BibTex RIS Kaynak Göster

Some new traveling wave solutions of the modified Benjamin-Bona-Mahony equation via the improved (G/G)-expansion method

Yıl 2015, Cilt: 3 Sayı: 1, 78 - 89, 22.12.2014

Hasibun Naher Farah Aini Abdullah Ahmet Bekir

Öz

The improved (G’/G)-expansion method is a powerful mathematical tool for solving nonlinear evolution equations whicharise in mathematical physics, engineering sciences and other technical arena. In this article, we construct some new exact travelingwave solutions for the modified Benjamin-Bona-Mahony equation by applying the improved (G’/G)-expansion method. In themethod, the general solution of the second order linear ordinary differential equation with constant coefficients is used for studyingnonlinear partial differential equations. The solution procedure of this method is executed by algebraic software, such as, Maple. Theobtained solutions including solitary and periodic wave solutions are presented in terms of the hyperbolic function, the trigonometricfunction and the rational forms. It is noteworthy to reveal that some of our solutions are in good agreement with the published resultsfor special cases which certifies our other solutions. Furthermore, the graphical presentations of some solutions are illustrated in thefigures

Anahtar Kelimeler

Traveling wave solutions the modified Benjamin-Bona-Mahony equation the improved (G’/G)-expansion method nonlinear evolution equations

Kaynakça

  • Malfliet, W., Solitary wave solutions of nonlinear wave equations,Am. J. Physics, 60, 650-654 (1992)
  • Wang, M.L., Zhou, Y.B., Li, Z.B., Application of homogeneous balance method to exact solutions of nonlinear equations in
  • mathematical physics Phys. Let. A, 216, 67-75 (1996)
  • He, J. ,Variational iteration method for delay differential equations, Communication in Nonlinear Science and Numerical
  • Simulation, 2(4), 235-236 (1997)
  • Abdou, M.A., Soliman, A.A., Variational iteration method for solving Burger’s and coupled Burger’s equations, J. Comput. Appl.
  • Math. 181, 245-251 (2005)
  • Yusufo˘glu, E., Bekir, A., The variational iteration method for solitary patterns solutions of gBBM equation, Physics Letter A, 367 (6) 461-464 (2007)
  • Younesian, D., Askari, H., Saadatnia, Z., Yildirim, A., Analytical solution for nonlinear wave propagation in shallow media using
  • the variational iteration method, Waves in Random and Complex Media, 22 (2), 133-142 (2012)
  • Abbasbandy, S., Numerical solution of non-linear Klein-Gordon equations by variational iteration method, Int. J. Numer. Meth.
  • Engng 70, 876-881 (2007)
  • Rogers, C., Shadwick, W.F., Backlund Transformations, Academic Press, New York. (1982)
  • Hirota, R., Exact solution of the KdV equation for multiple collisions of solitons, Phys. Rev. Lett. 27, 1192-1194 (1971)
  • Liu, S., Fu, Z., Liu, S., Zhao, Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289, 69–74 (2001)
  • Ebadi, G., Yousefzadeh, N., Triki, H., Yildirim, A., & Biswas, A., Envelope solitons, periodic waves and other solutions to Boussinesq-Burgers equation, Romanian Reports in Physics, 64(4), 915-932 (2012)
  • Salas, A.H., Gomez, C.A., Application of the Cole-Hopf transformation for finding exact solutions to several forms of the seventh- order KdV equation, Mathematical Problems in Engineering, 14, 194329 (2010)
  • Soliman, A.A. Abdo, H.A., New exact Solutions of nonlinear variants of the RLW, the PHI-four and Boussinesq equations based on modified extended direct algebraic method, International Journal of Nonlinear Science, 7, 3 274-282 (2009)
  • He, J.H., Wu, X.H., Exp-function method for nonlinear wave equations, Chaos Solitons and Fractals, 30 700-708 (2006)
  • Bekir, A., Boz, A., Exact solutions for nonlinear evolution equations using Exp-function method, Phys. Letters A, 372 1619-1625 (2008)
  • Naher, H., Abdullah, F.A., Akbar, M.A., The exp-function method for new exact solutions of the nonlinear partial differential equations, International Journal of the Physical Sciences, 6, 29 6706-6716 (2011)
  • Naher, H., Abdullah, F.A., Akbar, M.A., New travelling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method, Journal of Applied Mathematics, 575387 (2012)
  • Mohyud-Din, S.T., Noor, M.A., Noor, K.I., Exp-function method for traveling wave solutions of modified Zakharov-Kuznetsov equation, Journal of King Saud University, 22 213-216 (2010)
  • Ebadi, G., Krishnan, E. V., Labidi, M., Zerrad, E., & Biswas, A., Analytical and numerical solutions to the Davey–Stewartson equation with power-law nonlinearity, Waves in Random and Complex Media, 21(4), 559-590 (2011).
  • Bruzn, M. S., Gandarias, M. L., Camacho, J. C., Symmetry Analysis and Solutions for a Generalization of a Family of BBM Equations, Journal of Nonlinear Mathematical Physics, 15, 81-90 (2008)
  • Bruzn, M. S., Camacho, J. C., Gandarias, M. L., Symmetry Analysis and Solutions for a Family of BBM Equations, Nonlinear Evolution Equations and Dynamical Systems, (2007).
  • Jafari, H., Tajadodi, H., Biswas, A., Homotopy analysis method for solving a couple of evolution equations and comparison with Adomian’s decomposition method, Waves in Random and Complex Media, 21 (4), 657-667 (2011)
  • Bruzn, M. S., Gandarias, M. L., Symmetry reductions and exact solutions of Benjamin-Bona-Mahony-Burgers equation. In 4th Workshop Group Analysis of Differential Equations & Integrable Systems, 45-61 (2009).
  • Johnpillai, A. G., Kara, A. H., & Biswas, A., Symmetry reduction, exact group-invariant solutions and conservation laws of Benjamin-Bona-Mahoney equation, Applied Mathematics Letters, 26 (3), 376-381 (2013)
  • Wazwaz, A.M., A new (2+1)-dimensional Korteweg-de-Vries equation and its extension to a new (3+1)-dimensional Kadomtsev- Petviashvili equation, Physica Scripta, 84, 035010.(2011)
  • Ebadi, G., Kara, A.H., Petkovic, M.D., Biswas, A., Soliton solutions and conservation laws of Gilson-Pickering equation, Waves in Random and Complex Media, 21 (2) 378-385.(2011)
  • Wang, M., Li, X., Zhang, J., The (G’/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A, 372 417-423 (2008)
  • Bruzn, M. S., Gandarias, M. I., New exact solutions for a Benjamin-Bona-Mahony equation, World Scientific and Engineering Academy and Society (WSEAS), 189-194 (2008).
  • Zayed, E.M.E., Gepreel, K.A., The (G’/G)-expansion method for finding travelling wave solutions of nonlinear partial differential equations in mathematical physics, Journal of Mathematical Physics, 50 013502-013513 (2009)
  • Zayed, E.M.E., Al-Joudi, S., Applications of an Extended -Expansion Method to find Exact Solutions of Nonlinear PDEs in Mathematical Physics, Mathematical Problems in Engineering, 768573 (2010)
  • Ebadi, G., Biswas, A., Application of the (G’/G)-expansion method for nonlinear diffusion equations with nonlinear source, Journal of the Franklin Institute, 347(7), 1391-1398 (2010)
  • Ebadi, G., Biswas, A., The method and topological soliton solution of the K(m, n) equation, Communications in Nonlinear Science and Numerical Simulation, 16(6), 2377-2382 (2011)
  • Naher, H., Abdullah, F.A., Akbar, M.A., The (G’/G)-expansion method for abundant traveling wave solutions of Caudrey-Dodd- Gibbon equation, Mathematical Problems in Engineering, 218216. (2011)
  • Ebadi, G., Biswas, A., The method and 1-soliton solution of the Davey–Stewartson equation. Mathematical and Computer Modelling, 53(5), 694-698 (2011)
  • Ebadi, G., Krishnan, E. V., Biswas, A., Solitons and cnoidal waves of the Klein–Gordon–Zakharov equation in plasmas, Pramana, 79(2), 185-198 (2012)
  • Ebadi, G., Yildirim, A., Biswas, A., Chiral solitons with Bohm potential using method and exp-function method, Romanian Reports in Physics, 64(2), 357-366 (2012)
  • Ebadi, G., Mojaver, A., Johnson, S., Kumar, S., Biswas, A., Dynamics of dispersive topological solitons and its perturbations, Indian Journal of Physics, 86(12), 1115-1129 (2012)
  • Bekir, A., Aksoy, E., Exact solutions of shallow water wave equations by using the (G’/G)-expansion method Waves in Random and Complex Media, 22 (3), 317-331 (2012)
  • Zhang, H., Application of the (G’/G)-expansion method for the complex KdV equation, Commun Nonlinear Sci Numer Simulat, 15 1700-1704 (2010)
  • Liu, X., Tian, L., Wu, Y., Application of (G’/G)-expansion method to two nonlinear evolution equations, Applied Mathematics and Computation, 217 1376-1384 (2010)
  • Feng, J., Li, W., Wan, Q., Using (G’/G)-expansion method to seek the travelling wave solution of Kolmogorov-Petrovskii-Piskunov equation, Applied Mathematics and Computation, 217, 5860-5865 (2011)
  • Zhang, J., Jiang, F., Zhao, X., An improved (G’/G)-expansion method for solving nonlinear evolution equations, International Journal of Computer Mathematics, 87, 8 1716-1725.(2010)
  • Zhao, Y.M., Yang, Y.J., Li, W., Application of the improved (G’/G)-expansion method for the Variant Boussinesq equations, Applied Mathematics Sciences, 5, 58 2855-2861 (2011)
  • Hamad, Y.S., Sayed, M., Elagan, S.K., El-Zahar, E.R., The improved (G’/G)-expansion method for solving (3+1)-dimensional potential-YTSF equation, Journal of Modern Methods in Numerical Mathematics, 2, 1-2 32-38 (2011)
  • Naher, H., Abdullah, F.A., Some new traveling wave solutions of the nonlinear reaction diffusion equation by using the improved (G’/G)-expansion method, Mathematical Problems in Engineering, 871724.(2012)
  • Naher, H., Abdullah, F.A., The improved (G’/G)-expansion method for the (2+1)-dimensional Modified Zakharov-Kuznetsov equation, Journal of Applied Mathematics, 438928 (2012)
  • Nofel, T.A., Sayed, M., Hamad, Y.S., Elagan, S.K., The improved (G’/G)-expansion method for solving the fifth-order KdV equation, Annals of Fuzzy Mathematics and Informatics, 3, 1 9-17 (2011)
  • Yusufoglu, E., Bekir, A., The tanh and the sine-cosine methods for exact solutions of the MBBMN and the Vakhnenko equations, Chaos, Solitons and Fractals, 38 1126-1133 (2008)
  • Yusufoglu, E., New solitonary solutions for the MBBM Equation using Exp-function method, Physics Lett. A, 372 442-446.(2008)
  • Taghizadeh, N., Mirzazadeh, M., Exact solutions of modified Benjamin-Bona-Mahony equation and Zakharov-Kuznetsov equation by modified extended tanh method, International Journal of Applied Mathematics and Computation, 3, (2) 151-157 (2011)
  • Abbasbandy, S., Shirzadi, A., The first integral method for modified Benjamin-Bona-Mahony equation, Commun Nonlinear Sci Numer Simulat, 15 1759-1764 (2010)
  • Aslan, I., Exact and explicit solutions to some nonlinear evolution equations by utilizing the (G’/G)-expansion method, Applied Mathematics and Computation, 215 857-863 (2009)

Hasibun Naher1,2, Farah Aini Abdullah2and Ahmet Bekir3

Yıl 2015, Cilt: 3 Sayı: 1, 78 - 89, 22.12.2014

Hasibun Naher Farah Aini Abdullah Ahmet Bekir

Öz

Kaynakça

  • Malfliet, W., Solitary wave solutions of nonlinear wave equations,Am. J. Physics, 60, 650-654 (1992)
  • Wang, M.L., Zhou, Y.B., Li, Z.B., Application of homogeneous balance method to exact solutions of nonlinear equations in
  • mathematical physics Phys. Let. A, 216, 67-75 (1996)
  • He, J. ,Variational iteration method for delay differential equations, Communication in Nonlinear Science and Numerical
  • Simulation, 2(4), 235-236 (1997)
  • Abdou, M.A., Soliman, A.A., Variational iteration method for solving Burger’s and coupled Burger’s equations, J. Comput. Appl.
  • Math. 181, 245-251 (2005)
  • Yusufo˘glu, E., Bekir, A., The variational iteration method for solitary patterns solutions of gBBM equation, Physics Letter A, 367 (6) 461-464 (2007)
  • Younesian, D., Askari, H., Saadatnia, Z., Yildirim, A., Analytical solution for nonlinear wave propagation in shallow media using
  • the variational iteration method, Waves in Random and Complex Media, 22 (2), 133-142 (2012)
  • Abbasbandy, S., Numerical solution of non-linear Klein-Gordon equations by variational iteration method, Int. J. Numer. Meth.
  • Engng 70, 876-881 (2007)
  • Rogers, C., Shadwick, W.F., Backlund Transformations, Academic Press, New York. (1982)
  • Hirota, R., Exact solution of the KdV equation for multiple collisions of solitons, Phys. Rev. Lett. 27, 1192-1194 (1971)
  • Liu, S., Fu, Z., Liu, S., Zhao, Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289, 69–74 (2001)
  • Ebadi, G., Yousefzadeh, N., Triki, H., Yildirim, A., & Biswas, A., Envelope solitons, periodic waves and other solutions to Boussinesq-Burgers equation, Romanian Reports in Physics, 64(4), 915-932 (2012)
  • Salas, A.H., Gomez, C.A., Application of the Cole-Hopf transformation for finding exact solutions to several forms of the seventh- order KdV equation, Mathematical Problems in Engineering, 14, 194329 (2010)
  • Soliman, A.A. Abdo, H.A., New exact Solutions of nonlinear variants of the RLW, the PHI-four and Boussinesq equations based on modified extended direct algebraic method, International Journal of Nonlinear Science, 7, 3 274-282 (2009)
  • He, J.H., Wu, X.H., Exp-function method for nonlinear wave equations, Chaos Solitons and Fractals, 30 700-708 (2006)
  • Bekir, A., Boz, A., Exact solutions for nonlinear evolution equations using Exp-function method, Phys. Letters A, 372 1619-1625 (2008)
  • Naher, H., Abdullah, F.A., Akbar, M.A., The exp-function method for new exact solutions of the nonlinear partial differential equations, International Journal of the Physical Sciences, 6, 29 6706-6716 (2011)
  • Naher, H., Abdullah, F.A., Akbar, M.A., New travelling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method, Journal of Applied Mathematics, 575387 (2012)
  • Mohyud-Din, S.T., Noor, M.A., Noor, K.I., Exp-function method for traveling wave solutions of modified Zakharov-Kuznetsov equation, Journal of King Saud University, 22 213-216 (2010)
  • Ebadi, G., Krishnan, E. V., Labidi, M., Zerrad, E., & Biswas, A., Analytical and numerical solutions to the Davey–Stewartson equation with power-law nonlinearity, Waves in Random and Complex Media, 21(4), 559-590 (2011).
  • Bruzn, M. S., Gandarias, M. L., Camacho, J. C., Symmetry Analysis and Solutions for a Generalization of a Family of BBM Equations, Journal of Nonlinear Mathematical Physics, 15, 81-90 (2008)
  • Bruzn, M. S., Camacho, J. C., Gandarias, M. L., Symmetry Analysis and Solutions for a Family of BBM Equations, Nonlinear Evolution Equations and Dynamical Systems, (2007).
  • Jafari, H., Tajadodi, H., Biswas, A., Homotopy analysis method for solving a couple of evolution equations and comparison with Adomian’s decomposition method, Waves in Random and Complex Media, 21 (4), 657-667 (2011)
  • Bruzn, M. S., Gandarias, M. L., Symmetry reductions and exact solutions of Benjamin-Bona-Mahony-Burgers equation. In 4th Workshop Group Analysis of Differential Equations & Integrable Systems, 45-61 (2009).
  • Johnpillai, A. G., Kara, A. H., & Biswas, A., Symmetry reduction, exact group-invariant solutions and conservation laws of Benjamin-Bona-Mahoney equation, Applied Mathematics Letters, 26 (3), 376-381 (2013)
  • Wazwaz, A.M., A new (2+1)-dimensional Korteweg-de-Vries equation and its extension to a new (3+1)-dimensional Kadomtsev- Petviashvili equation, Physica Scripta, 84, 035010.(2011)
  • Ebadi, G., Kara, A.H., Petkovic, M.D., Biswas, A., Soliton solutions and conservation laws of Gilson-Pickering equation, Waves in Random and Complex Media, 21 (2) 378-385.(2011)
  • Wang, M., Li, X., Zhang, J., The (G’/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A, 372 417-423 (2008)
  • Bruzn, M. S., Gandarias, M. I., New exact solutions for a Benjamin-Bona-Mahony equation, World Scientific and Engineering Academy and Society (WSEAS), 189-194 (2008).
  • Zayed, E.M.E., Gepreel, K.A., The (G’/G)-expansion method for finding travelling wave solutions of nonlinear partial differential equations in mathematical physics, Journal of Mathematical Physics, 50 013502-013513 (2009)
  • Zayed, E.M.E., Al-Joudi, S., Applications of an Extended -Expansion Method to find Exact Solutions of Nonlinear PDEs in Mathematical Physics, Mathematical Problems in Engineering, 768573 (2010)
  • Ebadi, G., Biswas, A., Application of the (G’/G)-expansion method for nonlinear diffusion equations with nonlinear source, Journal of the Franklin Institute, 347(7), 1391-1398 (2010)
  • Ebadi, G., Biswas, A., The method and topological soliton solution of the K(m, n) equation, Communications in Nonlinear Science and Numerical Simulation, 16(6), 2377-2382 (2011)
  • Naher, H., Abdullah, F.A., Akbar, M.A., The (G’/G)-expansion method for abundant traveling wave solutions of Caudrey-Dodd- Gibbon equation, Mathematical Problems in Engineering, 218216. (2011)
  • Ebadi, G., Biswas, A., The method and 1-soliton solution of the Davey–Stewartson equation. Mathematical and Computer Modelling, 53(5), 694-698 (2011)
  • Ebadi, G., Krishnan, E. V., Biswas, A., Solitons and cnoidal waves of the Klein–Gordon–Zakharov equation in plasmas, Pramana, 79(2), 185-198 (2012)
  • Ebadi, G., Yildirim, A., Biswas, A., Chiral solitons with Bohm potential using method and exp-function method, Romanian Reports in Physics, 64(2), 357-366 (2012)
  • Ebadi, G., Mojaver, A., Johnson, S., Kumar, S., Biswas, A., Dynamics of dispersive topological solitons and its perturbations, Indian Journal of Physics, 86(12), 1115-1129 (2012)
  • Bekir, A., Aksoy, E., Exact solutions of shallow water wave equations by using the (G’/G)-expansion method Waves in Random and Complex Media, 22 (3), 317-331 (2012)
  • Zhang, H., Application of the (G’/G)-expansion method for the complex KdV equation, Commun Nonlinear Sci Numer Simulat, 15 1700-1704 (2010)
  • Liu, X., Tian, L., Wu, Y., Application of (G’/G)-expansion method to two nonlinear evolution equations, Applied Mathematics and Computation, 217 1376-1384 (2010)
  • Feng, J., Li, W., Wan, Q., Using (G’/G)-expansion method to seek the travelling wave solution of Kolmogorov-Petrovskii-Piskunov equation, Applied Mathematics and Computation, 217, 5860-5865 (2011)
  • Zhang, J., Jiang, F., Zhao, X., An improved (G’/G)-expansion method for solving nonlinear evolution equations, International Journal of Computer Mathematics, 87, 8 1716-1725.(2010)
  • Zhao, Y.M., Yang, Y.J., Li, W., Application of the improved (G’/G)-expansion method for the Variant Boussinesq equations, Applied Mathematics Sciences, 5, 58 2855-2861 (2011)
  • Hamad, Y.S., Sayed, M., Elagan, S.K., El-Zahar, E.R., The improved (G’/G)-expansion method for solving (3+1)-dimensional potential-YTSF equation, Journal of Modern Methods in Numerical Mathematics, 2, 1-2 32-38 (2011)
  • Naher, H., Abdullah, F.A., Some new traveling wave solutions of the nonlinear reaction diffusion equation by using the improved (G’/G)-expansion method, Mathematical Problems in Engineering, 871724.(2012)
  • Naher, H., Abdullah, F.A., The improved (G’/G)-expansion method for the (2+1)-dimensional Modified Zakharov-Kuznetsov equation, Journal of Applied Mathematics, 438928 (2012)
  • Nofel, T.A., Sayed, M., Hamad, Y.S., Elagan, S.K., The improved (G’/G)-expansion method for solving the fifth-order KdV equation, Annals of Fuzzy Mathematics and Informatics, 3, 1 9-17 (2011)
  • Yusufoglu, E., Bekir, A., The tanh and the sine-cosine methods for exact solutions of the MBBMN and the Vakhnenko equations, Chaos, Solitons and Fractals, 38 1126-1133 (2008)
  • Yusufoglu, E., New solitonary solutions for the MBBM Equation using Exp-function method, Physics Lett. A, 372 442-446.(2008)
  • Taghizadeh, N., Mirzazadeh, M., Exact solutions of modified Benjamin-Bona-Mahony equation and Zakharov-Kuznetsov equation by modified extended tanh method, International Journal of Applied Mathematics and Computation, 3, (2) 151-157 (2011)
  • Abbasbandy, S., Shirzadi, A., The first integral method for modified Benjamin-Bona-Mahony equation, Commun Nonlinear Sci Numer Simulat, 15 1759-1764 (2010)
  • Aslan, I., Exact and explicit solutions to some nonlinear evolution equations by utilizing the (G’/G)-expansion method, Applied Mathematics and Computation, 215 857-863 (2009)
Toplam 57 adet kaynakça vardır.

Ayrıntılar

Bölüm Articles
Yazarlar

Hasibun Naher Bu kişi benim

Farah Aini Abdullah Bu kişi benim

Ahmet Bekir Bu kişi benim

Yayımlanma Tarihi 22 Aralık 2014
Yayımlandığı Sayı Yıl 2015 Cilt: 3 Sayı: 1

Kaynak Göster

APA Naher, H., Abdullah, F. A., & Bekir, A. (2014). Hasibun Naher1,2, Farah Aini Abdullah2and Ahmet Bekir3. New Trends in Mathematical Sciences, 3(1), 78-89.
AMA Naher H, Abdullah FA, Bekir A. Hasibun Naher1,2, Farah Aini Abdullah2and Ahmet Bekir3. New Trends in Mathematical Sciences. Aralık 2014;3(1):78-89.
Chicago Naher, Hasibun, Farah Aini Abdullah, ve Ahmet Bekir. “Hasibun Naher1,2, Farah Aini Abdullah2and Ahmet Bekir3”. New Trends in Mathematical Sciences 3, sy. 1 (Aralık 2014): 78-89.
EndNote Naher H, Abdullah FA, Bekir A (01 Aralık 2014) Hasibun Naher1,2, Farah Aini Abdullah2and Ahmet Bekir3. New Trends in Mathematical Sciences 3 1 78–89.
IEEE H. Naher, F. A. Abdullah, ve A. Bekir, “Hasibun Naher1,2, Farah Aini Abdullah2and Ahmet Bekir3”, New Trends in Mathematical Sciences, c. 3, sy. 1, ss. 78–89, 2014.
ISNAD Naher, Hasibun vd. “Hasibun Naher1,2, Farah Aini Abdullah2and Ahmet Bekir3”. New Trends in Mathematical Sciences 3/1 (Aralık 2014), 78-89.
JAMA Naher H, Abdullah FA, Bekir A. Hasibun Naher1,2, Farah Aini Abdullah2and Ahmet Bekir3. New Trends in Mathematical Sciences. 2014;3:78–89.
MLA Naher, Hasibun vd. “Hasibun Naher1,2, Farah Aini Abdullah2and Ahmet Bekir3”. New Trends in Mathematical Sciences, c. 3, sy. 1, 2014, ss. 78-89.
Vancouver Naher H, Abdullah FA, Bekir A. Hasibun Naher1,2, Farah Aini Abdullah2and Ahmet Bekir3. New Trends in Mathematical Sciences. 2014;3(1):78-89.