University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
3
3
2015
07
01
Solving high-order partial differential equations in unbounded domains by means of double exponential second kind Chebyshev approximation
147
162
EN
Mohamed Abdel-Latif
Ramadan
Mathematics Department, Faculty of Science, Menoufia University, Shebein El-Koom, Egypt
ramadanmohamed13@yahoo.com
Kamal Mohamed
Raslsn
Mathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo, Egypt
kamal_raslan@yahoo.com
Talaat El-Sayed
El-Danaf
Department of Mathematics and Statistics, Taibah University Madinah Munawwarah, KSA
talaat11@yahoo.com
Mohamed
Abd El Salam
Mathematics Department, Faculty of Science
Al-Azhar University, Nasr-City, 11884, Cairo, Egypt
mohamed_salam1985@yahoo.com
In this paper, a collocation method for solving high-order linear partial differential equations (PDEs) with variable coefficients under more general form of conditions is presented. This method is based on the approximation of the truncated double exponential second kind Chebyshev (ESC) series. The definition of the partial derivative is presented and derived as new operational matrices of derivatives. All principles and properties of the ESC functions are derived and introduced by us as a new basis defined in the whole range. The method transforms the PDEs and conditions into block matrix equations, which correspond to system of linear algebraic equations with unknown ESC coefficients, by using ESC collocation points. Combining these matrix equations and then solving the system yield the ESC coefficients of the solution function. Numerical examples are included to test the validity and applicability of the method.
Exponential second kind Chebyshev functions,High-order partial differential equations,Collocation method
http://cmde.tabrizu.ac.ir/article_4716.html
http://cmde.tabrizu.ac.ir/article_4716_865a972e969ce0256b6db9b8006f6073.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
3
3
2015
07
01
Iterative scheme to a coupled system of highly nonlinear fractional order differential equations
163
176
EN
Kamal
Shah
Department of Mathematics, University of Malakand, Chakadara Dir(L), Khyber Pakhtunkhwa, Pakistan
kamalshah408@gmail.com
Rahmat
Khan
Department of Mathematics, University of Malakand, Chakadara Dir(L), Khyber Pakhtunkhwa, Pakistan
rahmat_alipk@yahoo.com
In this article, we investigate sufficient conditions for existence of maximal and minimal solutions to a coupled system of highly nonlinear differential equations of fractional order with mixed type boundary conditions. To achieve this goal, we apply monotone iterative technique together with the method of upper and lower solutions. Also an error estimation is given to check the accuracy of the method. We provide an example to illustrate our main results.
Coupled system,Mixed type boundary conditions, Upper and lower solutions, Monotone iterative technique, Existence and uniqueness results
http://cmde.tabrizu.ac.ir/article_4771.html
http://cmde.tabrizu.ac.ir/article_4771_2caa42796fa9abd0485bb9672a8dd0b4.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
3
3
2015
07
01
Solution of Bang-Bang Optimal Control Problems by Using Bezier Polynomials
177
191
EN
Ayatollah
Yari
Department of Applied Mathematics,
Faculty of Mathematical Sciences,
Payame Noor University, PO BOX 19395-3697, Tehran, Iran
a_yary@yahoo.com
Mirkamal
Mirnia
Department of Applied Mathematics,
Faculty of Mathematical Sciences,
University of Tabriz
mirnia-kam@tabrizu.ac.ir
Aghileh
Heydari
Department of Applied Mathematics,
Faculty of Mathematical Sciences,
Payame Noor University, PO BOX 19395-3697, Tehran,Iran
a_heidari@pnu.ac.ir
In this paper, a new numerical method is presented for solving the optimal control problems of Bang-Bang type with free or fixed terminal time. The method is based on Bezier polynomials which are presented in any interval as $[t_0,t_f]$. The problems are reduced to a constrained problems which can be solved by using Lagrangian method. The constraints of these problems are terminal state and conditions. Illustrative examples are included to demonstrate the validity and applicability of the method.
Optimal control,Bang-Bang control,Minimum-time,Bezier polynomials family,Best approximation
http://cmde.tabrizu.ac.ir/article_4770.html
http://cmde.tabrizu.ac.ir/article_4770_a00ee50be84f54611ea5e5335b492e2d.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
3
3
2015
07
01
Explicit exact solutions for variable coefficient Broer-Kaup equations
192
199
EN
Manjit
Singh
Yadawindra College of Engineering
Punjabi University Guru Kashi Campus
Talwandi Sabo-151302, Punjab, India
manjitcsir@gmail.com
R.K.
Gupta
Centre for Mathematics and Statistics
School of Basic and Applied Sciences, Central University of Punjab,
Bathinda-151001, Punjab, India
rajeshgupt@thapar.edu
Based on symbolic manipulation program Maple and using Riccati equation mapping method several explicit exact solutions including kink, soliton-like, periodic and rational solutions are obtained for (2+1)-dimensional variable coefficient Broer-Kaup system in quite a straightforward manner. The known solutions of Riccati equation are used to construct new solutions for variable coefficient Broer-Kaup system.
Broer-Kaup equations,Riccati equation mapping method,Explicit exact solutions
http://cmde.tabrizu.ac.ir/article_4774.html
http://cmde.tabrizu.ac.ir/article_4774_835457e718ceb27933e9085f35bb7eb9.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
3
3
2015
07
01
An application of differential transform method for solving nonlinear optimal control problems
200
217
EN
Alireza
Nazemi
Department of Mathematics,
School of Mathematical Sciences,
Shahrood University of Technology,
P.O. Box 3619995161-316, Tel-Fax No:+98 23-32300235,
Shahrood, Iran
nazemi20042003@gmail.com
Saiedeh
Hesam
Department of Mathematics,
School of Mathematical Sciences,
Shahrood University of Technology,
P.O. Box 3619995161-316, Tel-Fax No:+98 23-32300235,
Shahrood, Iran
taranome2009@yahoo.com
Ahmad
Haghbin
Department of Mathematics, Gorgan branch,
Islamic Azad University, Gorgan, Iran
ahmadbin@yahoo.com
In this paper, we present a capable algorithm for solving a class of nonlinear optimal control problems (OCP's). The approach rest mainly on the differential transform method (DTM) which is one of the approximate methods. The DTM is a powerful and efficient technique for finding solutions of nonlinear equations without the need of a linearization process. Utilizing this approach, the optimal control and the corresponding trajectory of the OCP's are found in the form of rapidly convergent series with easily computed components. Numerical results are also given for several test examples to demonstrate the applicability and the efficiency of the method.
Optimal Control Problems,Differential transform method,Hamiltonian system
http://cmde.tabrizu.ac.ir/article_4972.html
http://cmde.tabrizu.ac.ir/article_4972_22d60c5610c1acd953ee761bc55ca38e.pdf
University of Tabriz
Computational Methods for Differential Equations
2345-3982
2383-2533
3
3
2015
07
01
Non-polynomial Spline Method for Solving Coupled Burgers Equations
218
230
EN
Khalid
K. Ali
Department, Faculty of Science,
Al-Azhar University, Nasr City (11884), Cairo, Egypt
khalidkaram2012@yahoo.com
K. R.
Raslan
Department, Faculty of Science,
Al-Azhar University, Nasr City (11884), Cairo, Egypt.
kamal_raslan@yahoo.com
Talaat S.
El-Danaf
Mathematics Department, Faculty of Science, Menoufia University, Shebein El-Koom, Egypt
talaat11@yahoo.com
In this paper, non-polynomial spline method for solving Coupled Burgers Equations are presented. We take a new spline function. The stability analysis using Von-Neumann technique shows the scheme is unconditionally stable. To test accuracy the error norms 2L, L are computed and give two examples to illustrate the sufficiency of the method for solving such nonlinear partial differential equations. These results show that the technique introduced here is accurate and easy to apply.
Non-polynomial,spline method,Coupled,Burger’s,Equations
http://cmde.tabrizu.ac.ir/article_4974.html
http://cmde.tabrizu.ac.ir/article_4974_e3d373c5ccd58d93abdaae09981caee2.pdf