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An Application of Galerkin Method to Generalized Benjamin-Bona-Mahony-Burgers Equation

Yıl 2018, Cilt: 8 Sayı: 2, 53 - 69, 28.12.2018

Öz

In this study, Galerkin finite element method is applied to
generalized Benjamin-Bona-Mahony-Burgers (gBBM-B) equation. Quadratic B-spline
functions are used as interpolation function. Stability analysis is
investigated based on von Neumann theory. The performance of the proposed
method is checked by two test problems with zero boundary conditions. As a
result, it is observed that applied method is successful and efficient to show
motions of some waves
.

Kaynakça

  • Korteweg, D.J., de Vries, G., On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary wave, Philosophical Magazine, 39, 422–443, 1895.
  • Triki, H., Ak, T., Moshokoa, S.P., Biswas, A., Soliton solutions to KdV equation with spatio-temporal dispersion, Ocean Engineering, 114, 192-203, 2016.
  • Triki, H., Ak, T., Biswas, A., New types of soliton-like solutions for a second order wave equation of Korteweg-de Vries type, Applied and Computational Mathematics, 16(2), 168-176, 2017.
  • Karakoc, S.B.G., Zeybek, H., Ak, T., Numerical solutions of the Kawahara equation by the septic B-spline collocation method, Statistics, Optimizatin and Information Computing, 2, 211-221, 2014.
  • Raslan, K.R., Collocation method using quartic B-spline for the equal width (EW) equation, Applied Mathematics and Computation, 168(2), 795-805, 2005.
  • Saka, B, Dag, I., Dereli, Y, Korkmaz, A., Three different methods for numerical solution of the EW equation, Engineering Analysis with Boundary Elements, 32(7), 556-566, 2008.
  • Inan, B., Bahadir, A.R., A fully implicit finite difference scheme for the regularized long wave equation, General Mathematics Notes, 33(2), 40-59, 2016.
  • Lu, C., Huang, W., Qiu, J., An adaptive moving mesh finite element solution of the regularized long wave equation, Journal of Scientific Computing, 74(1), 122-144, 2018.
  • Inan, B., Bahadir, A.R., Numerical solution of the one-dimensional Burgers’ equation: Implicit and fully implicit exponential finite difference methods, Pramana-Journal of Physics, 81(4), 547-556, 2013.
  • Mittal, R.C., Tripathi, A., Numerical solutions of two-dimensional Burgers’ equations using modified Bi-cubic B-spline finite elements, Engineering Computations, 32(5), 1275-1306, 2014.
  • Ak, T., Karakoc, S.B.G., A numerical technique based on collocation method for solving modified Kawahara equation, Journal of Ocean Engineering and Science, 3(1), 67-75, 2018.
  • Ak, T., Aydemir, T., Saha, A., Kara, A.H., Propagation of nonlinear shock waves for the generalized Oskolkov equation and its dynamic motions in presence of an external periodic perturbation, Pramana-Journal of Physics, 90:78, 1-16, 2018.
  • Sierra, C.A.G., Closed form solutions for a generalized Benjamin-Bona-Mahony-Burgers equation with higher-order nonlinearity, Applied Mathematics and Computation, 234, 618-622, 2014.
  • Omrani, K., Ayadi, M., Finite difference discretization of the Benjamin-Bona-Mahony-Burgers equation, Numerical Methods for Partial Differential Equations, 24(1), 239-248, 2008.
  • Dehghan, M., Abbaszadeh, M., Mohebbi, A., The numerical solution of nonlinear high dimensional generalized Benjamin-Bona-Mahony-Burgers equation via the meshless method of radial basis functions, Journal Computers & Mathematics with Applications, 68(3), 212-237, 2014.
  • Alquran, M., Al-Khaled, K., Sinc and solitary wave solutions to the generalized Benjamin–Bona–Mahony–Burgers equations, Physica Scripta, 83, 065010, 2011.
  • Fardi, M., Sayevand, K., Homotopy analysis method: A fresh view on Benjamin-Bona-Mahony-Burgers equation, The Journal of Mathematics and Computer Science, 4(3), 494-501, 2012.
  • Ganji, Z.Z., Ganji, D.D., Bararnia, H., Approximate general and explicit solutions of nonlinear BBMB equations by Exp-function method, Applied Mathematical Modelling, 33, 1836-1841, 2009.
  • Jawad, A.J.M., New exact solutions of nonlinear partial differential equations using tan-cot function method, Studies in Mathematical Sciences, 5(2), 13-25, 2012.
  • Arora, G., Mittal, R.C., Singh, B.K., Numerical soltion of BBM-Burger equation with quartic B-spline collocation method, Journal of Engineering Science and Technology, Special Issue on (ICMTEA 2013), 104-116, 2014.
  • Prenter, P.M., Splines and Variational Methods, John Wiley, New York, 1975.

Genelleştirilmiş Benjamin-Bona-Mahony Denklemine Galerkin Metodunun Bir Uygulaması

Yıl 2018, Cilt: 8 Sayı: 2, 53 - 69, 28.12.2018

Öz

Bu çalışmada, genelleştirilmiş
Benjamin-Bona-Mahony
(gBBM-B) denklemine Galerkin
sonlu elemanlar yöntemi uygulanmıştır. İnterpolasyon fonksiyonu olarak
kuadratik B-spline fonksiyonlar kullanılmıştır. Von Neumann teorisine bağlı
olarak kararlılık analizi incelenmiştir. Sıfır sınır şartları ile iki test
problemi yardımıyla önerilen yöntemin performansı kontrol edilmiştir. Sonuç
olarak, uygulanan yöntemin bazı dalga hareketlerini göstermek için başarılı ve
etkili olduğu gözlenmiştir.

Kaynakça

  • Korteweg, D.J., de Vries, G., On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary wave, Philosophical Magazine, 39, 422–443, 1895.
  • Triki, H., Ak, T., Moshokoa, S.P., Biswas, A., Soliton solutions to KdV equation with spatio-temporal dispersion, Ocean Engineering, 114, 192-203, 2016.
  • Triki, H., Ak, T., Biswas, A., New types of soliton-like solutions for a second order wave equation of Korteweg-de Vries type, Applied and Computational Mathematics, 16(2), 168-176, 2017.
  • Karakoc, S.B.G., Zeybek, H., Ak, T., Numerical solutions of the Kawahara equation by the septic B-spline collocation method, Statistics, Optimizatin and Information Computing, 2, 211-221, 2014.
  • Raslan, K.R., Collocation method using quartic B-spline for the equal width (EW) equation, Applied Mathematics and Computation, 168(2), 795-805, 2005.
  • Saka, B, Dag, I., Dereli, Y, Korkmaz, A., Three different methods for numerical solution of the EW equation, Engineering Analysis with Boundary Elements, 32(7), 556-566, 2008.
  • Inan, B., Bahadir, A.R., A fully implicit finite difference scheme for the regularized long wave equation, General Mathematics Notes, 33(2), 40-59, 2016.
  • Lu, C., Huang, W., Qiu, J., An adaptive moving mesh finite element solution of the regularized long wave equation, Journal of Scientific Computing, 74(1), 122-144, 2018.
  • Inan, B., Bahadir, A.R., Numerical solution of the one-dimensional Burgers’ equation: Implicit and fully implicit exponential finite difference methods, Pramana-Journal of Physics, 81(4), 547-556, 2013.
  • Mittal, R.C., Tripathi, A., Numerical solutions of two-dimensional Burgers’ equations using modified Bi-cubic B-spline finite elements, Engineering Computations, 32(5), 1275-1306, 2014.
  • Ak, T., Karakoc, S.B.G., A numerical technique based on collocation method for solving modified Kawahara equation, Journal of Ocean Engineering and Science, 3(1), 67-75, 2018.
  • Ak, T., Aydemir, T., Saha, A., Kara, A.H., Propagation of nonlinear shock waves for the generalized Oskolkov equation and its dynamic motions in presence of an external periodic perturbation, Pramana-Journal of Physics, 90:78, 1-16, 2018.
  • Sierra, C.A.G., Closed form solutions for a generalized Benjamin-Bona-Mahony-Burgers equation with higher-order nonlinearity, Applied Mathematics and Computation, 234, 618-622, 2014.
  • Omrani, K., Ayadi, M., Finite difference discretization of the Benjamin-Bona-Mahony-Burgers equation, Numerical Methods for Partial Differential Equations, 24(1), 239-248, 2008.
  • Dehghan, M., Abbaszadeh, M., Mohebbi, A., The numerical solution of nonlinear high dimensional generalized Benjamin-Bona-Mahony-Burgers equation via the meshless method of radial basis functions, Journal Computers & Mathematics with Applications, 68(3), 212-237, 2014.
  • Alquran, M., Al-Khaled, K., Sinc and solitary wave solutions to the generalized Benjamin–Bona–Mahony–Burgers equations, Physica Scripta, 83, 065010, 2011.
  • Fardi, M., Sayevand, K., Homotopy analysis method: A fresh view on Benjamin-Bona-Mahony-Burgers equation, The Journal of Mathematics and Computer Science, 4(3), 494-501, 2012.
  • Ganji, Z.Z., Ganji, D.D., Bararnia, H., Approximate general and explicit solutions of nonlinear BBMB equations by Exp-function method, Applied Mathematical Modelling, 33, 1836-1841, 2009.
  • Jawad, A.J.M., New exact solutions of nonlinear partial differential equations using tan-cot function method, Studies in Mathematical Sciences, 5(2), 13-25, 2012.
  • Arora, G., Mittal, R.C., Singh, B.K., Numerical soltion of BBM-Burger equation with quartic B-spline collocation method, Journal of Engineering Science and Technology, Special Issue on (ICMTEA 2013), 104-116, 2014.
  • Prenter, P.M., Splines and Variational Methods, John Wiley, New York, 1975.
Toplam 21 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Matematik
Yazarlar

Turgut Ak 0000-0001-8368-8506

Yayımlanma Tarihi 28 Aralık 2018
Gönderilme Tarihi 27 Mayıs 2018
Kabul Tarihi 1 Ocak 2019
Yayımlandığı Sayı Yıl 2018 Cilt: 8 Sayı: 2

Kaynak Göster

APA Ak, T. (2018). An Application of Galerkin Method to Generalized Benjamin-Bona-Mahony-Burgers Equation. Adıyaman University Journal of Science, 8(2), 53-69.
AMA Ak T. An Application of Galerkin Method to Generalized Benjamin-Bona-Mahony-Burgers Equation. ADYU J SCI. Aralık 2018;8(2):53-69.
Chicago Ak, Turgut. “An Application of Galerkin Method to Generalized Benjamin-Bona-Mahony-Burgers Equation”. Adıyaman University Journal of Science 8, sy. 2 (Aralık 2018): 53-69.
EndNote Ak T (01 Aralık 2018) An Application of Galerkin Method to Generalized Benjamin-Bona-Mahony-Burgers Equation. Adıyaman University Journal of Science 8 2 53–69.
IEEE T. Ak, “An Application of Galerkin Method to Generalized Benjamin-Bona-Mahony-Burgers Equation”, ADYU J SCI, c. 8, sy. 2, ss. 53–69, 2018.
ISNAD Ak, Turgut. “An Application of Galerkin Method to Generalized Benjamin-Bona-Mahony-Burgers Equation”. Adıyaman University Journal of Science 8/2 (Aralık 2018), 53-69.
JAMA Ak T. An Application of Galerkin Method to Generalized Benjamin-Bona-Mahony-Burgers Equation. ADYU J SCI. 2018;8:53–69.
MLA Ak, Turgut. “An Application of Galerkin Method to Generalized Benjamin-Bona-Mahony-Burgers Equation”. Adıyaman University Journal of Science, c. 8, sy. 2, 2018, ss. 53-69.
Vancouver Ak T. An Application of Galerkin Method to Generalized Benjamin-Bona-Mahony-Burgers Equation. ADYU J SCI. 2018;8(2):53-69.

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