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