University of TabrizComputational Methods for Differential Equations2345-398211120230101A third-order weighted essentially non-oscillatory-flux limiter scheme for two-dimensional incompressible Navier-Stokes equations1111443410.22034/cmde.2022.48228.2014ENRooholahAbedianSchool of Engineering Science, College of Engineering, University of Tehran, Iran.Journal Article20210929In this paper, the 2D incompressible Navier-Stokes (INS) equations in terms of vorticity and stream function are considered. These equations describe the physics of many phenomena of scientific and engineering. By combining monotone upwind methods and weighted essentially non-oscillatory (WENO) procedures, a new numerical algorithm is proposed to approximate the solution of INS equations. To design this algorithm, after obtaining an optimal polynomial, it is rewritten as a convex combination of second-order modified ENO polynomials. Following the methodology of the traditional WENO procedure, the new non-linear weights are calculated. The performance of the new scheme on a number of numerical examples is illustrated.https://cmde.tabrizu.ac.ir/article_14434_ff50ca539c74582df3eb90624a74fe5e.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211120230101Modified Lucas polynomials for the numerical treatment of second-order boundary value problems12311489210.22034/cmde.2022.50891.2115ENYoussri HassanYoussriDepartment of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt.ShahendaSayedDepartment of Mathematics, Faculty of Science, Helwan University, Cairo 11795, Egypt.0000-0002-0909-781XAmany SaadMohamedDepartment of Mathematics, Faculty of Science, Helwan University, Cairo 11795, Egypt.EmadAboeldahabDepartment of Mathematics, Faculty of Science, Helwan University, Cairo 11795, Egypt.Waleed MohamedAbd-ElhameedDepartment of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt.0000-0002-6102-671XJournal Article20220325This paper is devoted to the construction of certain polynomials related to Lucas polynomials, namely, modified Lucas polynomials. The constructed modified Lucas polynomials are utilized as basis functions for the numerical treatment of the linear and non-linear second-order boundary value problems (BVPs) involving some specific important problems such as singular and Bratu-type equations. To derive our proposed algorithms, the operational matrix of derivatives of the modified Lucas polynomials is established by expressing the first-order derivative of these polynomials in terms of their original ones. The convergence analysis of the modified Lucas polynomials is deeply discussed by establishing some inequalities concerned with these modified polynomials. Some numerical experiments accompanied by comparisons with some other articles in the literature are presented to demonstrate the applicability and accuracy of the presented algorithms.https://cmde.tabrizu.ac.ir/article_14892_22f9d595bb3640943656ffc4fdbe62e1.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211120230101Extremal solutions for multi-term nonlinear fractional differential equations with nonlinear boundary conditions32411414710.22034/cmde.2021.48310.2018ENHosseinFazliFaculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.0000-0001-7987-3776FaribaBahramiFaculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.SedaghatShahmoradFaculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.0000-0002-2476-1176Journal Article20211007This paper is devoted to prove the existence of extremal solutions for multi-term nonlinear fractional differential equations with nonlinear boundary conditions. The fractional derivative is of Caputo type and the inhomogeneous term depends on the fractional derivatives of lower orders. By establishing a new comparison theorem and applying the monotone iterative technique, we show the existence of extremal solutions. The method is a constructive method that yields monotone sequences that converge to the extremal solutions. As an application, some examples are presented to illustrate the main results.https://cmde.tabrizu.ac.ir/article_14147_bcdaf68ffaf4cb098ee1253cb5ecb31f.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211120230101A robust numerical scheme for singularly perturbed delay parabolic initial-boundary-value problems involving mixed space shifts42511468710.22034/cmde.2022.50833.2109ENNarahariRaji ReddyDepartment of Mathematics and Humanities, Kakatiya Institute of Technology and Science, Warangal, India.JugalMohapatraDepartment of Mathematics, National Institute of Technology Rourkela, Odisha, India.0000-0001-5118-3933Journal Article20220317This article proposes a parameter uniform numerical method for solving a singularly perturbed delay parabolic initial boundary-value problem involving mixed space shifts. The model also involves a large delay in time. Taylor’s series expansion is applied to approximate the retarded terms in the spatial direction. For the time discretization, the implicit trapezoidal scheme is applied on uniform mesh, and for the spatial discretization, we use a proper combination of the mid-point upwind and the central difference scheme on Shishkin mesh. The proposed scheme provides a second-order convergence rate uniformly with respect to the perturbation parameter. Some comparison results are presented by using the proposed method to support our claim.https://cmde.tabrizu.ac.ir/article_14687_d8dca433b35f796ddb9a11c965bb3473.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211120230101Optimal control of Volterra integro-differential equations based on interpolation polynomials and collocation method52641467910.22034/cmde.2022.50643.2100ENMaryamAlipourDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran.SamanehSoradi ZeidFaculty of Industry and Mining (Khash), University of Sistan and Baluchestan, Zahedan, Iran.Journal Article20220302In this paper, we introduce a new direct scheme based on Dickson polynomials and collocation points to solve a class of optimal control problems (OCPs) governed by Volterra integro-differential equations namely Volterra integro-OCPs (VI-OCPs). This topic requires to calculating the corresponding operational matrices for expanding the solution of this problem in terms of Dickson polynomials. Further, the highlighted method allows us to transform the VI-OCP into a system of algebraic equations for choosing the coefficients and control parameters optimally. The error estimation of this technique is also investigated which given the high efficiency of the Dickson polynomials to deal with these problems. Finally, some examples are brought to confirm the validity and applicability of this approach in comparison with those obtained from other methods. https://cmde.tabrizu.ac.ir/article_14679_2e98e108789f2d0d30dd3d7e15d39c47.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211120230101Stability for neutral-type integro-differential neural networks with random switches in noise and delay65801446310.22034/cmde.2022.49283.2056ENChafaiImzegouanISTI Lab, ENSA PO Box 1136, Ibn Zohr University, Agadir, Morocco.AzizZouineISTI Lab, ENSA PO Box 1136, Ibn Zohr University, Agadir, Morocco.HassaneBouzahirISTI Lab, ENSA PO Box 1136, Ibn Zohr University, Agadir, Morocco.CemilTunçDepartment of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080, Van, Turkey.0000-0003-2909-8753Journal Article20211205This paper focuses on existence, uniqueness, and stability analysis of solutions for a new kind of delayed integro-differential neural networks with Markovian switches in delays and noises. The studied system combines many types of integro-differential neural network treatises in the literature. After having presented the studied system, the existence and uniqueness of solutions are shown under Lipschitz condition. By using the Lyapunov-Krasovskii functional, some stochastic analysis techniques and the M-matrix approach, stochastic stability, and general decay stability are established. Finally, a numerical example is given to validate the main established theoretical results.https://cmde.tabrizu.ac.ir/article_14463_c5c76cbb3b772cdaeaa192daf87fd006.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211120230101Numerical solution of space-time fractional PDEs with variable coefficients using shifted Jacobi collocation method81941443710.22034/cmde.2022.49901.2077ENSamiraBonyadiMathematics Department, Tabriz Branch, Islamic Azad University, Tabriz, Iran.YaghoubMahmoudiMathematics Department, Tabriz Branch, Islamic Azad University, Tabriz, Iran.MehrdadLakestaniDepartment of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.0000-0002-2752-0167MohammadJahangiri RadMathematics Department, Tabriz Branch, Islamic Azad University, Tabriz, Iran.Journal Article20220114The paper reports a spectral method for generating an approximate solution for the space-time fractional PDEs with variable coefficients based on the spectral shifted Jacobi collocation method in conjunction with the shifted Jacobi operational matrix of fractional derivatives. The spectral collocation method investigates both temporal and spatial discretizations. By applying the shifted Jacobi collocation method, the problem reduces to a system of algebraic equations, which greatly simplifies the problem. Numerical results are given to establish the validity and accuracy of the presented procedure for space-time fractional PDE. https://cmde.tabrizu.ac.ir/article_14437_d9eaf1c9bbcc1e97fa17a670800d869c.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211120230101A new application for numerical computations of the modified equal width equation (MEW) based on Lumped Galerkin method with the cubic B-spline951071489110.22034/cmde.2022.51278.2133ENMelikeKartaDepartment of Mathematics, Faculty of Science and Arts, Ağrı İbrahim Çeçen University, Ağrı, Turkey.0000-0003-3412-4370Journal Article20220424In this paper, numerical computation of the modified equal width equation (MEW), which is one of the equations used to model nonlinear events, will be carried out. For this equation, numerical computations have been obtained by many researchers using different methods. The goal of the new approach is to check how well it performs with respect to the numerical calculations the researchers found. For this, the proposed study presents a Lie-Trotter splitting algorithm in accordance with the time-splitting technical rules combined with Lumped Galerkin FEM based on the basis function of the cubic B-spline. Two valid test examples are given to determine the validity and effectiveness of the current technique. The results obtained in a new way with the Matlab computational software are compared with the studies of other authors in the literature and are shown graphically. Based on these new results, it can clearly be stated that the benefit of the proposed approach is to demonstrate that reliability is achieved in obtaining approximate computations. https://cmde.tabrizu.ac.ir/article_14891_5e45c114ccfa006420fa58c8f8923b65.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211120230101Choosing the best value of shape parameter in radial basis functions by Leave-P-Out Cross Validation1081291497810.22034/cmde.2022.46208.1939ENMohammad RezaYaghoutiFaculty of Mathematical Sciences, University of Guilan, Rasht, Iran.0000-0003-3137-5799FarnazFarshadmoghadamFaculty of Mathematical Sciences, University of Guilan, Rasht, Iran.Journal Article20210523The radial basis functions (RBFs) meshless method has high accuracy for the interpolation of scattered data in high dimensions. Most of the RBFs depend on a parameter, called shape parameter which plays a significant role to specify the accuracy of the RBF method. In this paper, we present three algorithms to choose the optimal value of the shape parameter. These are based on Rippa’s theory to remove data points from the data set and results obtained by Fasshauer and Zhang for the iterative approximate moving least square (AMLS) method. Numerical experiments confirm stable solutions with high accuracy compared to other methods.https://cmde.tabrizu.ac.ir/article_14978_e696f73550c3717f11d3153baea56e01.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211120230101A new numerical algorithm based on Quintic B-Spline and adaptive time integrator for Coupled Burger’s equation1301421476710.22034/cmde.2022.50940.2121ENYesimCicekEngineering Sciences, Faculty of Architecture and Engineering, Izmir Katip Celebi University, Izmir, Turkey.NurcanGucuyenen KaymakManagement Information Systems, Faculty of Economics and Administrative Sciences, Dogus University, Istanbul, Turkey.0000-0001-8226-8315ErsinBaharDepartment of Civil Engineering, Pamukkale University, Denizli, Turkey.GurhanGurarslanDepartment of Civil Engineering, Pamukkale University, Denizli, Turkey.0000-0002-9796-3334GamzeTangoluDepartment of Mathematics, Izmir Institute of Technology, Izmir, Turkey.Journal Article20220401In this article, the coupled Burger’s equation which is one of the known systems of the nonlinear parabolic partial differential equations is studied. The method presented here is based on a combination of the quintic B-spline and a high order time integration scheme known as adaptive Runge-Kutta method. First of all, the application of the new algorithm on the coupled Burger’s equation is presented. Then, the convergence of the algorithm is studied in a theorem. Finally, to test the efficiency of the new method, coupled Burger’s equations in literature are studied. We observed that the presented method has better accuracy and efficiency compared to the other methods in the literature.https://cmde.tabrizu.ac.ir/article_14767_a26becd708959dc729c32f6a3e061f54.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211120230101An ABC algorithm based approach to solve a nonlinear inverse reaction-diffusion problem associate with the ecological invasions1431601443610.22034/cmde.2022.49491.2060ENParastooReihaniDepartment of Mathematics, Payame Noor University (PNU), P. O. Box: 19395-4697, Tehran, Iran.HosseinEsmailpourDepartment of Mathematics, Payame Noor University (PNU), P. O. Box: 19395-4697, Tehran, Iran.FahimehSoltanianDepartment of Mathematics, Payame Noor University (PNU), P. O. Box: 19395-4697, Tehran, Iran.0000-0003-3068-3378Journal Article20211219In the present study, we consider an important mathematical model of the spread of two competing species in an ecological system with two species considering the interactions between these species. This model is derived from a system of nonlinear reaction-diffusion equations. We investigate this model as an inverse problem. Using appropriate initial and boundary conditions, the finite difference method in the time variable and the Quartic Bspline collocation method in the spatial variable are used to develop a numerical method. The proposed numerical approach results in an ill-posed linear system of equations and to overcome the ill-posedness, the Tikhonov regularization method is implemented. An effective approach based on the ABC algorithm is established to determine the regularization parameter. To show the robustness and ability of the present approach, for a test case, the results are compared with the results of the L-curve and GCV methods.https://cmde.tabrizu.ac.ir/article_14436_fde14e26d9a8ef412b1546532eefc399.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211120230101Linear B-spline finite element Method for solving delay reaction diffusion equation1611741467810.22034/cmde.2022.49678.2066ENGemeda TolessaLuboDepartment of Mathematics, Wollega University, Nekemte, Ethiopia.0000-0003-1254-8295Gemechis FileDuressaDepartment of Mathematics, Jimma University, Jimma, Ethiopia.0000-0003-1889-4690Journal Article20211229This paper is concerned with the numerical treatment of delay reaction-diffusion with the Dirichlet boundary condition. The finite element method with linear B-spline basis functions is utilized to discretize the space variable. The Crank-Nicolson method is used for the processes of time discretization. Sufficient and necessary conditions for the numerical method to be asymptotically stable are investigated. The convergence of the numerical method is studied. Some numerical experiments are performed to verify the applicability of the numerical method. https://cmde.tabrizu.ac.ir/article_14678_5b765da509a2de5c42d7dab0e23090c5.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211120230101Approximate symmetry group analysis and similarity reductions of the perturbed mKdV-KS equation1751821443510.22034/cmde.2022.48341.2022ENMehdiJafariDepartment of Mathematics, Payame Noor University, PO BOX 19395-4697, Tehran, Iran.RazieDarvazebanzadeDepartment of Mathematics, Payame Noor University, PO BOX 19395-4697, Tehran, Iran.Journal Article20211009In this paper, we apply the approximate symmetry transformation group to obtain the approximate symmetry group of the perturbed mKdV-KS equation which is a modified Korteweg-de Vries (mKdV) equation with a higher singularity perturbed term as the Kuramoto-Sivashinsky (KS) equation. Also, an optimal system of one-dimensional subalgebras of symmetry algebra is constructed and the corresponding differential invariants and some approximately invariant solutions of the equation are computed. https://cmde.tabrizu.ac.ir/article_14435_2acb98a54724c87f05f5ed80210cf245.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211120230101Hybrid collocation method for some classes of second-kind nonlinear weakly singular integral equations1831961443310.22034/cmde.2022.47657.1992ENNargesMahmoodi DaraniDepartment of mathematics, Hashtgerd branch, Islamic Azad University, Hashtgerd, Iran.Journal Article20210828In the current study, a fast, accurate, and reliable numerical scheme for approximating second-kind nonlinear Fredholm, Volterra, and Fredholm-Volterra integral equations with a weakly singular kernel and invertible nonlinearity is presented. The computational approach is based upon function, especially the hybrid one. Hybrid functions give us the opportunity to achieve an appropriate solution by adjusting a suitable order for polynomials’ degrees and block-pulse functions. The basic idea of this method is based on using the invertibility of the nonlinear function as a benefit to reduce the total error and simplify the procedure. The scheme reduces these types of equations to nonlinear systems of algebraic equations. Convergence analysis of the method under the infinity norm is wellstudied. Numerical results indicate the superiority of the present method compared with another existing method in the literaturehttps://cmde.tabrizu.ac.ir/article_14433_ef029be561e99d1fefac05424086a167.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211120230101Holder estimates of solutions degenerate nonlinear parabolic equations1972061546710.22034/cmde.2022.46575.1959ENTahir SadiGadjievAzerbaijan Architecture and Construction University, Baku, Azerbaijan.Institute of Mathematics and Mechanics of NAS Azerbaijan, Baku, Azerbaijan.Sardar YahyaAlievBaku State University, Department of Higher Mathematics, Baku, Azerbaijan.Aybeniz HYagnalievaSumqait State University, Sumqait, Azerbaijan.Journal Article20210617Holder estimates of solutions of initial-boundary problem degenerate nonlinear parabolic equations are obtained. Estimates for solutions and parabolic Harnack inequality are proved. Also, one variant of weighted Poincare inequality is shown.https://cmde.tabrizu.ac.ir/article_15467_c4730f8a9747cf46bebc8491ee6e9fdd.pdf