Ferdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697713Issue 420231201A generalized form of the parametric spline methods of degree (2k + 1) for solving a variety of two-point boundary value problems5786034368410.22067/ijnao.2023.79288.1192ENZ. SarvariDepartment of Applied Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.Journal Article20221020In this paper, a high order accuracy method is developed for finding the approximate solution of two-point boundary value problems. The present approach is based on a special algorithm, taken from Pascal’s triangle, for obtaining a generalized form of the parametric splines of degree (2k + 1), k = 1, 2, . . . , which has a lower computational cost and gives the better ap-proximation. Some appropriate band matrices are used to obtain a matrix form for this algorithm.<br />The approximate solution converges to the exact solution of order O(h4k ), where k is a quantity related to the degree of parametric splines and the number of matrix bands that are applied in this paper. Some examples are given to illustrate the applicability of the method, and we compare the computed results with other existing known methods. It is<br />observed that our approach produced better results.https://ijnao.um.ac.ir/article_43684_5df402f8cb2364e5eee61b0e658bb547.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697713Issue 420231201Collection-based numerical method for multi-order fractional integro-differential equations6046264382410.22067/ijnao.2023.81586.1232ENG. AjileyeDepartment of Mathematics and Statistics, Federal University Wukari, Taraba State, Nigeria.0000-0002-4161-686xT. OyedepoFederal College of Dental Technology and Therapy, Enugu, Nigeria.0000-0001-90638806L. AdikuDepartment of Mathematics and Statistics, Federal University Wukari, Taraba State, Nigeria.J. SaboDepartment of Mathematics, Adamawa State University, Mubi, Nigeria.0000-0002-8402-9219Journal Article20230312In this paper, the standard collocation approach is used to solve multi-order fractional integro-differential equations using Caputo sense. We obtain the integral form of the problem and transform it into a system of linear alge-braic equations using standard collocation points. The algebraic equations are then solved using the matrix inversion method. By substituting the algebraic equation solutions into the approximate solution, the numerical result is obtained. We establish the method’s uniqueness as well as the convergence of the method. Numerical examples show that the developed method is efficient in problem-solving and competes favorably with the existing method.https://ijnao.um.ac.ir/article_43824_4ece47a2046a6ce53a3e6f618f7fc82a.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697713Issue 420231201A robust uniformly convergent scheme for two parameters singularly perturbed parabolic problems with time delay6276454390710.22067/ijnao.2023.80721.1214ENN.T. NegeroDepartment of Mathematics, Wollega University, Nekemte, Ethiopia.Journal Article20230121A singularly perturbed time delay parabolic problem with two small pa-rameters is considered. The paper develops a finite difference scheme that is exponentially fitted on a uniform mesh in the spatial direction and uses the implicit-Euler method to discretize the time derivative in the temporal direction in order to obtain a better numerical approximation to the solu-tions of this class of problems. We establish the parameter-uniform error estimate and discuss the stability of the suggested approach. In order to demonstrate the improvement in terms of accuracy, numerical results are also shown to validate the theoretical conclusions and are contrasted with the current hybrid scheme.https://ijnao.um.ac.ir/article_43907_c6b2d1382308555f419bf9a91322e100.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697713Issue 420231201Numerical nonlinear model solutions for the hepatitis C transmission between people and medical equipment using Jacobi wavelets method6466714341010.22067/ijnao.2023.79648.1198ENN. HamidatLaboratory of pure and applied mathematics, Faculty of exact science and computer science, University of Abdelhamid Ibn Badis, Mostaganem -Algeria.S.M. BahriLaboratory of pure and applied mathematics, Faculty of exact science and computer science, University of Abdelhamid Ibn Badis, Mostaganem -Algeria.N. AbbassaLaboratory of pure and applied mathematics, Faculty of exact science and computer science, University of Abdelhamid Ibn Badis, Mostaganem -Algeria.Journal Article20221117In this work, we present a new mathematical model for the spread of hepatitis C disease in two populations: human population and medical equipment population. Then, we apply the Jacobi wavelets method com-bined with the decoupling and quasi-linearization technique to solve this set of nonlinear differential equations for numerical simulation.https://ijnao.um.ac.ir/article_43410_c1ae77c60d4a251498fcef1fe7723a67.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697713Issue 420231201A shifted fractional-order Hahn functions Tau method for time-fractional PDE with nonsmooth solution6726944379010.22067/ijnao.2023.81716.1238ENN. MollahasaniDepartment of applied mathematics, Graduate university of advanced technology, Kerman, Iran.0000-0001-9291-1394Journal Article20230323In this paper, a new orthogonal system of nonpolynomial basis functions is introduced and used to solve a class of time-fractional partial differential equations that have nonsmooth solutions. In fact, unlike polynomial bases, such basis functions have singularity and are constructed with a fractional variable change on Hahn polynomials. This feature leads to obtaining more accurate spectral approximations than polynomial bases. The introduced method is a spectral method that uses the operational matrix of fractional order integral of fractional-order shifted Hahn functions and finally converts<br />the equation into a matrix equation system. In the introduced method, no collocation method has been used, and initial and boundary conditions are applied during the execution of the method. Error and convergence analysis of the numerical method has been investigated in a Sobolev space. Finally, some numerical experiments are considered in the form of tables and figures to demonstrate the accuracy and capability of the proposed method.https://ijnao.um.ac.ir/article_43790_b95901b0f2c3c04836a9c2ed82d8b5d4.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697713Issue 420231201Numerical solution of fractional Bagley–Torvik equations using Lucas polynomials6957104413610.22067/ijnao.2023.81548.1230ENM. AskariDepartment of Mathematics, Professor Hesabi Branch, Islamic Azad University, Tafresh, Iran.0000-0001-5924-2555Journal Article20230311The aim of this article is to present a new method based on Lucas poly-nomials and residual error function for a numerical solution of fractional Bagley–Torvik equations. Here, the approximate solution is expanded as a linear combination of Lucas polynomials, and by using the collocation method, the original problem is reduced to a system of linear equations. So, the approximate solution to the problem could be found by solving this system. Then, by using the residual error function and approximating the error function by utilizing the same approach, we achieve more accurate results. In addition, the convergence analysis of the method is investi-gated. Numerical examples demonstrate the validity and applicability of the method.https://ijnao.um.ac.ir/article_44136_d4a475ff85dc768135a202da8dfadeea.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697713Issue 420231201Singularly perturbed two-point boundary value problem by applying exponential fitted finite difference method7117274417110.22067/ijnao.2023.83070.1283ENN. KumarDepartment of Mathematics, National Institute of Technology Patna, Patna - 800005, India.R. Kumar SinhaDepartment of Mathematics, National Institute of Technology Patna, Patna - 800005, India.R. RanjanDepartment of Science and Technology, Bihar, Government Polytechnic, Lakhisarai, Lakhisarai- 811311, India.0000-0001-5735-545XJournal Article20230622The present study addresses an exponentially fitted finite difference method to obtain the solution of singularly perturbed two-point boundary value problems (BVPs) having a boundary layer at one end (left or right) point on uniform mesh. A fitting factor is introduced in the derived scheme using the theory of singular perturbations. Thomas algorithm is employed to solve the resulting tri-diagonal system of equations. The convergence of the presented method is investigated. Several model example problems are solved using the proposed method. The results are presented with terms of maximum absolute errors, which demonstrate the accuracy and efficiency of the method. It is observed that the proposed method is capable of producing highly accurate results with minimal computational effort for a fixed value of step size h, when the perturbation parameter tends to zero. From the graphs, we also observed that a numerical solution approximates the exact solution very well in the boundary layers for smaller value of ε.https://ijnao.um.ac.ir/article_44171_e4113223bcf920a6994845cda431a10f.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697713Issue 420231201Numerical study of sine-Gordon equations using Bessel collocation method7287464396710.22067/ijnao.2023.81484.1229ENS. AroraDepartment of Mathematics, Punjabi University, Patiala, 147002, Punjab, India.I. BalaDepartment of Mathematics, Punjabi University, Patiala, 147002, Punjab, India.Journal Article20230309The nonlinear space time dynamics have been discussed in terms of a hyper-bolic equation known as a sine-Gordon equation. The proposed equation has been discretized using the Bessel collocation method with Bessel poly-nomials as base functions. The proposed hyperbolic equation has been transformed into a system of parabolic equations using a continuously dif-ferentiable function. The system of equations involves one linear and the other nonlinear diffusion equation. The convergence of the present tech-nique has been discussed through absolute error, L2-norm, and L∞-norm.<br />The numerical values obtained from the Bessel collocation method have been compared with the values already given in the literature. The present technique has been applied to different problems to check its applicability. Numerical values obtained from the Bessel collocation method have been presented in tabular as well as in graphical form.https://ijnao.um.ac.ir/article_43967_8adfe440c357caaf5961e7f520cb9d67.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697713Issue 420231201Optimal control analysis for modeling HIV transmission7477624382510.22067/ijnao.2023.78096.1165ENK. R. ChenekeDepartment of Mathematics, Wollega University, Nekemte, Ethiopia.Journal Article20220805In this study, a modified model of HIV with therapeutic and preventive controls is developed. Moreover, a simple evaluation of the optimal control problem is investigated. We construct the Hamiltonian function by way of integrating Pontryagin’s maximal principle to achieve the point-wise optimal solution. The effects obtained from the version analysis strengthen public health education to a conscious population, PrEP for early activation of HIV infection prevention, and early treatment with artwork for safe life after HIV infection. Moreover, numerical simulations are done using the MATLAB platform to illustrate the qualitative conduct of the HIV infection. In the end, we receive that adhering to ART protective prone people, the usage of PrEP along with different prevention control is safer control measures.https://ijnao.um.ac.ir/article_43825_7531be5681b694b17125c3478946e18b.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697713Issue 420231201Improving the performance of the FCM algorithm in clustering using the DBSCAN algorithm7637744411710.22067/ijnao.2023.82361.1260ENS. Barkhordari FirozabadiPhD candidate, Department of Computer Science, Yazd University , Yazd, Iran.S.A. Shahzadeh FazeliParallel Processing Lab, Department of Computer Science, Yazd University, Yazd, Iran.0000-0002-3724-8689J. Zarepour AhmadabadiDepartment of Computer Science, Yazd University, Yazd, Iran.S.M. KarbassiDepartment of Applied Mathematics, Faculty of Mathematical Sciences, Yazd University, Yazd, Iran.Journal Article20230511The fuzzy-C-means (FCM) algorithm is one of the most famous fuzzy clus-tering algorithms, but it gets stuck in local optima. In addition, this algo-rithm requires the number of clusters. Also, the density-based spatial of the application with noise (DBSCAN) algorithm, which is a density-based clus-tering algorithm, unlike the FCM algorithm, should not be pre-numbered. If the clusters are specific and depend on the number of clusters, then it can determine the number of clusters. Another advantage of the DBSCAN clus-tering algorithm over FCM is its ability to cluster data of different shapes. In this paper, in order to overcome these limitations, a hybrid approach for clustering is proposed, which uses FCM and DBSCAN algorithms. In this method, the optimal number of clusters and the optimal location for the centers of the clusters are determined based on the changes that take place according to the data set in three phases by predicting the possibility of the problems stated in the FCM algorithm. With this improvement, the values of none of the initial parameters of the FCM algorithm are random, and in the first phase, it has been tried to replace these random values to the optimal in the FCM algorithm, which has a significant effect on the convergence of the algorithm because it helps to reduce iterations. The proposed method has been examined on the Iris flower and compared the results with basic FCM algorithm and another algorithm. Results shows the better performance of the proposed method.https://ijnao.um.ac.ir/article_44117_c35d5a744ce9c31c04abbea89eda4c41.pdf