University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
8
3
2020
08
01
A new operational matrix of Muntz-Legendre polynomials and Petrov-Galerkin method for solving fractional Volterra-Fredholm integro-differential equations
408
423
EN
Sedigheh
Sabermahani
0000-0002-7320-8908
Department of Mathematics, Faculty of Mathematical Sciences,
Alzahra University, Tehran, Iran.
s.saber@alzahra.ac.ir
Yadollah
Ordokhani
0000-0002-5167-6874
Department of Mathematics, Faculty of Mathematical Sciences,
Alzahra University, Tehran, Iran.
ordokhani@alzahra.ac.ir
10.22034/cmde.2020.32623.1515
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.
Muntz-Legendre polynomia,Petrov-Galerkin method,Laplace transform
https://cmde.tabrizu.ac.ir/article_9916.html
https://cmde.tabrizu.ac.ir/article_9916_558e32fbe7c52487502fdbb5cf28cd15.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
8
3
2020
08
01
On the new extensions to the Benjamin-Ono equation
424
445
EN
Khalid
Karam Ali
Mathematics Department, Faculty of Science,
Al-Azhar University, Nasr-City, Cairo, Egypt.
khalidkaram2012@yahoo.com
Rahmatullah
Ibrahim
Nuruddeen
Department of Mathematics, Faculty of Science,
Federal University Dutse, Jigawa State, Nigeria.
rahmatullah.n@fud.edu.ng
Ahmet
Yildirim
Department of Mathematics, Faculty of Science,
Ege University,Bornova, Izmir, Turkey.
ahmetyildirim80@gmail.com
10.22034/cmde.2020.32382.1505
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.
Solitary wave solutions,Benjamin-Ono equations extensions,Modified extended tanh method
https://cmde.tabrizu.ac.ir/article_9929.html
https://cmde.tabrizu.ac.ir/article_9929_ad3a99b6800934face9dc443e21b206d.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
8
3
2020
08
01
Lyapunov exponents for discontinuous dynamical systems of Filippov type
446
453
EN
Zahra
Monfared
Department of Applied Mathematics,
Ferdowsi University of Mashhad(FUM), Mashhad, Iran.
zahra.monfared@mail.um.ac.ir
Zohreh
Dadi
Department of Mathematics, Faculty of
Basic Sciences, University of Bojnord, Bojnord, Iran.
z.dadi@ub.ac.ir
Zahra
Afsharnezhad
Department of Applied Mathematics,
Ferdowsi University of Mashhad(FUM), Mashhad, Iran.
afsharnezhad@math.um.ac.ir
10.22034/cmde.2020.30174.1446
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.
Chaos,Lyapunov exponents,Filippov systems
https://cmde.tabrizu.ac.ir/article_9913.html
https://cmde.tabrizu.ac.ir/article_9913_c2e071945eab2425900a20d83bbd6ece.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
8
3
2020
08
01
A pseudo-spectral based method for time-fractional advection-diffusion equation
454
467
EN
Ali
Shokri
0000-0002-5942-5260
Department of Mathematics, Faculty of Sciences,
University of Zanjan, Zanjan, Iran.
a.shokri@znu.ac.ir
Soheila
Mirzaei
Department of Mathematics, Faculty of Sciences,
University of Zanjan, Zanjan, Iran.
soheila.mirzaie@znu.ac.ir
10.22034/cmde.2020.29307.1414
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.
Time-fractional advection-diffusion equations,Mittag-Leffler functions,Fractional derivative,Pseudo-spectral method
https://cmde.tabrizu.ac.ir/article_9918.html
https://cmde.tabrizu.ac.ir/article_9918_fac33f02e9c071d7cc438c6d8004d0c4.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
8
3
2020
08
01
Preserving asymptotic mean-square stability of stochastic theta scheme for systems of stochastic delay differential equations
468
479
EN
Omid
Farkhondeh Rouz
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.
omid_farkhonde_7088@yahoo.com
10.22034/cmde.2020.32139.1502
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.
Stochastic delay differential equations,Stochastic linear theta scheme,Asymptotic mean-square stability
https://cmde.tabrizu.ac.ir/article_9921.html
https://cmde.tabrizu.ac.ir/article_9921_3f10a4526577f99bc18606248bc855b6.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
8
3
2020
08
01
Some Results on Reflected Forward-Backward Stochastic differential equations
480
492
EN
Zahra
Poursepahi Samian
Faculty of Mathematical Sciences,
University of Guilan, Rasht, Iran.
z.poursepahi@yahoo.com
Mohammad Reza
Yaghouti
0000-0003-3137-5799
Faculty of Mathematical Sciences,
University of Guilan, Rasht, Iran.
yaghouti@guilan.ac.ir
10.22034/cmde.2020.26327.1337
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.
Forward-backward stochastic differential equations,Increasing processes,Monotonicity condition
https://cmde.tabrizu.ac.ir/article_9928.html
https://cmde.tabrizu.ac.ir/article_9928_149a9fe87857c689a6d64aefadf8d12a.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
8
3
2020
08
01
Solving the Fokker-Planck equation via the compact finite difference method
493
504
EN
Behnam
Sepehrian
Department of Mathematics, Faculty of Science,
Arak University, Arak 38156-8-8349, Iran.
b-sepehrian@araku.ac.ir
Marzieh
Karimi Radpoor
Department of Mathematics, Hamedan Branch,
Islamic Azad University, Hamedan, Iran.
m.karimiradpoor@gmail.com
10.22034/cmde.2020.28609.1396
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.
Boundary value method,Collocation method,Compact method,Cubic C$^1$-spline,Fokker-Planck equation
https://cmde.tabrizu.ac.ir/article_9917.html
https://cmde.tabrizu.ac.ir/article_9917_45d1be4183ace74eb0cd5ae72befb378.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
8
3
2020
08
01
Haar wavelet iteration method for solving time fractional Fisher's equation
505
522
EN
Ghader
Ahmadnezhad
Department of Mathematics,
Azarbaijan Shahid Madani University, Tabriz, Iran.
hazharahmadnezhad@gmail.com
Naser
Aghazadeh
0000-0003-2705-8942
Department of Mathematics,
Azarbaijan Shahid Madani University, Tabriz, Iran.
aghazadeh@azaruniv.ac.ir
Shahram
Rezapour
Department of Mathematics,
Azarbaijan Shahid Madani University, Tabriz, Iran.
rezapour@azaruniv.ac.ir
10.22034/cmde.2020.31527.1475
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.
fractional differential equation,Haar wavelet,Operational matrices,Numerical solution,iterative technique,Sylvester equation
https://cmde.tabrizu.ac.ir/article_9908.html
https://cmde.tabrizu.ac.ir/article_9908_98f964ee0ab0c0fc98b954412e9b9b57.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
8
3
2020
08
01
Symmetry analysis and exact solutions of acoustic equation
523
536
EN
Ahmad
Motamednezhad
Faculty of mathematical sciences, Shahrood university
of technology, Shahrood, Semnan, Iran.
a.motamedne@gmail.com
Fariba
Khajevand
Faculty of mathematical sciences, Shahrood university
of technology, Shahrood, Semnan, Iran.
khajevandfariba@yahoo.com
10.22034/cmde.2020.28975.1407
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.
Lie symmetry,Group-invariant solution,acoustic equation
https://cmde.tabrizu.ac.ir/article_9915.html
https://cmde.tabrizu.ac.ir/article_9915_f6c12047230cc509ccf57ea4b7cbea0b.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
8
3
2020
08
01
Analytical approximations of one-dimensional hyperbolic equation with non-local integral conditions by reduced differential transform method
537
552
EN
Seyyedeh Roodabeh
Moosavi
0000-0003-2981-5599
Department of Pure Mathematics,
Faculty of Mathematical Sciences,
University of Guilan, P.O.Box 1914, Rasht, Iran.
moosavinoori@gmail.com
Nasir
Taghizadeh
0000-0002-3865-7943
Department of Pure Mathematics,
Faculty of Mathematical Sciences,
University of Guilan, P.O.Box 1914, Rasht, Iran.
taghizadeh@guilan.ac.ir
Jalil
Manafian
0000-0001-7201-6667
Department of Applied Mathematics,
Faculty of Mathematical Sciences,
University of Tabriz, Tabriz, Iran.
manafeian2@gmail.com
10.22034/cmde.2020.29576.1424
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.
Reduced Differential Transform Method,Non-classic condition,Hyperbolic partial differential equation,Approximate solutions,Adomian’s polynomial
https://cmde.tabrizu.ac.ir/article_10364.html
https://cmde.tabrizu.ac.ir/article_10364_922fb130c5837367ed518b69ef07c08e.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
8
3
2020
08
01
An efficient high-order compact finite difference method for the Helmholtz equation
553
563
EN
Jafar
Biazar
0000-0001-8026-2999
Department of Applied Mathematics, University of Guilan,
P. O. Box. 41635-19141, P. C. 41938336997, Rasht, Iran.
biazar@guilan.ac.ir
Roxana
Asayesh
Department of Applied Mathematics, University of Guilan,
P. O. Box. 41635-19141, P. C. 41938336997, Rasht, Iran.
roksana.asayesh@gmail.com
10.22034/cmde.2020.27993.1382
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.
Helmholtz equation,Compact finite difference method,Fast discrete sine transform
https://cmde.tabrizu.ac.ir/article_9910.html
https://cmde.tabrizu.ac.ir/article_9910_3b6fe6a49e477ad98e3912767be69d04.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
8
3
2020
08
01
Stability analysis of third derivative multi-step methods for stiff initial value problems
564
572
EN
Zohreh
Eskandari
Department of Mathematical sciences, Shahrekord University, Shahrekord, Iran.
zohre.eskandari.math.iut@gmail.com
Mohammad Shafi
Dahaghin
Department of Mathematical sciences, Shahrekord University, Shahrekord, Iran.
mshdahaghin@gmail.com
10.22034/cmde.2020.28604.1395
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.
Stiff ODEs,Multi-step methods,Super-future point technique,Stability analysis
https://cmde.tabrizu.ac.ir/article_9912.html
https://cmde.tabrizu.ac.ir/article_9912_4dd7eff787d471ef6024ad42e78223f2.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
8
3
2020
08
01
A numerical scheme for diffusion-convection equation with piecewise constant arguments
573
584
EN
Mojgan
Esmaeilzadeh
Department of Applied Mathematics, Lahijan Branch,
Islamic Azad University, Lahijan, Iran.
m.esmailzadeh11@gmail.com
Hashem
Saberi Najafi
0000-0001-6723-8126
Department of Applied Mathematics, Lahijan Branch,
Islamic Azad University, Lahijan, Iran.
hnajafi@guilan.ac.ir
Hossein
Aminikhah
Faculty of Mathematical Sciences, Department of Applied Mathematics
and Computer Science, University of Guilan, Rasht, Iran.
aminikhah@guilan.ac.ir
10.22034/cmde.2020.31155.1468
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.
Diffusion-Convection equation,piecewise constant arguments,theta-methods,asymptotically stability
https://cmde.tabrizu.ac.ir/article_9919.html
https://cmde.tabrizu.ac.ir/article_9919_b30d6ee562b774995c8fb26dfba2b864.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
8
3
2020
08
01
Complex Wave Surfaces to the Extended Shallow Water Wave Model with (2+1)-dimensional
585
596
EN
Haci Mehmet
Baskonus
0000-0003-4085-3625
Harran University, Faculty of Education, Sanliurfa, Turkey.
hmbaskonus@gmail.com
Esin Inan
Eskitascioglu
Van Yuzuncu Yil University, Faculty of Science, Van, Turkey.
inancinar@yyu.edu.tr
10.22034/cmde.2020.31374.1471
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.
Extended shallow water wave model,Sine-Gordon expansion method,Complex mixed-dark and bright solitons
https://cmde.tabrizu.ac.ir/article_9923.html
https://cmde.tabrizu.ac.ir/article_9923_3b8c9050d94e3f2f9fcb40718e44a055.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
8
3
2020
08
01
Symbolic methods to construct a cusp, breathers, kink, rogue waves and some soliton waves solutions of nonlinear partial differential equations
597
609
EN
Md
Nur
ALAM
School of Mathematical Sciences,
University of Science and Technology of China,
230026, Hefei, China.
nuralam23@mail.ustc.edu.cn
Xin
Li
School of Mathematical Sciences,
University of Science and Technology of China,
230026, Hefei, China.
lixustc@ustc.edu.cn
10.22034/cmde.2020.31942.1489
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.
The $exp(-phi(xi))$-expansion method,the fourth order Benjamin-Ono equation,BBM equation,traveling wave solutions,Nonlinear evolution equation
https://cmde.tabrizu.ac.ir/article_10523.html
https://cmde.tabrizu.ac.ir/article_10523_2917ea233324ddc3d1709c1f21ead5d7.pdf