Mathematics, Vellore Institute of Technology, Vellore, Tamilnadu, 632014, India.0000-0002-5664-0045V. DeviDepartment of Mathematics, Bhakt Darshan Govt. P.G. College, Pauri Garhwal, Uttrakhand, 246193, India.Journal Article20240409This research presents a nonuniform 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 nonuniform Haar wavelets basis on an arbitrary nonuniform mesh. To resolve the multiscale nature of the problem, adaptive mesh is generated using the equidistribution principle. This approach allows for the dynamical adjustment of the mesh based on the solution’s behavior without requiring any information about the solution. The combination of nonuniform 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. https://ijnao.um.ac.ir/article_45503_47c8dae28264bd95c7aed2197de00964.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 420241201Mathematical modeling and optimal control of customer’s behavior toward e-commerce116812024554210.22067/ijnao.2024.88052.1448ENH. HassouniLaboratory of Fundamental Mathematics and their Applications, Department of Mathematics, Faculty of Sciences El Jadida, Chouaib Doukkali University, El Jadida, Morocco.S. BelhdidLaboratory of Fundamental Mathematics and their Applications, Department of Mathematics, Faculty of Sciences El Jadida, Chouaib Doukkali University, El Jadida, Morocco.0009-0002-3532-9028O. BalatiLaboratory of Fundamental Mathematics and their Applications, Department of Mathematics, Faculty of Sciences El Jadida, Chouaib Doukkali University, El Jadida, Morocco.0000-0003-1887-5350Journal Article20240513The 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 parameter sensitivity. Furthermore, control theory has been adopted to optimize the adoption of e-commerce using Pontryagin’s maximum principle, with numerical simulations to evaluate the effectiveness of our proposed strategies.https://ijnao.um.ac.ir/article_45542_1bd0800e40747aa1b83962cb07388108.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 420241201Modified hat functions: Application in space-time-fractional differential equations with Caputo derivative120312234555610.22067/ijnao.2024.87189.1405ENE. SedighiDepartment of Applied Mathematics, Faculty of Mathematics and Computer Science, Hakim Sabzevari University, Iran.O. BaghaniDepartment of Applied Mathematics, Faculty of Mathematics and Computer
Science, Hakim Sabzevari University, Iran.0000-0002-5429-9373H. AzinDepartment of Applied Mathematics, Faculty of Mathematics and Computer Science, Hakim Sabzevari University, Iran.Journal Article20240308The 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 modified hat functions. We use the operational matrix of the Riemann–Liouville fractional integral of modified hat functions to approximate the Caputo fractional derivative in order to reduce the problem to a system of Sylvester equations. The error of the mentioned method is of the order O(h3). In addition, we examine several numerical examples to confirm the ability of the proposed approach. https://ijnao.um.ac.ir/article_45556_340031334795eed9f0588de360ee0444.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 420241201An efficient collocation scheme for new type of variable-order fractional Lane–Emden equation122412464560010.22067/ijnao.2024.87501.1419ENH. AzinDepartment of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.A. HabibiradDepartment of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.E. HesameddiniDepartment of Mathematics, Shiraz University of Technology, Shiraz, Iran.M.H. HeydariDepartment of Mathematics, Shiraz University of Technology, Shiraz, Iran.Journal Article20240406The 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 equation (linear and nonlinear). The operational matrix of the 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 and those obtained via the Bernoulli polynomials. Overall, this paper introduces a new methodology for solving V-O fractional Lane–Emden equations using Vt-L 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.https://ijnao.um.ac.ir/article_45600_3b69a4d72b3a78971e09c9288d5fc077.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 420241201A novel mid-point upwind scheme for fractional-order singularly perturbed convection-diffusion delay differential equation124712794558810.22067/ijnao.2024.88781.1472ENN.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, Arba Minch University, Arba Minch, Ethiopia.Department of Mathematics, College of Natural and Computational Science, Jimma University, Jimma, Ethiopia.0000-0003-1889-4690Journal Article20240702This study presents a numerical approach for solving temporal fractionalorder 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 ε are explored.https://ijnao.um.ac.ir/article_45588_994a451c41f05a5de1f91d5b86fb19f3.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 420241201Advanced mathematical modeling and prognosication of regulated spatio-temporal dynamics of Monkeypox128013084567710.22067/ijnao.2024.89077.1484ENM. BaroudiLaboratory LMACS, Sultan Moulay Slimane University, MATIC Research Team: Applied Mathematics and Information and Communication Technologie,Polydisciplinary Faculty, Morocco.Department of Mathematics and Computer Science, Khouribga Polydisciplinary Faculty, Morocco.0009-0006-8669-5764I. Smouni1Laboratory LMACS, Sultan Moulay Slimane University, MATIC Research Team: Applied Mathematics and Information and Communication TechnologieDepartment of Mathematics and Computer Science, Khouribga Polydisciplinary Faculty, Morocco.H. GourramLaboratory LMACS, Sultan Moulay Slimane University, MATIC Research Team: Applied Mathematics and Information and Communication TechnologieDepartment of Mathematics and Computer Science, Khouribga Polydisciplinary Faculty, Morocco.0009-0001-2404-6404A. Labzai2Laboratory of Analysis Modeling and Simulation, Department of Mathematics and Computer ScienceFaculty of Sciences Ben M’sik, Hassan II University of Casablanca, Morocco.M. BelamLaboratory LMACS, Sultan Moulay Slimane University, MATIC Research Team: Applied Mathematics and Information and Communication TechnologieDepartment of Mathematics and Computer Science, Khouribga Polydisciplinary Faculty, Morocco.Journal Article20240725This study explores a continuous spatio-temporal 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 are 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.https://ijnao.um.ac.ir/article_45677_84ecea37cc7cd8708049a9f5549be265.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697714Issue 420241201A Petrov–Galerkin approach for the numerical analysis of soliton and multi-soliton solutions of the Kudryashov–Sinelshchikov equation130913354554110.22067/ijnao.2024.87548.1422ENH. SamyDepartment of Mathematics and Computer Sciences, Faculty of Science, Port-Said University, Egypt.W. AdelLaboratoire Interdisciplinaire de l’Universite Francaise d’Egypte (UFEID Lab), Universite Francaise d’Egypte, Cairo 11837, Egypt. . Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, Mansoura, 35516, Egypt.0000-0002-0557-8536I. HanafyDepartment of Mathematics and Computer Sciences, Faculty of Science, Port-Said University, Egypt.M. RamadanDepartment of Mathematics and Computer Sciences, Faculty of Science, Port-Said University, Egypt.Journal Article20240409This 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 L2 and L∞, 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 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.https://ijnao.um.ac.ir/article_45541_2d9bfdc7f14705609f784d1034bc22b7.pdf