University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
6
4
2018
10
01
A numerical approach for variable-order fractional unified chaotic systems with time-delay
396
410
EN
Sholeh
Yaghoobi
Department of Mathematics, Lahijan Branch,
Islamic Azad University, Lahijan, Iran
sholeyaghoobi91@stumail.liau.ac.ir
Behrouz
Parsa Moghaddam
0000-0003-4957-9028
Department of Mathematics, Lahijan Branch,
Islamic Azad University, Lahijan, Iran
parsa.math@gmail.com
Karim
Ivaz
Tabriz university
ivaz@tabriz.ac.ir
This paper proposes a new computational scheme for approximating variable-order fractional integral operators by means of finite element scheme. This strategy is extended to approximate the solution of a class of variable-order fractional nonlinear systems with time-delay. Numerical simulations are analyzed in the perspective of the mean absolute error and experimental convergence order. To illustrate the effectiveness of the proposed scheme, dynamical behaviors of the variable-order fractional unified chaotic systems with time-delay are investigated in the time domain.
Variable-order fractional calculus,Finite element scheme,Unified chaotic systems
https://cmde.tabrizu.ac.ir/article_7678.html
https://cmde.tabrizu.ac.ir/article_7678_56a30376a636faf293a45a8899fd1e33.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
6
4
2018
10
01
Solving singular integral equations by using orthogonal polynomials
411
425
EN
Samad
Ahdiaghdam
Department of Mathematics, Marand Branch,
Islamic Azad University, Marand, Iran
s_ahdiaghdam@tabrizu.ac.ir
In this paper, a special technique is studied by using the orthogonal Chebyshev polynomials to get approximate solutions for singular and hyper-singular integral equations of the first kind. A singular integral equation is converted to a system of algebraic equations based on using special properties of Chebyshev series. The error bounds are also stated for the regular part of approximate solution of singular integral equations. The efficiency of the method is illustrated through some examples.
Singular integral equations,Chebyshev systems,Approximate quadratures
https://cmde.tabrizu.ac.ir/article_7703.html
https://cmde.tabrizu.ac.ir/article_7703_bfffa64cdafff69a1db80fda554e03a3.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
6
4
2018
10
01
Space-time radial basis function collocation method for one-dimensional advection-diffusion problem
426
437
EN
Marzieh
Khaksarfard
Department of Mathematics, Faculty of Mathematical Sciences,
Alzahra University, Tehran, Iran
khaksarfard.m@gmail.com
Yadollah
Ordokhani
0000-0002-5167-6874
Department of Mathematics, Faculty of Mathematical Sciences,
Alzahra University, Tehran, Iran
ordokhani@alzahra.ac.ir
Mir Sajjad
Hashemi
Department of Mathematics, Basic Science Faculty,
Universiry of Bonab, Bonab, Iran
hashemi@bonabu.ac.ir
Kobra
Karimi
Department of Mathematics, Buin Zahra Technical University,
Buin Zahra, Qazvin, Iran
kobra.karimi@yahoo.com
The parabolic partial differential equation arises in many application of technologies. In this paper, we propose an approximate method for solution of the heat and advection-diffusion equations using Laguerre-Gaussians radial basis functions (LG-RBFs). The results of numerical experiments are compared with the other radial basis functions and the results of other schemes to confirm the validity of the presented method.
Radial basis functions,Heat and advection-diffusion equations,Partial differential equation
https://cmde.tabrizu.ac.ir/article_7670.html
https://cmde.tabrizu.ac.ir/article_7670_41fdd142453538f57cb04e13bb5ed69d.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
6
4
2018
10
01
An extended complete Chebyshev system of 3 Abelian integrals related to a non-algebraic Hamiltonian system
438
447
EN
Pegah
Moghimi
Department of Mathematical Sciences,
Isfahan University of Technology,
Isfahan, Iran, 84156-83111
p.moghimi@math.iut.ac.ir
Rasoul
Asheghi
Department of Mathematical Sciences,
Isfahan University of Technology,
Isfahan, Iran, 84156-83111
r.asheghi@cc.iut.ac.ir
Rasool
Kazemi
Department of Mathematical Sciences,
University of Kashan, Kashan, Iran, 87317-53153
r.kazemi@kashanu.ac.ir
In this paper, we study the Chebyshev property of the 3-dimentional vector space $E =\langle I_0, I_1, I_2\rangle$, where $I_k(h)=\int_{H=h}x^ky\,dx$ and $H(x,y)=\frac{1}{2}y^2+\frac{1}{2}(e^{-2x}+1)-e^{-x}$ is a non-algebraic Hamiltonian function. Our main result asserts that $E$ is an extended complete Chebyshev space for $h\in(0,\frac{1}{2})$. To this end, we use the criterion and tools developed by Grau et al. in \cite{Grau} to investigate when a collection of Abelian integrals is Chebyshev.
Non-algebraic Hamiltonian,Abelian integral,Chebyshev property,ECT-system
https://cmde.tabrizu.ac.ir/article_7715.html
https://cmde.tabrizu.ac.ir/article_7715_658154616797cb8e27927c86cb9b9d39.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
6
4
2018
10
01
Chebyshev finite difference method for solving a mathematical model arising in wastewater treatment plants
448
455
EN
Abbas
Saadatmandi
0000-0002-7744-7770
Department of Applied Mathematics, Faculty of Mathematical Sciences,
University of Kashan, Kashan 87317-53153, Iran
a.saadatmandi@gmail.com
Samaneh
Fayyaz
Department of Chemical industries, Faculty of Valiasr, Tehran Branch,
Technical and Vocational University (TVU), Tehran, Iran
samfayyaz@gmail.com
The Chebyshev finite difference method is applied to solve a system of two coupled nonlinear Lane-Emden differential equations arising in mathematical modelling of the excess sludge production from wastewater treatment plants. This method is based on a combination of the useful properties of Chebyshev polynomials approximation and finite difference method. The approach consists of reducing the problem to a set of algebraic equations. Numerical results are included to demonstrate the validity and applicability of the technique and a comparison is made with the existing results.
Chebyshev finite difference method,Gauss-Lobatto nodes,Excess sludge production,Activated sludge,Carbon substrate
https://cmde.tabrizu.ac.ir/article_7669.html
https://cmde.tabrizu.ac.ir/article_7669_f61c52f9072cc754cbc511507c3f21cc.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
6
4
2018
10
01
A total variation diminishing high resolution scheme for nonlinear conservation laws
456
470
EN
Javad
Farzi
Sahand university Of Technology,
P.O. Box 51335-1996, Tabriz, Iran
farzi@sut.ac.ir
Fayyaz
Khodadosti
0000-0003-4315-3907
Sahand university Of Technology,
P.O. Box 51335-1996, Tabriz, Iran
fayyaz64dr@gmail.com
In this paper we propose a novel high resolution scheme for scalar nonlinear hyperbolic conservation laws. The aim of high resolution schemes is to provide at least second order accuracy in smooth regions and produce sharp solutions near the discontinuities. We prove that the proposed scheme that is derived by utilizing an appropriate flux limiter is nonlinear stable in the sense of total variation diminishing (TVD). The TVD schemes are robust against the spurious oscillations and preserve the sharpness of the solution near the sharp discontinuities and shocks. We also, prove the positivity and maximum-principle properties for this scheme. The numerical results are presented for both of the advection and Burger’s equation. A comparison of numerical results with some classical limiter functions is also provided.
High resolution schemes,Flux limiter,Total variation diminishing,Nonlinear conservation laws
https://cmde.tabrizu.ac.ir/article_7677.html
https://cmde.tabrizu.ac.ir/article_7677_5d459469989103afafdfcf84fa49e253.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
6
4
2018
10
01
A numerical technique for solving a class of 2D variational problems using Legendre spectral method
471
482
EN
Kamal
Mamehrashi
0000-0003-3663-2424
Department of Mathematics, Payame Noor University,
P. O. BOX 19395-3697, Tehran, Iran
k_mamehrashi@pnu.ac.ir
Fahimeh
Soltanian
null
Department of Mathematics, Payame Noor University,
P. O. BOX 19395-3697, Tehran, Iran
f_soltanian@pnu.ac.ir
An effective numerical method based on Legendre polynomials is proposed for the solution of a class of variational problems with suitable boundary conditions. The Ritz spectral method is used for finding the approximate solution of the problem. By utilizing the Ritz method, the given nonlinear variational problem reduces to the problem of solving a system of algebraic equations. The advantage of the Ritz method is that it provides greater flexibility in which the boundary conditions are imposed at the end points of the interval. Furthermore, compared with the exact and eigenfunction solutions of the presented problems, the satisfactory results are obtained with low terms of basis elements. The convergence of the method is extensively discussed and finally two illustrative examples are included to demonstrate the validity and applicability of the proposed technique.
Ritz method,Legendre polynomials,2D Variational problems,Eigenfunction method
https://cmde.tabrizu.ac.ir/article_7648.html
https://cmde.tabrizu.ac.ir/article_7648_c2032555f7cdaa961e31d638d213cfd2.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
6
4
2018
10
01
Rational Chebyshev Collocation approach in the solution of the axisymmetric stagnation flow on a circular cylinder
483
500
EN
Ahmad
Golbabai
School of Mathematics,
Iran University of Science and Technology, Tehran, Iran
golbabai@iust.ac.ir
Sima
Samadpour
School of Mathematics,
Iran University of Science and Technology, Tehran, Iran
samadpoor@cmps2.iust.ac.ir
In this paper, a spectral collocation approach based on the rational Chebyshev functions for solving the axisymmetric stagnation point flow on an infinite stationary circular cylinder is suggested. The Navier-Stokes equations which govern the flow, are changed to a boundary value problem with a semi-infinite domain and a third-order nonlinear ordinary differential equation by applying proper similarity transformations. The approach is named the rational Chebyshev collocation (RCC) method. This method reduces this nonlinear ordinary differential equation to an algebraic equations system. RCC method is a strong kind of the collocation technique to solve the problems of boundary value over a semi-infinite interval without truncating them to a finite domain. We also present the comparison of this work with others and show that the present method is more effective and precise.
Axisymmetric flow,Stagnation point,Collocation method,Rational Chebyshev functions,Boundary value problem
https://cmde.tabrizu.ac.ir/article_7702.html
https://cmde.tabrizu.ac.ir/article_7702_893b99143ce525e40e03f36056999f13.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
6
4
2018
10
01
Homotopy perturbation method for eigenvalues of non-definite Sturm-Liouville problem
501
507
EN
Farhad
Dastmalchi Saei
Department of Mathematics, Tabriz Branch,
Islamic Azad University, Tabriz, Iran
farhadsaei@gmail.com
In this paper, we consider the application of the homotopy perturbation method (HPM) to compute the eigenvalues of the Sturm-Liouville problem (SLP) which is called non-definite SLP. Two important Examples show that HPM is reliable method for computing the eigenvalues of SLP.
Turning point,Sturm-Liouville,Homotopy perturbation method,eigenvalues
https://cmde.tabrizu.ac.ir/article_7708.html
https://cmde.tabrizu.ac.ir/article_7708_88ba55bd7aa2e537894b27bdd1c7c54c.pdf