University of TabrizComputational Methods for Differential Equations2345-39828320200801A new operational matrix of Muntz-Legendre polynomials and Petrov-Galerkin method for solving fractional Volterra-Fredholm integro-differential equations408423991610.22034/cmde.2020.32623.1515ENSedighehSabermahaniDepartment of Mathematics, Faculty of Mathematical Sciences,
Alzahra University, Tehran, Iran.0000-0002-7320-8908YadollahOrdokhaniDepartment of Mathematics, Faculty of Mathematical Sciences,
Alzahra University, Tehran, Iran.0000-0002-5167-6874Journal Article20190320This manuscript is devoted to present an efficient numerical method for finding numerical solution of Volterra-Fredholm integro-differential equations of fractional-order. The technique is based on the M\"{u}ntz-Legendre polynomials and Petrov-Galerkin method. A new Riemann-Liouville operational matrix for M\"{u}ntz-Legendre polynomials is proposed using Laplace transform. Employing this operational matrix and Petrov-Galerkin method, the problem transforms to a system of algebraic equations. Next, we solve this system by applying any iterative method. An estimation of the error is proposed. The efficiency and accuracy of the present scheme is illustrated using several examples.https://cmde.tabrizu.ac.ir/article_9916_558e32fbe7c52487502fdbb5cf28cd15.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39828320200801On the new extensions to the Benjamin-Ono equation424445992910.22034/cmde.2020.32382.1505ENKhalidKaram AliMathematics Department, Faculty of Science,
Al-Azhar University, Nasr-City, Cairo, Egypt.Rahmatullah IbrahimNuruddeenDepartment of Mathematics, Faculty of Science,
Federal University Dutse, Jigawa State, Nigeria.AhmetYildirimDepartment of Mathematics, Faculty of Science,
Ege University,Bornova, Izmir, Turkey.Journal Article20190307In this paper, we analytically study the newly developed (2+1)-dimensional BenjaminOno equation by Wazwaz and propose its (3+1)-dimensional version. For this purpose, we successfully employed the modified extended tanh expansion method to construct certain hyperbolic, periodic and complex solitary wave structures simulated with the aid of symbolic computation using Mathematica. Also, we have depicted graphically the constructed solutions.https://cmde.tabrizu.ac.ir/article_9929_ad3a99b6800934face9dc443e21b206d.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39828320200801Lyapunov exponents for discontinuous dynamical systems of Filippov type446453991310.22034/cmde.2020.30174.1446ENZahraMonfaredDepartment of Applied Mathematics,
Ferdowsi University of Mashhad(FUM), Mashhad, Iran.ZohrehDadiDepartment of Mathematics, Faculty of
Basic Sciences, University of Bojnord, Bojnord, Iran.ZahraAfsharnezhadDepartment of Applied Mathematics,
Ferdowsi University of Mashhad(FUM), Mashhad, Iran.Journal Article20181104The area of discontinuous dynamical systems is almost a young research area, and the enthusiasm and necessity for analysing these systems have been growing. On the other hand, chaos appears in a rather wide class of discontinuous systems. One of the most important properties of chaos is sensitive dependence on initial conditions. Also, the most effective way to diagnosis chaotic systems is defining Lyapunov exponents of these systems. In addition, defining and calculating Lyapunov exponents for all discontinuous systems are real challenges. This paper is devoted to define Lyapunov exponents for discontinuous dynamical systems of Filippov type in order to investigate chaos for these systems.https://cmde.tabrizu.ac.ir/article_9913_c2e071945eab2425900a20d83bbd6ece.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39828320200801A pseudo-spectral based method for time-fractional advection-diffusion equation454467991810.22034/cmde.2020.29307.1414ENAliShokriDepartment of Mathematics, Faculty of Sciences,
University of Zanjan, Zanjan, Iran.0000-0002-5942-5260SoheilaMirzaeiDepartment of Mathematics, Faculty of Sciences,
University of Zanjan, Zanjan, Iran.Journal Article20180912In this paper, a pseudo-spectral method with the Lagrange polynomial basis is proposed to solve the time-fractional advection-diffusion equation. A semi-discrete approximation scheme is used for conversion of this equation to a system of ordinary fractional differential equations. Also, to protect the high accuracy of the spectral approximation, the Mittag-Leffler function is used for the integration along the time variable. Some examples are performed to illustrate the accuracy and efficiency of the proposed method.https://cmde.tabrizu.ac.ir/article_9918_fac33f02e9c071d7cc438c6d8004d0c4.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39828320200801Preserving asymptotic mean-square stability of stochastic theta scheme for systems of stochastic delay differential equations468479992110.22034/cmde.2020.32139.1502ENOmidFarkhondeh RouzFaculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.Journal Article20190224This article examines asymptotic mean-square stability analysis of stochastic linear theta (SLT) scheme for n-dimensional stochastic delay differential equations (SDDEs). We impose some conditions on drift and diffusion terms, which admit that the diffusion coefficient can be highly nonlinear and does not necessarily satisfy a linear growth or global Lipschitz condition. We prove that the proposed scheme is asymptotically mean square stable if the employed stepsize is smaller than a given and easily computable upper bound. In particular, based on our investigation in the case θ ∈[ 1/2 , 1], the stepsize is arbitrary. Eventually, numerical examples are given to demonstrate the effectiveness of our work.https://cmde.tabrizu.ac.ir/article_9921_3f10a4526577f99bc18606248bc855b6.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39828320200801Some Results on Reflected Forward-Backward Stochastic differential equations480492992810.22034/cmde.2020.26327.1337ENZahraPoursepahi SamianFaculty of Mathematical Sciences,
University of Guilan, Rasht, Iran.Mohammad RezaYaghoutiFaculty of Mathematical Sciences,
University of Guilan, Rasht, Iran.0000-0003-3137-5799Journal Article20180307This paper is concerned with the reflected forward-backward stochastic differential equations with continuous monotone coefficients. Using the continuity approach, we prove that there exists at least one solution for the reflected forward-backward stochastic differential equations. The distinct character of our result is that the coefficient of the reflected forward SDEs contains the solution variable of the reflected BSDEs.https://cmde.tabrizu.ac.ir/article_9928_149a9fe87857c689a6d64aefadf8d12a.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39828320200801Solving the Fokker-Planck equation via the compact finite difference method493504991710.22034/cmde.2020.28609.1396ENBehnamSepehrianDepartment of Mathematics, Faculty of Science,
Arak University, Arak 38156-8-8349, Iran.MarziehKarimi RadpoorDepartment of Mathematics, Hamedan Branch,
Islamic Azad University, Hamedan, Iran.Journal Article20180728In this study, we solve the Fokker-Planck equation by a compact finite difference method. By the finite difference method the computation of Fokker-Planck equation is reduced to a system of ordinary differential equations. Two different methods, boundary value method and cubic $C^1$-spline collocation method, for solving the resulting system are proposed. Both methods have fourth order accuracy in time variable. By the boundary value method some pointwise approximate solutions are only obtained. But, $C^1$-spline method gives a closed form approximation in each space step, too. Illustrative examples are included to demonstrate the validity and efficiency of the methods. A comparison is made with existing results.https://cmde.tabrizu.ac.ir/article_9917_45d1be4183ace74eb0cd5ae72befb378.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39828320200801Haar wavelet iteration method for solving time fractional Fisher's equation505522990810.22034/cmde.2020.31527.1475ENGhaderAhmadnezhadDepartment of Mathematics,
Azarbaijan Shahid Madani University, Tabriz, Iran.NaserAghazadehDepartment of Mathematics,
Azarbaijan Shahid Madani University, Tabriz, Iran.0000-0003-2705-8942ShahramRezapourDepartment of Mathematics,
Azarbaijan Shahid Madani University, Tabriz, Iran.Journal Article20190114In this work, we investigate fractional version of the Fisher equation and solve it by using an efficient iteration technique based on the Haar wavelet operational matrices. In fact, we convert the nonlinear equation into a Sylvester equation by the Haar wavelet collocation iteration method (HWCIM) to obtain the solution. We provide four numerical examples to illustrate the simplicity and efficiency of the technique.https://cmde.tabrizu.ac.ir/article_9908_98f964ee0ab0c0fc98b954412e9b9b57.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39828320200801Symmetry analysis and exact solutions of acoustic equation523536991510.22034/cmde.2020.28975.1407ENAhmadMotamednezhadFaculty of mathematical sciences, Shahrood university
of technology, Shahrood, Semnan, Iran.FaribaKhajevandFaculty of mathematical sciences, Shahrood university
of technology, Shahrood, Semnan, Iran.Journal Article20180822The Lie symmetry method for differential equations is applied to study the exact solutions of the acoustic PDE. This study is based on two methods: Kudryashov and direct method for reduction's process. By using the symmetry operators some exact solutions are found with their graphs are plotted.https://cmde.tabrizu.ac.ir/article_9915_f6c12047230cc509ccf57ea4b7cbea0b.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39828320200801Analytical approximations of one-dimensional hyperbolic equation with non-local integral conditions by reduced differential transform method5375521036410.22034/cmde.2020.29576.1424ENSeyyedeh RoodabehMoosaviDepartment of Pure Mathematics,
Faculty of Mathematical Sciences,
University of Guilan, P.O.Box 1914, Rasht, Iran.0000-0003-2981-5599NasirTaghizadehDepartment of Pure Mathematics,
Faculty of Mathematical Sciences,
University of Guilan, P.O.Box 1914, Rasht, Iran.0000-0002-3865-7943JalilManafianDepartment of Applied Mathematics,
Faculty of Mathematical Sciences,
University of Tabriz, Tabriz, Iran.0000-0001-7201-6667Journal Article20180928In this work, an initial-boundary value problem with a non-classic condition for the one-dimensional wave equation is presented and the reduced differential transform method is applied to ascertain the solution of the problem. We will investigate a new kind of non-local boundary value problems in which are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The advantage of this method is its simplicity in using, it solves the problem directly and straightforward without using perturbation, linearization, Adomian’s polynomial or any other transformation and gives the solution in the form of convergent power series with simply determinable components. Also, the convergence of the method is proved and seven examples are tested to shows the competency of our study.https://cmde.tabrizu.ac.ir/article_10364_922fb130c5837367ed518b69ef07c08e.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39828320200801An efficient high-order compact finite difference method for the Helmholtz equation553563991010.22034/cmde.2020.27993.1382ENJafarBiazarDepartment of Applied Mathematics, University of Guilan,
P. O. Box. 41635-19141, P. C. 41938336997, Rasht, Iran.0000-0001-8026-2999RoxanaAsayeshDepartment of Applied Mathematics, University of Guilan,
P. O. Box. 41635-19141, P. C. 41938336997, Rasht, Iran.Journal Article20180619This paper is devoted to applying the sixth-order compact finite difference approach to the Helmholtz equation. Instead of using matrix inversion, a discrete sinusoidal transform is used as a quick solver to solve the discretized system resulted from the compact finite difference method. Through this way, the computational costs of the method with large numbers of nodes are greatly reduced. The efficiency and accuracy of the scheme are investigated by solving some illustrative examples, having the exact solutions.https://cmde.tabrizu.ac.ir/article_9910_3b6fe6a49e477ad98e3912767be69d04.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39828320200801Stability analysis of third derivative multi-step methods for stiff initial value problems564572991210.22034/cmde.2020.28604.1395ENZohrehEskandariDepartment of Mathematical sciences, Shahrekord University, Shahrekord, Iran.Mohammad ShafiDahaghinDepartment of Mathematical sciences, Shahrekord University, Shahrekord, Iran.Journal Article20180728In this paper we present two class of third derivative multistep methods (TDMMs) that have good stability properties. Stability analysis of this method is examined and our numerical results are compared with the results of the existing method.https://cmde.tabrizu.ac.ir/article_9912_4dd7eff787d471ef6024ad42e78223f2.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39828320200801A numerical scheme for diffusion-convection equation with piecewise constant arguments573584991910.22034/cmde.2020.31155.1468ENMojganEsmaeilzadehDepartment of Applied Mathematics, Lahijan Branch,
Islamic Azad University, Lahijan, Iran.HashemSaberi NajafiDepartment of Applied Mathematics, Lahijan Branch,
Islamic Azad University, Lahijan, Iran.0000-0001-6723-8126HosseinAminikhahFaculty of Mathematical Sciences, Department of Applied Mathematics
and Computer Science, University of Guilan, Rasht, Iran.Journal Article20181227This article is concerned with using a ﬁnite difference method, namely the theta-methods, to solve the diffusion-convection equation with piecewise constant arguments.The stability of this scheme is also obtained. Since there are not many published results on the numerical solution of this sort of differential equation and because of the importance of the above equation in the physics and engineering sciences, we have decided to study and present a stable numerical solution for the above mentioned problem. At the end of article some experiments are done to demonstrate the stability of the scheme. We also draw the ﬁgures for the numerical and analytical solutions which conﬁrm ou results.The numerical solutions have also been compared with analytical solutions.https://cmde.tabrizu.ac.ir/article_9919_b30d6ee562b774995c8fb26dfba2b864.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39828320200801Complex Wave Surfaces to the Extended Shallow Water Wave Model with (2+1)-dimensional585596992310.22034/cmde.2020.31374.1471ENHaci MehmetBaskonusHarran University, Faculty of Education, Sanliurfa, Turkey.0000-0003-4085-3625Esin InanEskitasciogluVan Yuzuncu Yil University, Faculty of Science, Van, Turkey.Journal Article20190106In this paper, we apply an analytical method, namely, the sine-Gordon expansion method and extract some complex optical soliton solutions to the (2+1)-dimensional extended shallow water wave model, which describes the evolution of shallow water wave propagation. We obtain some complex mixed-dark and bright soliton solutions to this nonlinear model. Considering some suitable values of parameters, we plot the various dimensional simulations of every results found in this manuscript. We observe that our result may be useful in detecting some complex waves behaviors.https://cmde.tabrizu.ac.ir/article_9923_3b8c9050d94e3f2f9fcb40718e44a055.pdfUniversity of TabrizComputational Methods for Differential Equations2345-39828320200801Symbolic methods to construct a cusp, breathers, kink, rogue waves and some soliton waves solutions of nonlinear partial differential equations5976091052310.22034/cmde.2020.31942.1489ENMd NurALAMSchool of Mathematical Sciences,
University of Science and Technology of China,
230026, Hefei, China.XinLiSchool of Mathematical Sciences,
University of Science and Technology of China,
230026, Hefei, China.Journal Article20190207A cusp, bright breathers, dark breathers, kink, bright rogue waves and some soliton waves solutions are obtained by using the $exp(-\phi(\xi))$-expansion method for the fourth order Benjamin-Ono equation and BBM equations. The obtained solutions might be indicated and meaningful for narrating the physical phenomena in the real-world. For compatible values of the arbitrary parameter included in the solution, We plot the 3D surface of the all obtained solutions in this paper which are shown in Figures 1 to 10.https://cmde.tabrizu.ac.ir/article_10523_2917ea233324ddc3d1709c1f21ead5d7.pdf