University of TabrizComputational Methods for Differential Equations2345-39829320210701Mixed reproducing kernel-based iterative approach for nonlinear boundary value problems with nonlocal conditions6496581077510.22034/cmde.2020.38153.1681ENXiuyingLiSchool of Mathematics and Statistics, University of South Florida, Tampa, Florida 33620, USA.YangGaoSchool of Mathematics and Statistics, Changshu Institute of Technology,
Suzhou, Jiangsu 215500, China.BoyingWuSchool of Mathematical Sciences, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China.Journal Article20200130In this paper, a mixed reproducing kernel function (RKF) is introduced. The kernel function consists of piecewise polynomial kernels and polynomial kernels. On the basis of the mixed RKF, a new numerical technique is put forward for solving non-linear boundary value problems (BVPs) with nonlocal conditions. Compared with the classical RKF-based methods, our method is simpler since it is unnecessary to convert the original equation to an equivalent equation with homogeneous boundary conditions. Also, it is not required to satisfy the homogeneous boundary conditions for the used RKF. Finally, the higher accuracy of the method is shown via several numerical tests.https://cmde.tabrizu.ac.ir/article_10775_53c288d273b4e594b1a3d2058ec69778.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701Numerical analysis of fractional differential equation by TSI-wavelet method6596691038510.22034/cmde.2020.29679.1429ENFarhadShariffarDepartment of Applied Mathematics,
Faculty of Mathematical Sciences,
Lahijan Branch, Islamic Azad University, Lahijan, Iran.Amir HosseinRefahi SheikhaniDepartment of Applied Mathematics,
Faculty of Mathematical Sciences,
Lahijan Branch, Islamic Azad University, Lahijan, Iran.MohammadMashoofDepartment of Applied Mathematics,
Faculty of Mathematical Sciences,
Lahijan Branch, Islamic Azad University, Lahijan, Iran.Journal Article20181007In this paper, we propose a new numerical algorithm for the approximate solution of non-homogeneous fractional differential equation. Using this algorithm the fractional differential equations are transformed into a system of algebraic linear equations by operational matrices of block-pulse and hybrid functions. Based on our new algorithm, this system of algebraic linear equations can be solved by a proposed (TSI) method. Further, some numerical examples are given to illustrate and establish the accuracy and reliability of the proposed algorithm.https://cmde.tabrizu.ac.ir/article_10385_dfa31644a29622f65563f1aad887c291.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701New exact solutions and numerical approximations of the generalized KdV equation6706911032810.22034/cmde.2020.36253.1628ENSeydi BattalKarakocDepartment of Mathematics, Faculty of Science and Art,
Nevsehir Haci Bektas Veli University, Nevsehir, 50300, Turkey.Khalid KaramAliDepartment of Mathematics, Faculty of Science,
AL-Azhar University, Nasr City, P.N.Box: 11884- Cairo, Egypt.Journal Article20191017This paper is devoted to create new exact and numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with ansatz method and Galerkin finite element method based on cubic B splines over finite elements. Propagation of a single solitary wave is investigated to show the efficiency and applicability of the proposed methods. The performance of the numerical algorithm is proved by computing L2 and L∞ error norms. Also, three invariants I1, I2, and I3 have been calculated to determine the conservation properties of the presented algorithm. The obtained numerical solutions are compared with some earlier studies for similar parameters. This comparison clearly shows that the obtained results are better than some earlier results and they are found to be in good agreement with exact solutions. Additionally, a linear stability analysis based on Von Neumann's theory is surveyed and indicated that our method is unconditionally stable.https://cmde.tabrizu.ac.ir/article_10328_6373388662fb5de84854045e38e43e35.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701Stability and bifurcation of fractional tumor-immune model with time delay6927091033410.22034/cmde.2020.37915.1672ENJavadAlidoustiDepartment of Mathematical Sciences, Shahrekord University, Shahrekord, Iran.ElhamGhafariDepartment of Mathematical Sciences, Shahrekord University, Shahrekord, Iran.Journal Article20200115The present study aims are to analyze a delay tumor-immune fractional-order system to describe the rivalry among the immune system and tumor cells. Given that the dynamics of this system depend on the time delay parameter, we examine the impact of time delay on this system to attain better compatibility with actuality. For this purpose, we analytically evaluated the stability of the system’s equilibrium points. It is shown that Hopf bifurcation occurs in the fractional system when the delay parameter passes a certain value. Finally, by using numerical simulations, the analytical results were compared to the numerical results to acquire several dynamical behaviors of this system.https://cmde.tabrizu.ac.ir/article_10334_9ab6cb7a552dab94cf95c3adb92a2caa.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701Constructing an efficient multi-step iterative scheme for nonlinear system of equations7107211033610.22034/cmde.2020.34989.1596ENTaherLotfiDepartment of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran.MohammadMomenzadehDepartment of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran.Journal Article20190801The objective of this research is to propose a new multi-step method in tackling a system of nonlinear equations. The constructed iterative scheme achieves a higher rate of convergence whereas only one LU decomposition per cycle is required to proceed. This makes the efficiency index to be high as well in contrast to the existing solvers. The usefulness of the presented approach for tackling differential equations of nonlinear type with partial derivatives is also given.https://cmde.tabrizu.ac.ir/article_10336_7b0b3eafd0f6714a1f7488d2e97575b3.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701An efficient technique based on the HAM with Green's function for a class of nonlocal elliptic boundary value problems7227351075710.22034/cmde.2020.32673.1519ENRandhirSinghDepartment of Mathematics, Birla Institute of Technology Mesra, Ranchi-835215, India.Journal Article20190329In this paper, we propose an efficient technique-based optimal homotopy analysis method with Green’s function technique for the approximate solutions of nonlocal elliptic boundary value problems. We first transform the nonlocal boundary value problems into the equivalent integral equations. We then apply the optimal homotopy analysis method for the approximate solution of the considered problems. Several examples are considered to compare the results with the existing technique. The numerical results confirm the reliability of the present method as it tackles such nonlocal problems without any limiting assumptions. We also provide the convergence and the error estimation of the proposed method.https://cmde.tabrizu.ac.ir/article_10757_9fcde846d85b483defd9f9662056b21c.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701Fourth-order numerical method for the Riesz space fractional diffusion equation with a nonlinear source term7367481075910.22034/cmde.2020.36930.1643ENAkbarMohebbiDepartment of Applied Mathematics,
Faculty of Mathematical Science,
University of Kashan, Kashan, Iran.Journal Article20191125This paper aims to propose a high-order and accurate numerical scheme for the solution of the nonlinear diffusion equation with Riesz space fractional derivative. To this end, we first discretize the Riesz fractional derivative with a fourth-order finite difference method, then we apply a boundary value method (BVM) of fourth-order for the time integration of the resulting system of ordinary differential equations. The proposed method has a fourth-order of accuracy in both space and time components and is unconditionally stable due to the favorable stability property of BVM. The numerical results are compared with analytical solutions and with those provided by other methods in the literature. Numerical experiments obtained from solving several problems including fractional Fisher and fractional parabolic-type sine-Gordon equations show that the proposed method is an efficient algorithm for solving such problems and can arrive at the high-precision.https://cmde.tabrizu.ac.ir/article_10759_550274ba9a1a18e4b93024fcda905f89.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701VIM-Pad´e technique for solving nonlinear and delay initial value problems7497611075810.22034/cmde.2020.35417.1606ENAsadollahTorabi GiklouDepartment of Mathematics,
Guilan Science and Research Branch,
Islamic Azad University, Rasht, Iran.Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran.0000-0001-7296-6531MojtabaRanjbarDepartment of Mathematics,
Azarbaijan Shahid Madani University, Tabriz, Iran.Faculty of Finance Sciences, Kharazmi University, Tehran, Iran.0000-0003-0491-526XMahmoudShafieeDepartment of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran.0000-0001-8974-3895VahidRoomiDepartment of Mathematics,
Azarbaijan Shahid Madani University, Tabriz, Iran.Journal Article20190904In this work, we employ a combination of variational iteration method (VIM) and Pad´e approximation method, called the VIM-Pad´e technique, to solve some nonlinear initial value problems and a delay differential equation (DDE). Some examples are provided to illustrate the capability and reliability of the technique. The obtained results by using the VIM are compared to the results of this technique. This comparison shows that VIM-Pad´e technique is more effective than VIM and yields faster convergence compared to the VIM.https://cmde.tabrizu.ac.ir/article_10758_ea230fd1473563b6d52910c673ee8809.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701Alpert wavelet system for solving fractional nonlinear Fredholm integro-differential equations7627731076010.22034/cmde.2020.35252.1604ENShabnamPaseban HagFaculty of Science,
Urmia University of Technology, Urmia, Iran.ElnazOsgooeiFaculty of Science, Urmia University of Technology, Urmia, Iran.0000-0002-1110-9410ElmiraAshpazzadehDepartment of Applied Mathematics,
Faculty of Mathematical Sciences,
University of Tabriz, Tabriz, Iran.Journal Article20190903In this paper, we first construct Alpert wavelet system and propose a computational method for solving a fractional nonlinear Fredholm integro-differential equation. Then we create an operational matrix of fractional integration and use it to simplify the equation to a system of algebraic equations. By using Newtons iterative method, this system is solved and the solution of the fractional nonlinear Fredholm integro-differential equations is achieved. Thresholding parameter is used to increase the sparsity of matrix coefficients and the speed of computations. Finally, the method is demonstrated by examples, and then compared results with CAS wavelet method show that our proposed method is more effective and accurate.https://cmde.tabrizu.ac.ir/article_10760_1dcaaff1c50512a583544a6189f75a33.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701An accurate method for nonlinear local fractional Wave-Like equations with variable coefficients7747871093510.22034/cmde.2020.32950.1532ENDjelloulZianeLaboratory of mathematics and its applications (LAMAP),
University of Oran1 Ahmed Ben Bella, Oran, 31000, Algeria.MountassirHamdi CherifLaboratory of mathematics and its applications (LAMAP), University of Oran1 Ahmed Ben Bella, Oran, 31000, Algeria.KacemBelghabaLaboratory of mathematics and its applications (LAMAP),
University of Oran1 Ahmed Ben Bella, Oran, 31000, Algeria.Fethi Bin MuhammadBelgacemDepartment of Mathematics, Faculty of Basic Education,
PAAET, Al-Ardhiya, Kuwait.Journal Article20190420The basic motivation of the present study is to apply the local fractional Sumudu variational iteration method (LFSVIM) for solving nonlinear wave-like equations with variable coefficients and within local fractional derivatives. The derivatives operators are taken in the local fractional sense. The results show that the LFSVIM is an appropriate method to find non-differentiable solutions for similar problems. The results of the solved examples showed the flexibility of applying this method and its ability to reach accurate results even with these new differential equations.https://cmde.tabrizu.ac.ir/article_10935_13ef96beb668f088b5428287e7a55265.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701Sensitivity analytic and synchronization of a new fractional-order financial system7887981077410.22034/cmde.2020.36747.1637ENMalekKarimianDepartment of mathematics, Payame Noor, Tehran, Iran.BashirNaderiDepartment of mathematics, Payame Noor, Tehran, Iran.YousefEdrisi TabrizDepartment of mathematics, Payame Noor, Tehran, Iran.Journal Article20191110In this paper, we present a new fractional-order financial system (FOFS) with the new parameters. We study the synchronization for commensurate order of the fractional-order financial system with disturbance observer (FOFSDO) on the new parameters. Also, the sensitivity analysis of the synchronization error was investigated by using the feedback control technique for the FOFSDO. The stability of the used method demonstrates by Lyapunov stability theorem. Numerical simulations are presented to ensure the validity and influence of the target feedback control design in the presence of extrinsic bounded unknown disturbance.https://cmde.tabrizu.ac.ir/article_10774_340ae8a46378af4f4b76f0ad4d9346b5.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701Solving of partial differential equations with distributed order in time using fractional-order Bernoulli-Legendre functions7998171077310.22034/cmde.2020.36904.1642ENParisaRahimkhaniDepartment of Mathematics,
Faculty of Mathematical Sciences,
Alzahra University, Tehran, Iran.YadollahOrdokhaniDepartment of Mathematics,
Faculty of Mathematical Sciences,
Alzahra University, Tehran, Iran.0000-0002-5167-6874Journal Article20191125In this paper, an efficient numerical method is used to provide the approximate solution of distributed-order fractional partial differential equations (DFPDEs). The proposed method is based on the fractional integral operator of fractional-order Bernoulli-Legendre functions and the collocation scheme. In our technique, by approximating functions that appear in the DFPDEs by fractional-order Bernoulli functions in space and fractional-order Legendre functions in time using Gauss-Legendre numerical integration, the under study problem is converted to a system of algebraic equations. This system is solved by using Newton's iterative scheme, and the numerical solution of DFPDEs is obtained. Finally, some numerical experiments are included to show the accuracy, efficiency, and applicability of the proposed method.https://cmde.tabrizu.ac.ir/article_10773_7d644191868d557429759ee8f614c243.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701Multiple solutions for a fourth-order elliptic equation involving singularity8188291093610.22034/cmde.2020.34709.1584ENRezaMahdavi KhanghahiDepartment of Pure Mathematics, Faculty of Science,
Imam Khomeini International University, Qazvin, Iran.AbdolrahmanRazaniDepartment of Pure Mathematics, Faculty of Science,
Imam Khomeini International University, Qazvin, Iran.0000-0002-3092-3530Journal Article20190723Here, we consider a fourth-order elliptic problem involving singularity and p(x)- biharmonic operator. Using Hardy’s inequality, S+-condition, and Palais-Smale condition, the existence of weak solutions in a bounded domain in RN is proved. Finally, we percent some examples.https://cmde.tabrizu.ac.ir/article_10936_9dd71b38fdf9ba8c297e4a697067cedf.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701The numerical approximation for the solution of linear and nonlinear integral equations of the second kind by interpolating moving least squares8308451093410.22034/cmde.2020.31729.1483ENMortezaAsgariFaculty of Mathematical Sciences, Shahrood
University of Technology, Shahrood, Iran.AliMesforushFaculty of Mathematical Sciences, Shahrood
University of Technology, Shahrood, Iran.0000-0001-9098-8953AlirezaNazemiFaculty of Mathematical Sciences, Shahrood
University of Technology, Shahrood, Iran.Journal Article20190125In this paper, the interpolating moving least-squares (IMLS) method is discussed. The interpolating moving least square methodology is an effective technique for the approximation of an unknown function by using a set of disordered data. Then we apply the IMLS method for the numerical solution of Volterra–Fredholm integral equations, and finally some examples are given to show the accuracy and applicability of the method.https://cmde.tabrizu.ac.ir/article_10934_127c0500a26ee320bd025d8c9b6c3125.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701Bernoulli wavelet method for numerical solutions of system of fuzzy integral equations8468571283910.22034/cmde.2021.22093.1257ENMohamed A.RamadanDepartment of Mathematics, Faculty of Science, Menoua University, Egypt.Mohamed RedaAliDepartment of Mathematics, Faculty of Engineering,
Benha University, Egypt.Journal Article20170731In this paper, we have proposed an efficient numerical method to solve a system linear fuzzy Fredholm integral equations of the second kind based on Bernoulli wavelet method (BWM). Bernoulli wavelets have been generated by dilation and translation of Bernoulli polynomials. The aim of this paper is to apply Bernoulli wavelet method to obtain approximate solutions of a system of linear Fredholm fuzzy integral equations. First, we introduce properties of Bernoulli wavelets then we used it to transform the integral equations to the system of algebraic equations, the error estimates of the proposed method are given and compared by solving some numerical examples.https://cmde.tabrizu.ac.ir/article_12839_63506e0da7c9fdf6de865a09a0f910d1.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701Collocation method based on Chebyshev polynomials for solving distributed order fractional differential equations8588731093310.22034/cmde.2020.38506.1695ENMarziehPourbabaeeDepartment of Applied Mathematics,
Faculty of Mathematical Sciences,
University of Kashan, Kashan, Iran.AbbasSaadatmandiDepartment of Applied Mathematics,
Faculty of Mathematical Sciences,
University of Kashan, Kashan, Iran.Journal Article20200221This work presents a new approximation approach to solve the linear/nonlinear distributed order fractional differential equations using the Chebyshev polynomials. Here, we use the Chebyshev polynomials combined with the idea of the collocation method for converting the distributed order fractional differential equation into a system of linear/nonlinear algebraic equations. Also, fractional differential equations with initial conditions can be solved by the present method. We also give the error bound of the modified equation for the present method. Moreover, four numerical tests are included to show the effectiveness and applicability of the suggested method.https://cmde.tabrizu.ac.ir/article_10933_4c7179b9247b6162d7f98ec5e0bda8f8.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701Optimal control of double delayed HIV-1 infection model of fighting a virus with another virus8748851210910.22034/cmde.2020.31728.1482ENNigarAliDepartment of Mathematics,
University of Malakand, Chakdara Dir (Lower)
Khyber Pakhtunkhawa, Pakistan.GulZamanDepartment of Mathematics,
University of Malakand, Chakdara Dir (Lower)
Khyber Pakhtunkhawa, Pakistan.Journal Article20190125A double delayed- HIV-1 infection model with optimal controls is taken into account. The proposed model consists of double-time delays and the following five compartments: uninfected cells CD4+ T cells, infected CD4+ T cells, double infected CD4+ T cells, human immunodeficiency virus, and recombinant virus. Further, the optimal controls functions are introduced into the model. Objective functional is constituted which aims to (i) minimize the infected cells quantity; (ii) minimize free virus particles number; and (iii) maximize healthy cells density in blood Then, the existence and uniqueness results for the optimal control pair are established. The optimality system is derived and then solved numerically using an iterative method with Runge-Kutta fourth-order scheme.https://cmde.tabrizu.ac.ir/article_12109_04898e41542e6de71cda36442a13c61c.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701Accelerated fitted operator finite difference method for singularly perturbed parabolic reaction-diffusion problems8868981211310.22034/cmde.2020.39685.1737ENTesfaye AgaBulloDepartment of Mathematics, College of Natural science, Jimma University, Ethiopia.0000-0001-6766-4803Gemechis FileDuressaDepartment of Mathematics, College of Natural science, Jimma University, Ethiopia.0000-0003-1889-4690GuyDeglaInstitut De Mathematiques et de sciences physiques,
Universit D’Abomey Calavi, Benin.0000-0003-1162-6140Journal Article20200506This paper deals with the numerical treatment of singularly perturbed parabolic reaction-diffusion initial boundary value problems. Introducing a fitting parameter into the asymptotic solution and applying average finite difference approximation, a fitted operator finite difference method is developed for solving the problem. To accelerate the rate of convergence of the method, Richardson extrapolation technique is applied. The consistency and stability of the proposed method have been established very well to ensure the convergence of the method. Numerical experimentation is carried out on some model problems and both the results are presented in tables and graphs. The numerical results are compared with findings of some methods existing in the literature and found to be more accurate. Generally, the formulated method is consistent, stable, and more accurate than some methods existing in the literature for solving singularly perturbed parabolic reaction-diffusion initial boundary value problems.https://cmde.tabrizu.ac.ir/article_12113_b179c0405f7b8fbf2b02093213dcac2e.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701Solving free boundary problem for an initial cell layer in multispecies biofilm formation by Newton-Raphson method8999071090810.22034/cmde.2020.39940.1746ENKarimIvazFaculty of Mathematical sciences,
University of Tabriz, Tabriz, Iran.MohammadAsadpour FazlallahiFaculty of Mathematical sciences,
University of Tabriz, Tabriz, Iran.Journal Article20200523The initial attached cell layer in multispecies biofilm growth is studied in this paper. The corresponding mathematical model leads to discuss a free boundary problem for a system of nonlinear hyperbolic partial differential equations, where the initial biofilm thickness is equal to zero. No assumptions on initial conditions for biomass concentrations and biofilm thickness are required. The data that the problem needs are the concentration of biomass in the bulk liquid and biomass flux from the bulk liquid. The differential equations are converted into an equivalent system of Volterra integral equations. We use Newton-Raphson method to solve the nonlinear system of Volterra integral equations (SVIEs) of the second kind. This method converts the nonlinear system of integral equations into a linear integral equation at each step.https://cmde.tabrizu.ac.ir/article_10908_082747e2679c9a440406ec4a05720457.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39829320210701Conformable double Laplace transform method for solving conformable fractional partial differential equations9089181211210.22034/cmde.2020.38135.1678ENSulimanAlfaqeihFaculty of Science Department of Mathematics,
35100 Ege University, Izmir, Turkey.EmineMısırlıFaculty of Science Department of Mathematics,
35100 Ege University, Izmir, Turkey.Journal Article20200126In the present article, we utilize the Conformable Double Laplace Transform Method (CDLTM) to get the exact solutions of a wide class of Conformable fractional differential in mathematical physics. The results obtained show that the proposed method is efficient, reliable, and easy to be implemented on related linear problems in applied mathematics and physics. Moreover, the (CDLTM) has a small computational size as compared to other methods.https://cmde.tabrizu.ac.ir/article_12112_c19661f0a313154e121fe54e9abe2815.pdf