University of TabrizComputational Methods for Differential Equations2345-398211220230401Optimal fractional order PID controller performance in chaotic system of HIV disease: particle swarm and genetic algorithms optimization method2072241539710.22034/cmde.2022.51132.2127ENShabanMohammadiFaculty of mathematical sciences, Shahrood university of technology, Shahrood, Semnan, Iran.RezaHejaziFaculty of mathematical sciences, Shahrood university of technology, Shahrood, Semnan, Iran.Journal Article20220414The present study aims to investigate the optimal fractional order PID controller performance in the chaotic system of HIV disease fractional order using the Particle Swarm optimization and Genetic algorithm method. Differential equations were used to represent the chaotic behavior associated with HIV. The optimal fractional order of the PID controller was constructed, and its performance in the chaotic system with HIV fractional order was tested. Optimization methods were used to get PID control coefficients from particle swarm and genetic algorithms. Findings revealed that the equations for the HIV disease model are such that the system’s behavior is greatly influenced by the number of viruses produced by infected cells, such that if the number of viruses generated by infected cells exceeds 202, the disease’s behavior is such that the virus and disease spread. For varying concentrations of viruses, the controller created for this disease does not transmit the disease.https://cmde.tabrizu.ac.ir/article_15397_0f6d8480f313c3473f2049e8e741e8cd.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211220230401Non-polynomial cubic spline method for solution of higher order boundary value problems2252401500210.22034/cmde.2022.49827.2072ENShahnaKhanDepartment of Mathematics, Sri Venkateswara College, University of Delhi, New Delhi-21, India.ArshadKhanDepartment of Mathematics, Jamia Millia Islamia, New Delhi-25, India.Journal Article20220108In this paper, a new algorithm based on non-polynomial spline is developed for the solution of higher order boundary value problems(BVPs). Employment of the method is done by decomposing the higher order BVP into a system of third order BVPs. Convergence analysis of the developed method is also discussed. The method is tested on higher order linear as well as non-linear BVPs which shows the accuracy and efficiency of the method and also compared our results with some existing fourth order methods.https://cmde.tabrizu.ac.ir/article_15002_d581cd78048b059b5b4bf7e45cb1574f.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211220230401A numerical method for solving the Duffing equation involving both integral and non-integral forcing terms with separated and integral boundary conditions2412531500110.22034/cmde.2022.50054.2082ENMohammad RezaDoostdarDepartment of Mathematics, Zarandieh Branch, Islamic Azad University, Zarandieh, IranManochehrKazemiDepartment of Mathematics, Ashtian Branch, Islamic Azad University, Ashtian, IranAlirezaVahidiDepartment of Mathematics, Yadegar -e-Imam Khomeini (RAH) Shahre Rey Branch, Islamic Azad University, Tehran, IranJournal Article20220124This paper presents an efficient numerical method to solve two versions of the Duffing equation by the hybrid functions based on the combination of Block-pulse functions and Legendre polynomials. This method reduces the solution of the considered problem to the solution of a system of algebraic equations. Moreover, the convergence of the method is studied. Some examples are given to demonstrate the applicability and effectiveness of the proposed method. Also, the obtained results are compared with some other results. https://cmde.tabrizu.ac.ir/article_15001_c7c1ad5df26ae11cc861083caf87d8f8.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211220230401Steady state bifurcation in a cross diffusion prey-predator model2542621540110.22034/cmde.2022.52663.2213ENMarziehFarshidDepartment of Mathematics, Razi University, Kermanshah, Iran.YaghoubJalilianDepartment of Mathematics, Razi University, Kermanshah, Iran.Journal Article20220722In this paper, we study the bifurcation of nontrivial steady state solutions for a cross-diffusion prey-predator model with homogeneous Neumann boundary conditions. The existence of positive steady state solutions near a bifurcation point is proved using a crossing curve bifurcation theorem. We consider a situation where the transversality condition is not satisfied. Unlike the case in saddle-node bifurcation, the solution set is a pair of transversally intersecting curves.https://cmde.tabrizu.ac.ir/article_15401_d44135fc019221cc89f6b4855a63ba3d.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211220230401A novel scheme for SMCH equation with two different approaches2632801522510.22034/cmde.2022.50363.2093ENArzuAkbulutDepartment of Mathematics, Faculty of Arts and Science, Bursa Uludag University, Bursa, Turkey.0000-0003-2448-2481RayhanulIslamDepartment of Mathematics, Pabna University of Science and Technology, Pabna-Bangladesh.YiasirArafatDepartment of Mathematics, Pabna University of Science and Technology, Pabna-Bangladesh.FilizTaşcanFaculty of Arts and Science, Department of Mathematics and Computer, Eski¸sehir Osmangazi University, Eskisehir, Turkey.Journal Article20220212In this study, the unified and improved F-expansion methods are applied to derive exact traveling wave solutions of the simplified modified Camassa-Holm (SMCH) equation. The current methods can calculate all branches of solutions at the same time, even if several solutions are quite near and therefore impossible to identify via numerical methods. All obtained solutions are given by hyperbolic, trigonometric, and rational function solutions which obtained solutions are useful for real-life problems in fluid dynamics, optical fibers, plasma physics and so on. The two-dimensional (2D) and three-dimensional (3D) graphs of the obtained solutions are plotted. Finally, we can state that these strategies are extremely successful, dependable, and simple. These ideas might potentially be applied to many nonlinear evolution models in mathematics and physics. https://cmde.tabrizu.ac.ir/article_15225_2c395bd4ed2346e9a55ff505c84199e2.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211220230401Haar wavelet-based valuation method for pricing European options2812901525810.22034/cmde.2022.52027.2177ENSaeedVahdatiDepartment of Mathematics, Khansar Campus, University of Isfahan, Iran.0000-0001-6716-1050Mohammad RezaAhmadi DaraniDepartment of Applies Mathematics, Faculty of Mathematical Sciences, Shahrekord University, Shahrekord, Iran.Mohammad RezaGhaneiDepartment of Mathematics, Khansar Campus, University of Isfahan, Iran.Journal Article20220612A numerical method based on the Haar wavelet is introduced in this study for solving the partial differential equation which arises in the pricing of European options. In the first place, and due to the change of variables, the related partial differential equation (PDE) converts into a forward time problem with a spatial domain ranging from 0 to 1. In the following, the Haar wavelet basis is used to approximate the highest derivative order in the equation concerning the spatial variable. Then the lower derivative orders are approximated using the Haar wavelet basis. Finally, by substituting the obtained approximations in the main PDE and doing some computations using the finite differences approach, the problem reduces to a system of linear equations that can be solved to get an approximate solution. The provided examples demonstrate the effectiveness and precision of the method.https://cmde.tabrizu.ac.ir/article_15258_afeebb2e383bbbe40712a4d405fc86ee.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211220230401Numerical computation of exponential functions in frame of Nabla fractional calculus2913021523910.22034/cmde.2022.50918.2119ENJagan MohanJonnalagaddaDepartment of Mathematics, Birla Institute of Technology and Science Pilani, Hyderabad - 500078, Telangana, India.JehadAlzabutDepartment of Mathematics and Sciences, Prince Sultan University, Riyadh 12435, Saudi Arabia.Department of Industrial Engineering, OST_IM
Technical University, Ankara 06374, Turkiye.Journal Article20220329Exponential functions play an essential role in describing the qualitative properties of solutions of nabla fractional difference equations. In this article, we illustrate their asymptotic behavior. We know that these functions involve infinite series of ratios of gamma functions, and it is challenging to compute them. For this purpose, we propose a novel matrix technique to compute the addressed functions numerically. The results are supported by illustrative examples. The proposed method can be extended to obtain numerical solutions for non-homogeneous nabla fractional difference equations.https://cmde.tabrizu.ac.ir/article_15239_c443e283e494f0cb6f76f4c8db75594e.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211220230401A numerical method for KdV equation using rational radial basis functions3033181550010.22034/cmde.2022.51967.2171ENMansourShiralizadehDepartment of Mathematics, Payame Noor University (PNU), Tehran, Iran.AmjadAliPanahDepartment of Applied Mathematics, University of Kurdistan, Sanandaj, Iran.MaryamMohammadiFaculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran.Journal Article20220606In this paper, we use the rational radial basis functions ( RRBFs) method to solve the Korteweg-de Vries (KdV) equation, particularly when the equation has a solution with steep front or sharp gradients. We approximate the spatial derivatives by RRBFs method then we apply an explicit fourth-order Runge-Kutta method to advance the resulting semi-discrete system in time. Numerical examples show that the presented scheme preserves the conservation laws and the results obtained from this method are in good agreement with analytical solutions. https://cmde.tabrizu.ac.ir/article_15500_a836ec06443e749fe92e9f3da7d0080e.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211220230401On the existence of periodic solutions of third order delay differential equations3193311481210.22034/cmde.2022.50862.2112ENSultanErdurDepartment of Mathematics Faculty of Arts and Sciences Gaziantep University 27310, Gaziantep, Turkey.CemilTuncDepartment of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080, VAN, Turkey.0000-0003-2909-8753Journal Article20220320This work deals with the existence of periodic solutions (EPSs) to a third order nonlinear delay differential equation (DDE) with multiple constant delays. For the considered DDE, a theorem is proved, which includes sufficient criteria related to the EPSs. The technique of the proof depends on Lyapunov-Krasovskiˇı functional (LKF) approach. The obtained result extends and improves some results that can be found in the literature. In a particular case of the considered DDE, an example is provided to show the applicability of the main result of this paper. https://cmde.tabrizu.ac.ir/article_14812_9e2b9c80de0789b9a6684fd07c6fbe7e.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211220230401To study existence of unique solution and numerically solving for a kind of three-point boundary fractional high-order problem subject to Robin condition3323441525710.22034/cmde.2022.51628.2153ENElyasShivanianDepartment of Applied Mathematics, Imam Khomeini International University, Qazvin, 34149-16818, Iran.HedayatFatahiDepartment of Mathematics, Baneh Branch, Islamic Azad University, Baneh, Iran.Journal Article20220516In this paper, we prove the existence and uniqueness of the solutions for a non-integer high order boundary value problem which is subject to the Caputo fractional derivative. The boundary condition is a non-local type. Analytically, we introduce the fractional Green’s function. The main principle applied to simulate our results is the Banach contraction fixed point theorem. We deduce this paper by presenting some illustrative examples. Furthermore, it is presented a numerical based semi-analytical technique to approximate the unique solution to the desired order of precision. https://cmde.tabrizu.ac.ir/article_15257_205f33a2dfe9114356c177183de31b94.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211220230401Solution of time-fractional equations via Sumudu-Adomian decomposition method3453561500010.22034/cmde.2022.51421.2139ENShivaji AshokTarateDepartment of Mathematics, New Arts, Commerce and Science College, Ahmednagar, Maharashtra, India.0000-0001-7942-5682Ashok PBhadaneDepartment of Mathematics, Loknete Vyankatrao Hiray Arts&sbquo; Science and Commerce College, Nashik, Maharashtra, IndiaShrikisan BGaikwadDepartment of Mathematics, New Arts, Commerce and Science College, Ahmednagar, Maharashtra, India.Kishor AshokKshirsagarDepartment of Mathematics, New Arts, Commerce and Science College, Ahmednagar, Maharashtra, India.0000-0001-8123-8608Journal Article20220504This paper investigates the semi-analytical solutions of linear and non-linear Time Fractional Klein-Gordon equations with appropriate initial conditions to apply the New Sumudu-Adomian Decomposition method (NSADM). This paper shows the semi-analytical as well as a graphical interpretation of the solution by using mathematical software “Mathematica Wolform” and considering Caputo’s sense derivatives to semi-analytical results obtained by NSADM. https://cmde.tabrizu.ac.ir/article_15000_005c11a4fe0c1078226c56babfc38721.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211220230401A pseudospectral Sinc method for numerical investigation of the nonlinear time-fractional Klein-Gordon and sine-Gordon equations3573681548110.22034/cmde.2022.49999.2080ENShimaTaherkhaniDepartment of Mathematics, Qazvin Branch, Islamic Azad University, Qazvin, IranIrajNajafi KhalilsarayeDepartment of Mathematics, Qazvin Branch, Islamic Azad University, Qazvin, IranBakhtiyarGhayebiDepartment of Mathematics, Qazvin Branch, Islamic Azad University, Qazvin, IranJournal Article20220124In this paper, a pseudospectral method is proposed for solving the nonlinear time-fractional Klein-Gordon and sine-Gordon equations. The method is based on the Sinc operational matrices. A finite difference scheme is used to discretize the Caputo time-fractional derivative, while the spatial derivatives are approximated by the Sinc method. The convergence of the full discretization of the problem is studied. Some numerical examples are presented to confirm the accuracy and efficiency of the proposed method. The numerical results are compared with the analytical solution and the reported results in the literature. https://cmde.tabrizu.ac.ir/article_15481_20c3530fcae0c80dab6fa97e9e7015f6.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211220230401Numerical solution of nonlinear Sine-Gordon equation using modified cubic B-spline-based differential quadrature method3693861523810.22034/cmde.2022.50331.2091ENAthiraBabuDepartment of Mathematics, Cochin University of Science and Technology, Kerala, India.NoufalAsharafDepartment of Mathematics, Cochin University of Science and Technology, Kerala, India.Journal Article20220209In this article, we discuss the numerical solution of the nonlinear Sine-Gordon equation in one and two dimensions and its coupled form. A differential quadrature technique based on a modified set of cubic B-splines has been used. The chosen modification possesses the optimal accuracy order four in the spatial domain. The spatial derivatives are approximated by the differential quadrature technique, where the weight coefficients are calculated using this set of modified cubic B-splines. This approximation will lead to the discretization of the problem in the spatial domain that gives a system of first-order ordinary differential equations. This system is then solved using the SSP-RK54 scheme to progress the solution to the next time level. The convergence of this numerical scheme solely depends on the differential quadrature and is found to give a stable solution. The order of convergence is calculated and is observed to be four. The entire computation is performed up to a large time level with an efficient speed. It is found that the computed solution is in good agreement with the exact one and the error comparison with similar works in the literature indicates the scheme outperforms. https://cmde.tabrizu.ac.ir/article_15238_3c041bb25bf4374e1bf5ea1ad3fa7cff.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211220230401Jacobi wavelets method for numerical solution of fractional population growth model3873981447010.22034/cmde.2022.49041.2047ENYahyaRahroviMathematics Department, Tabriz Branch, Islamic Azad University, Tabriz, Iran.YaghoubMahmoudiMathematics Department, Tabriz Branch, Islamic Azad University, Tabriz, Iran.AliSalimi ShamlooMathematics Department, Shabestar Branch, Islamic Azad University, Shabestar, Iran.MohammadJahangiri RadMathematics Department, Tabriz Branch, Islamic Azad University, Tabriz, Iran.Journal Article20211118This paper deals with the generalization of the fractional operational matrix of Jacobi wavelets. The fractional population growth model was solved by using this operational matrix and compared with other existing methods to illustrate the applicability of the method. Then, convergence and error analysis of this procedure were studied. https://cmde.tabrizu.ac.ir/article_14470_fddd3e53f809a025504d7aad9bd73eed.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398211220230401Applying moving frames to finding conservation laws of the nonlinear Klein-Gordon equation3994111499910.22034/cmde.2022.50659.2101ENYousefMasoudiDepartment of Mathematics, Islamic Azad University, Naghadeh Branch, Naghadeh, Iran.0000-0001-8346-5695MehdiNadjafikhahDepartment of Pure Mathematics, School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, 16846-13114, Iran.0000-0002-1354-9786MegerdichToomanianDepartment of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.Journal Article20220303In this paper, we use a geometric approach based on the concepts of variational principle and moving frames to obtain the conservation laws related to the one-dimensional nonlinear Klein-Gordon equation. Noether’s First Theorem guarantees conservation laws, provided that the Lagrangian is invariant under a Lie group action. So, for calculating conservation laws of the Klein-Gordon equation, we first present a Lagrangian whose Euler-Lagrange equation is the Klein-Gordon equation, and then according to Gon¸calves and Mansfield’s method, we obtain the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame, for that Lagrangian, which is invariant under a hyperbolic group action. https://cmde.tabrizu.ac.ir/article_14999_a1db7d08cd72cfdabb168f0f0ca66806.pdf