@article { author = {Sabermahani, Sedigheh and Ordokhani, Yadollah}, title = {A new operational matrix of Muntz-Legendre polynomials and Petrov-Galerkin method for solving fractional Volterra-Fredholm integro-differential equations}, journal = {Computational Methods for Differential Equations}, volume = {8}, number = {3}, pages = {408-423}, year = {2020}, publisher = {University of Tabriz}, issn = {2345-3982}, eissn = {2383-2533}, doi = {10.22034/cmde.2020.32623.1515}, abstract = {This 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.}, keywords = {Muntz-Legendre polynomia,Petrov-Galerkin method,Laplace transform}, url = {https://cmde.tabrizu.ac.ir/article_9916.html}, eprint = {https://cmde.tabrizu.ac.ir/article_9916_558e32fbe7c52487502fdbb5cf28cd15.pdf} } @article { author = {Karam Ali, Khalid and Nuruddeen, Rahmatullah and Yildirim, Ahmet}, title = {On the new extensions to the Benjamin-Ono equation}, journal = {Computational Methods for Differential Equations}, volume = {8}, number = {3}, pages = {424-445}, year = {2020}, publisher = {University of Tabriz}, issn = {2345-3982}, eissn = {2383-2533}, doi = {10.22034/cmde.2020.32382.1505}, abstract = {In 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.}, keywords = {Solitary wave solutions,Benjamin-Ono equations extensions,Modified extended tanh method}, url = {https://cmde.tabrizu.ac.ir/article_9929.html}, eprint = {https://cmde.tabrizu.ac.ir/article_9929_ad3a99b6800934face9dc443e21b206d.pdf} } @article { author = {Monfared, Zahra and Dadi, Zohreh and Afsharnezhad, Zahra}, title = {Lyapunov exponents for discontinuous dynamical systems of Filippov type}, journal = {Computational Methods for Differential Equations}, volume = {8}, number = {3}, pages = {446-453}, year = {2020}, publisher = {University of Tabriz}, issn = {2345-3982}, eissn = {2383-2533}, doi = {10.22034/cmde.2020.30174.1446}, abstract = {‎The 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.}, keywords = {‎Chaos,Lyapunov exponents,Filippov systems}, url = {https://cmde.tabrizu.ac.ir/article_9913.html}, eprint = {https://cmde.tabrizu.ac.ir/article_9913_c2e071945eab2425900a20d83bbd6ece.pdf} } @article { author = {Shokri, Ali and Mirzaei, Soheila}, title = {A pseudo-spectral based method for time-fractional advection-diffusion equation}, journal = {Computational Methods for Differential Equations}, volume = {8}, number = {3}, pages = {454-467}, year = {2020}, publisher = {University of Tabriz}, issn = {2345-3982}, eissn = {2383-2533}, doi = {10.22034/cmde.2020.29307.1414}, abstract = {In 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.}, keywords = {Time-fractional advection-diffusion equations,Mittag-Leffler functions,Fractional derivative,Pseudo-spectral method}, url = {https://cmde.tabrizu.ac.ir/article_9918.html}, eprint = {https://cmde.tabrizu.ac.ir/article_9918_fac33f02e9c071d7cc438c6d8004d0c4.pdf} } @article { author = {Farkhondeh Rouz, Omid}, title = {Preserving asymptotic mean-square stability of stochastic theta scheme for systems of stochastic delay differential equations}, journal = {Computational Methods for Differential Equations}, volume = {8}, number = {3}, pages = {468-479}, year = {2020}, publisher = {University of Tabriz}, issn = {2345-3982}, eissn = {2383-2533}, doi = {10.22034/cmde.2020.32139.1502}, abstract = {This 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.}, keywords = {Stochastic delay differential equations,Stochastic linear theta scheme,Asymptotic mean-square stability}, url = {https://cmde.tabrizu.ac.ir/article_9921.html}, eprint = {https://cmde.tabrizu.ac.ir/article_9921_3f10a4526577f99bc18606248bc855b6.pdf} } @article { author = {Poursepahi Samian, Zahra and Yaghouti, Mohammad Reza}, title = {Some Results on Reflected Forward-Backward Stochastic differential equations}, journal = {Computational Methods for Differential Equations}, volume = {8}, number = {3}, pages = {480-492}, year = {2020}, publisher = {University of Tabriz}, issn = {2345-3982}, eissn = {2383-2533}, doi = {10.22034/cmde.2020.26327.1337}, abstract = {This 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.}, keywords = {Forward-backward stochastic differential equations,Increasing processes,Monotonicity condition}, url = {https://cmde.tabrizu.ac.ir/article_9928.html}, eprint = {https://cmde.tabrizu.ac.ir/article_9928_149a9fe87857c689a6d64aefadf8d12a.pdf} } @article { author = {Sepehrian, Behnam and Karimi Radpoor, Marzieh}, title = {Solving the Fokker-Planck equation via the compact finite difference method}, journal = {Computational Methods for Differential Equations}, volume = {8}, number = {3}, pages = {493-504}, year = {2020}, publisher = {University of Tabriz}, issn = {2345-3982}, eissn = {2383-2533}, doi = {10.22034/cmde.2020.28609.1396}, abstract = {In 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.}, keywords = {Boundary value method,Collocation method,Compact method,Cubic C$^1$-spline,Fokker-Planck equation}, url = {https://cmde.tabrizu.ac.ir/article_9917.html}, eprint = {https://cmde.tabrizu.ac.ir/article_9917_45d1be4183ace74eb0cd5ae72befb378.pdf} } @article { author = {Ahmadnezhad, Ghader and Aghazadeh, Naser and Rezapour, Shahram}, title = {Haar wavelet iteration method for solving time fractional Fisher's equation}, journal = {Computational Methods for Differential Equations}, volume = {8}, number = {3}, pages = {505-522}, year = {2020}, publisher = {University of Tabriz}, issn = {2345-3982}, eissn = {2383-2533}, doi = {10.22034/cmde.2020.31527.1475}, abstract = {In 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.}, keywords = {fractional differential equation,Haar wavelet,Operational matrices,Numerical solution,iterative technique,Sylvester equation}, url = {https://cmde.tabrizu.ac.ir/article_9908.html}, eprint = {https://cmde.tabrizu.ac.ir/article_9908_98f964ee0ab0c0fc98b954412e9b9b57.pdf} } @article { author = {Motamednezhad, Ahmad and Khajevand, Fariba}, title = {Symmetry analysis and exact solutions of acoustic equation}, journal = {Computational Methods for Differential Equations}, volume = {8}, number = {3}, pages = {523-536}, year = {2020}, publisher = {University of Tabriz}, issn = {2345-3982}, eissn = {2383-2533}, doi = {10.22034/cmde.2020.28975.1407}, abstract = {‎‎The 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.}, keywords = {Lie symmetry,Group-invariant solution,acoustic equation}, url = {https://cmde.tabrizu.ac.ir/article_9915.html}, eprint = {https://cmde.tabrizu.ac.ir/article_9915_f6c12047230cc509ccf57ea4b7cbea0b.pdf} } @article { author = {Moosavi, Seyyedeh Roodabeh and Taghizadeh, Nasir and Manafian, Jalil}, title = {Analytical approximations of one-dimensional hyperbolic equation with non-local integral conditions by reduced differential transform method}, journal = {Computational Methods for Differential Equations}, volume = {8}, number = {3}, pages = {537-552}, year = {2020}, publisher = {University of Tabriz}, issn = {2345-3982}, eissn = {2383-2533}, doi = {10.22034/cmde.2020.29576.1424}, abstract = {In 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.}, keywords = {Reduced Differential Transform Method,Non-classic condition,Hyperbolic partial differential equation,Approximate solutions,Adomian’s polynomial}, url = {https://cmde.tabrizu.ac.ir/article_10364.html}, eprint = {https://cmde.tabrizu.ac.ir/article_10364_922fb130c5837367ed518b69ef07c08e.pdf} } @article { author = {Biazar, Jafar and Asayesh, Roxana}, title = {An efficient high-order compact finite difference method for the Helmholtz equation}, journal = {Computational Methods for Differential Equations}, volume = {8}, number = {3}, pages = {553-563}, year = {2020}, publisher = {University of Tabriz}, issn = {2345-3982}, eissn = {2383-2533}, doi = {10.22034/cmde.2020.27993.1382}, abstract = {This 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.}, keywords = {Helmholtz equation,Compact finite difference method,Fast discrete sine transform}, url = {https://cmde.tabrizu.ac.ir/article_9910.html}, eprint = {https://cmde.tabrizu.ac.ir/article_9910_3b6fe6a49e477ad98e3912767be69d04.pdf} } @article { author = {Eskandari, Zohreh and Dahaghin, Mohammad Shafi}, title = {Stability analysis of third derivative multi-step methods for stiff initial value problems}, journal = {Computational Methods for Differential Equations}, volume = {8}, number = {3}, pages = {564-572}, year = {2020}, publisher = {University of Tabriz}, issn = {2345-3982}, eissn = {2383-2533}, doi = {10.22034/cmde.2020.28604.1395}, abstract = {In 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.}, keywords = {Stiff ODEs,Multi-step methods,Super-future point technique,Stability analysis}, url = {https://cmde.tabrizu.ac.ir/article_9912.html}, eprint = {https://cmde.tabrizu.ac.ir/article_9912_4dd7eff787d471ef6024ad42e78223f2.pdf} } @article { author = {Esmaeilzadeh, Mojgan and Saberi Najafi, Hashem and Aminikhah, Hossein}, title = {A numerical scheme for diffusion-convection equation with piecewise constant arguments}, journal = {Computational Methods for Differential Equations}, volume = {8}, number = {3}, pages = {573-584}, year = {2020}, publisher = {University of Tabriz}, issn = {2345-3982}, eissn = {2383-2533}, doi = {10.22034/cmde.2020.31155.1468}, abstract = {This 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.}, keywords = {Diffusion-Convection equation,piecewise constant arguments,theta-methods,asymptotically stability}, url = {https://cmde.tabrizu.ac.ir/article_9919.html}, eprint = {https://cmde.tabrizu.ac.ir/article_9919_b30d6ee562b774995c8fb26dfba2b864.pdf} } @article { author = {Baskonus, Haci Mehmet and Eskitascioglu, Esin Inan}, title = {Complex Wave Surfaces to the Extended Shallow Water Wave Model with (2+1)-dimensional}, journal = {Computational Methods for Differential Equations}, volume = {8}, number = {3}, pages = {585-596}, year = {2020}, publisher = {University of Tabriz}, issn = {2345-3982}, eissn = {2383-2533}, doi = {10.22034/cmde.2020.31374.1471}, abstract = {In 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.}, keywords = {Extended shallow water wave model,Sine-Gordon expansion method,Complex mixed-dark and bright solitons}, url = {https://cmde.tabrizu.ac.ir/article_9923.html}, eprint = {https://cmde.tabrizu.ac.ir/article_9923_3b8c9050d94e3f2f9fcb40718e44a055.pdf} } @article { author = {ALAM, Md and Li, Xin}, title = {Symbolic methods to construct a cusp, breathers, kink, rogue waves and some soliton waves solutions of nonlinear partial differential equations}, journal = {Computational Methods for Differential Equations}, volume = {8}, number = {3}, pages = {597-609}, year = {2020}, publisher = {University of Tabriz}, issn = {2345-3982}, eissn = {2383-2533}, doi = {10.22034/cmde.2020.31942.1489}, abstract = {A 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.}, keywords = {The $exp(-phi(xi))$-expansion method,the fourth order Benjamin-Ono equation,BBM equation,traveling wave solutions,Nonlinear evolution equation}, url = {https://cmde.tabrizu.ac.ir/article_10523.html}, eprint = {https://cmde.tabrizu.ac.ir/article_10523_2917ea233324ddc3d1709c1f21ead5d7.pdf} }