Abstract
In this study, the Rosenau-Korteweg-de Vries-Regular Longwave (Rosenau-KdV-RLW) equation has been converted into a partial differential equation system consisting of two equations using a splitting technique. Then, numerical solutions for the Rosenau-KdV-RLW equation system have been obtained using separately both cubic and quintic B-spline finite element collocation method. For the unknowns in those equations, B-spline functions at x-position and Crank-Nicolson type finite difference approaches at time positions are used. A test problem has been chosen to check the accuracy of the proposed discretized scheme. The basic conservation properties of the Rosenau-KdV-RLW equation have been shown to be protected by the proposed numerical scheme. The results are compared with the analytical solution of the problem and the results given in the literature. For the reliability of the method the error norms L_2 and L_∞ are calculated. It is seen that the proposed method gives harmonious results with exact solutions.
References
- Wongsaijai, B., Poochinapan, K., A three-level average implicit finite difference scheme to solve equation obtained by coupling the Rosenau-KdV equation and Rosenau-RLW equation, Applied Mathematics and Computation, 245, 289-304, (2014).
- Hu, J., Xu, Y., Hu, B., Conservative linear difference scheme for Rosenau-KdV equation, Advances in Mathematical Physics, (2013). DOI: https://dx.doi.org/10.1155/2013/423718
- Ucar, Y., Karaagac, B., Kutluay, S., A Numerical approach to the Rosenau-KdV equation using Galerkin Cubic Finite element method, International Journal of Applied Mathematics and Statistics, 56(3), 83-92, (2017).
- Wang, X., Dai, W., A three-level linear implicit conservative scheme for the Rosenau-KdV-RLW equation, Journal of Computational and Applied Mathematics , 330, 295-306, (2018).
- Mittal, R. C., Jain, R. K., Numerical solution of general Rosenau-RLW equations using quintic B-splines collocation method, Communications in Numerical Analysis, 16, 1-16, (2012).
- Wongsaijai, B., Poochinapan, K., Disyadej, T., A compact finite difference method for solving the general Rosenau-RLW equation. IAENG International Journal of Applied Mathematics, 44, 4, IJAM-44-4-05, (2014).
- Yagmurlu, N. M., Karaagac, B., Kutluay, S., Numerical solutions of Rosenau-RLW Equation using Galerkin Cubic B-Spline finite element method, American Journal of Computational and Applied Mathematics, 7(1), 1-10, (2017).
- Pan, X., Wang, Y., Zhang, L., Numerical analysis of a pseudo-compact C-N conservative scheme for the Rosenau-KdV equation coupling with the Rosenau-RLW equation, Boundary Value Problems, (2015). DOI: https://doi.org/10.1186/ s13661-015-0328-2
- Korkmaz, B., Dereli, Y., Numerical solution of the Rosenau KdV-RLW equation by using RBFs collocation method, International Journal of Modern Physics C, 27, 1650117 (11 pages), (2016).
- Ghiloufi, A., Omrani, K., New conservative difference schemes with fourth-order accuracy for some model equation for nonlinear dispersive waves, Numerical Methods Partial Differential Equation, (2017). DOI: https://doi.org/10.1002/num.22208
- Foroutan, M., Ebadian, A., Chebyshev rational approximations for the Rosenau-KdV-RLW equation on the whole line, International Journal of Analysis and Applications 16(1), 1-15, (2018).
- Fernández, A. A., Ramos, J.I., Numerical solution of the generalized, dissipative KdV-RLW-Rosenau equation with a compact method, Communications in Nonlinear Science and Numerical Simulation, 60, 165-183, (2018).
- P. M. Prenter, Splines and variational methods, John Wiley & Sons, New York, (1975).
- P. Razborova, B. Ahmed, A. Biswas, Solitons, shock waves and conservation laws of Rosenau-KdV-RLW equation with power law nonlinearity, Applied Mathematics and Information Science, 8, 485-491, (2014).
Abstract
Bu çalışmada, Rosenau Korteweg-de Vries düzenli uzun dalga (Rosenau-KdV-RLW) denklemi bir parçalama tekniği kullanılarak iki denklemden oluşan bir kısmi diferansiyel denklem sistemine dönüştürülmüştür. Daha sonra, Rosenau-KdV-RLW denklem sistemi için kübik ve kuintik B-spline sonlu eleman kollakasyon yöntemi kullanılarak sayısal çözümler önerilmiştir. Bu denklemlerdeki bilinmeyenler için x-konumunda B-spline fonksiyonlar ve zaman konumunda Crank-Nicolson tipi sonlu fark yaklaşımları kullanılmıştır. Önerilen sayısal şemaların doğruluğunu kontrol etmek için bir test problemi seçilmiştir. Rosenau-KdV-RLW denkleminin temel korunum özelliklerinin önerilen sayısal şemalar ile korunduğu görülmüştür. Elde edilen sonuçlar problemin analitik çözümü ve literatürde verilen sonuçlarla karşılaştırılmıştır. Yöntemin güvenilirliği için L_2 ve L_∞ hata normları hesaplanmıştır. Önerilen yöntemin tam çözümlerle uyumlu sonuçlar verdiği görülmüştür.
References
- Wongsaijai, B., Poochinapan, K., A three-level average implicit finite difference scheme to solve equation obtained by coupling the Rosenau-KdV equation and Rosenau-RLW equation, Applied Mathematics and Computation, 245, 289-304, (2014).
- Hu, J., Xu, Y., Hu, B., Conservative linear difference scheme for Rosenau-KdV equation, Advances in Mathematical Physics, (2013). DOI: https://dx.doi.org/10.1155/2013/423718
- Ucar, Y., Karaagac, B., Kutluay, S., A Numerical approach to the Rosenau-KdV equation using Galerkin Cubic Finite element method, International Journal of Applied Mathematics and Statistics, 56(3), 83-92, (2017).
- Wang, X., Dai, W., A three-level linear implicit conservative scheme for the Rosenau-KdV-RLW equation, Journal of Computational and Applied Mathematics , 330, 295-306, (2018).
- Mittal, R. C., Jain, R. K., Numerical solution of general Rosenau-RLW equations using quintic B-splines collocation method, Communications in Numerical Analysis, 16, 1-16, (2012).
- Wongsaijai, B., Poochinapan, K., Disyadej, T., A compact finite difference method for solving the general Rosenau-RLW equation. IAENG International Journal of Applied Mathematics, 44, 4, IJAM-44-4-05, (2014).
- Yagmurlu, N. M., Karaagac, B., Kutluay, S., Numerical solutions of Rosenau-RLW Equation using Galerkin Cubic B-Spline finite element method, American Journal of Computational and Applied Mathematics, 7(1), 1-10, (2017).
- Pan, X., Wang, Y., Zhang, L., Numerical analysis of a pseudo-compact C-N conservative scheme for the Rosenau-KdV equation coupling with the Rosenau-RLW equation, Boundary Value Problems, (2015). DOI: https://doi.org/10.1186/ s13661-015-0328-2
- Korkmaz, B., Dereli, Y., Numerical solution of the Rosenau KdV-RLW equation by using RBFs collocation method, International Journal of Modern Physics C, 27, 1650117 (11 pages), (2016).
- Ghiloufi, A., Omrani, K., New conservative difference schemes with fourth-order accuracy for some model equation for nonlinear dispersive waves, Numerical Methods Partial Differential Equation, (2017). DOI: https://doi.org/10.1002/num.22208
- Foroutan, M., Ebadian, A., Chebyshev rational approximations for the Rosenau-KdV-RLW equation on the whole line, International Journal of Analysis and Applications 16(1), 1-15, (2018).
- Fernández, A. A., Ramos, J.I., Numerical solution of the generalized, dissipative KdV-RLW-Rosenau equation with a compact method, Communications in Nonlinear Science and Numerical Simulation, 60, 165-183, (2018).
- P. M. Prenter, Splines and variational methods, John Wiley & Sons, New York, (1975).
- P. Razborova, B. Ahmed, A. Biswas, Solitons, shock waves and conservation laws of Rosenau-KdV-RLW equation with power law nonlinearity, Applied Mathematics and Information Science, 8, 485-491, (2014).
Details
| Primary Language | English |
|---|---|
| Journal Section | Research Articles |
| Authors | |
| Publication Date | October 29, 2018 |
| Submission Date | July 26, 2018 |
| Published in Issue | Year 2018 Volume: 20 Issue: 3 |