Research Article
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Year 2022, Volume: 10 Issue: 2, 220 - 232, 31.10.2022

Abstract

References

  • [1] P.J.Morrison, J.D. Meiss, J.R. Carey, Scattering of RLW solitary waves, Physica D., 11 (1981), 324–336.
  • [2] I.Dag, B.Saka, A cubic B-spline collocation method for the EW equation, Math. Comput. Appl., 9(3),2004,381–392.
  • [3] K. R.Raslan, A computational method for the equal width equation, Int. J. Comput. Math., 81 (1), 2004,63–72.
  • [4] D. Irk, B. Saka,I Dag, Cubic spline collocation method for the equal width equation, Hadronic Journal Supplement, 18 (2003), 201-214.
  • [5] H.Fazal-i, A. Inayet, A.Shakeel, Septic B-spline Collocation method for numerical solution of the Equal Width Wave (EW) equation, Life Science Journal, 10 (2013), 253-260.
  • [6] A.Dogan, Application of the Galerkin’s method to equal width wave equation, Appl. Math. Comput., 160 (2005), 65–76.
  • [7] L.R.T.Gardner, G.A.Gardner, Solitary waves of the equal width wave equation, J. Comput. Phys., 101 (1992), 218–223.
  • [8] S.G¨ulec¸, Numerical Solutions of partial differantial equations using Galerkin finite element method, Master Thesis, Nigde University, Turkey, 2007.
  • [9] M. A. Banaja, H. O. Bakodah Runge-Kutta integration of the equal width wave equation using the method of lines, Math. Probl. Eng., (2015), 1-9.
  • [10] B.Saka A finite element method for equal width equation, Appl. Math. Comput.,(175) 2006, 730–747.
  • [11] A.Esen,S. Kutluay, A linearized implicit finite difference method for solving the equal width wave equation, Int. J. Comp. Math., 83 (2006) 319–330.
  • [12] A.Esen, A numerical solution of the equal width wave equation by a lumped Galerkin method, Appl. Math. Comput., 168 (2005),270–282.
  • [13] S.I.Zaki, A least-squares finite element scheme for the EW equation, Comput. Meth. Appl. Mech. Eng., 189 (2000), 587–594.
  • [14] T.Roshan, A Petrov-Galerkin Method for Equal width equation, Applied Mathematics and Computation, 218 (2011), 2730-2739.
  • [15] A.H.A. Ali, Spectral method for solving the equal width equation based on Chebyshev polynomials, Nonlinear Dyn 51 (2008) 59-70.
  • [16] L.R.T.Gardner, G.A. Gardner, F.A.Ayoup, N.K. Amein, Simulations of the EW undular bore, Communications in Numerical Methods in Engineering, 13 (1997), 583-592.
  • [17] İ.Çelikkaya, Operator Splitting Solution of Equal Width Wave Equation Based on the Lie-Trotter and Strang Splitting Methods, Konuralp Journal of Mathematics, 6 (2) (2018) 200-208.
  • [18] NM.Ya˘gmurlu, AS.Karakas¸, Numerical Solutions of the EW Equation By Trigonometric Cubic B-spline Collocation Method Based on Rubin-Graves Type Linearization, Numerical Methods for Partial Differential Equations 36 (5),2020, 1170-1183.
  • [19] N. M.Yagmurlu, A. S.Karakas¸, A Novel Perspective for Simulations of the MEW Equation By Trigonometric Cubic B-spline Collocation Method Based on Rubin-Graves Type Linearization, Computational Methods for Differential Equations, DOI: 10.22034/CMDE.2021.47358.1981
  • [20] S.Ozer, Numerical solution of the Rosenau–KdV–RLW equation by operator splitting techniques based on B-spline collocation method,Numer Methods Partial Differential Eq., 35 (2019), 1928–1943 (2019),1–16, DOI: 10.1002/num.22387.
  • [21] Y. Dereli and R. Schaback, The Meshless Kernel-Based Method of Lines for solving the Equal Width Equation, Georg-August G¨ottingen University institut for Numerische and Angewandte Mathematik Preprint-Serie, Number:2010/27.
  • [22] B. Saka, I. Da˘g, Y. Dereli and A. Korkmaz, Three different methods for numerical solutions of the EW equation, Engineering Analysis with Boundary Elements, 32 (2008) 556-566.
  • [23] S.O¨ zer, Two efficient numerical methods for solving Rosenau-KdV-RLW equation, Kuwait J. Sci., 48 (1),2021, 14-24.
  • [24] A.Bas¸han, Y.Uc¸ar¸ NM.Ya˘gmurlu and A. Esen, A new perspective for quintic B-spline based Crank-Nicolson-differential quadrature method algorithm for numerical solutions of the nonlinear Schr¨odinger equation, Eur. Phys. J. Plus, 133 (12), (2018), https://doi.org/10.1140/epjp/i2018-11843-1.
  • [25] A.Bas¸han, NM.Ya˘gmurlu, Y.Uc¸ar¸ and A. Esen, A new perspective for the numerical solution of the Modified Equal Width wave equation, Math Meth Appl Sci. 44 (2021), 8925–8939, DOI: 10.1002/mma.7322.
  • [26] A.Bas¸han, An effective approximation to the dispersive soliton solutions of the coupled KdV equation via combination of two efficient methods, Computational and Applied Mathematics, (2020), 39 (80), https://doi.org/10.1007/s40314-020-1109-9.
  • [27] A. Bas¸han, A novel approach via mixed Crank–Nicolson scheme and differential quadrature method for numerical solutions of solitons of mKdV equation,Pramana – J. Phys. (2019) 92(84),https://doi.org/10.1007/s12043-019-1751-1
  • [28] PJ. Olver, Euler operators and conservation laws of the BBM equation, Math. Proc. Camb. Phil. Soc., 85 (1979) ,143-160.
  • [29] P. M. Prenter, Splines and variational methods, John Wiley, New York, NY, 1975.
  • [30] H. Holden et al., Splitting methods for partial differential equations with rough solutions, European Mathematical Society, Publishing House, Z¨urich, 2010.
  • [31] J.VonNeumann, R. D.Richtmyer, A Method for the Numerical Calculation of Hydrodynamic Shocks, J. Appl. Phys., Vol:21, (1950), 232-237.
  • [32] G. D. Smith, Numerical solutions of partial differential equations: Finite difference methods, Clarendon Press,Oxford, 1985.
  • [33] M.Seydaoglu, U. Erdog˘an, T.O¨ zis¸¸ Numerical solution of Burgers’ equation with high order splitting methods, J. Comput. Appl. Math., Vol: 291, (2016) 410–421.
  • [34] J. Geiser, Iterative Splitting Methods for Differential Equations, CHAPMAN and HALL/CRC., Numerical Analysis and Scientific Computing, Boca Raton, 2011.

Two Effective Numerical Approaches for Equal Width Wave (EW) Equation Using Lie-Trotter Splitting Technique

Year 2022, Volume: 10 Issue: 2, 220 - 232, 31.10.2022

Abstract

In this work, approximate solutions of the EW equation are obtained by two influential numerical schemes. For the first and second method, after splitting equal width wave (EW) equation in time, it is solved by Lie- Trotter splitting technique via quintic B-spline Collocation and cubic B-spline Lumped Galerkin FEMs and the suited finite difference approaches for space and time discretizations respectively. Stability analysis of schemes is shown and both schemes are implemented to two example. The acquired numerical results are compared with those in the literature with the help of the error norms and conservation features. It is seen that the error norms are quite small, the present conservation constants are consistent according to the results compared.

References

  • [1] P.J.Morrison, J.D. Meiss, J.R. Carey, Scattering of RLW solitary waves, Physica D., 11 (1981), 324–336.
  • [2] I.Dag, B.Saka, A cubic B-spline collocation method for the EW equation, Math. Comput. Appl., 9(3),2004,381–392.
  • [3] K. R.Raslan, A computational method for the equal width equation, Int. J. Comput. Math., 81 (1), 2004,63–72.
  • [4] D. Irk, B. Saka,I Dag, Cubic spline collocation method for the equal width equation, Hadronic Journal Supplement, 18 (2003), 201-214.
  • [5] H.Fazal-i, A. Inayet, A.Shakeel, Septic B-spline Collocation method for numerical solution of the Equal Width Wave (EW) equation, Life Science Journal, 10 (2013), 253-260.
  • [6] A.Dogan, Application of the Galerkin’s method to equal width wave equation, Appl. Math. Comput., 160 (2005), 65–76.
  • [7] L.R.T.Gardner, G.A.Gardner, Solitary waves of the equal width wave equation, J. Comput. Phys., 101 (1992), 218–223.
  • [8] S.G¨ulec¸, Numerical Solutions of partial differantial equations using Galerkin finite element method, Master Thesis, Nigde University, Turkey, 2007.
  • [9] M. A. Banaja, H. O. Bakodah Runge-Kutta integration of the equal width wave equation using the method of lines, Math. Probl. Eng., (2015), 1-9.
  • [10] B.Saka A finite element method for equal width equation, Appl. Math. Comput.,(175) 2006, 730–747.
  • [11] A.Esen,S. Kutluay, A linearized implicit finite difference method for solving the equal width wave equation, Int. J. Comp. Math., 83 (2006) 319–330.
  • [12] A.Esen, A numerical solution of the equal width wave equation by a lumped Galerkin method, Appl. Math. Comput., 168 (2005),270–282.
  • [13] S.I.Zaki, A least-squares finite element scheme for the EW equation, Comput. Meth. Appl. Mech. Eng., 189 (2000), 587–594.
  • [14] T.Roshan, A Petrov-Galerkin Method for Equal width equation, Applied Mathematics and Computation, 218 (2011), 2730-2739.
  • [15] A.H.A. Ali, Spectral method for solving the equal width equation based on Chebyshev polynomials, Nonlinear Dyn 51 (2008) 59-70.
  • [16] L.R.T.Gardner, G.A. Gardner, F.A.Ayoup, N.K. Amein, Simulations of the EW undular bore, Communications in Numerical Methods in Engineering, 13 (1997), 583-592.
  • [17] İ.Çelikkaya, Operator Splitting Solution of Equal Width Wave Equation Based on the Lie-Trotter and Strang Splitting Methods, Konuralp Journal of Mathematics, 6 (2) (2018) 200-208.
  • [18] NM.Ya˘gmurlu, AS.Karakas¸, Numerical Solutions of the EW Equation By Trigonometric Cubic B-spline Collocation Method Based on Rubin-Graves Type Linearization, Numerical Methods for Partial Differential Equations 36 (5),2020, 1170-1183.
  • [19] N. M.Yagmurlu, A. S.Karakas¸, A Novel Perspective for Simulations of the MEW Equation By Trigonometric Cubic B-spline Collocation Method Based on Rubin-Graves Type Linearization, Computational Methods for Differential Equations, DOI: 10.22034/CMDE.2021.47358.1981
  • [20] S.Ozer, Numerical solution of the Rosenau–KdV–RLW equation by operator splitting techniques based on B-spline collocation method,Numer Methods Partial Differential Eq., 35 (2019), 1928–1943 (2019),1–16, DOI: 10.1002/num.22387.
  • [21] Y. Dereli and R. Schaback, The Meshless Kernel-Based Method of Lines for solving the Equal Width Equation, Georg-August G¨ottingen University institut for Numerische and Angewandte Mathematik Preprint-Serie, Number:2010/27.
  • [22] B. Saka, I. Da˘g, Y. Dereli and A. Korkmaz, Three different methods for numerical solutions of the EW equation, Engineering Analysis with Boundary Elements, 32 (2008) 556-566.
  • [23] S.O¨ zer, Two efficient numerical methods for solving Rosenau-KdV-RLW equation, Kuwait J. Sci., 48 (1),2021, 14-24.
  • [24] A.Bas¸han, Y.Uc¸ar¸ NM.Ya˘gmurlu and A. Esen, A new perspective for quintic B-spline based Crank-Nicolson-differential quadrature method algorithm for numerical solutions of the nonlinear Schr¨odinger equation, Eur. Phys. J. Plus, 133 (12), (2018), https://doi.org/10.1140/epjp/i2018-11843-1.
  • [25] A.Bas¸han, NM.Ya˘gmurlu, Y.Uc¸ar¸ and A. Esen, A new perspective for the numerical solution of the Modified Equal Width wave equation, Math Meth Appl Sci. 44 (2021), 8925–8939, DOI: 10.1002/mma.7322.
  • [26] A.Bas¸han, An effective approximation to the dispersive soliton solutions of the coupled KdV equation via combination of two efficient methods, Computational and Applied Mathematics, (2020), 39 (80), https://doi.org/10.1007/s40314-020-1109-9.
  • [27] A. Bas¸han, A novel approach via mixed Crank–Nicolson scheme and differential quadrature method for numerical solutions of solitons of mKdV equation,Pramana – J. Phys. (2019) 92(84),https://doi.org/10.1007/s12043-019-1751-1
  • [28] PJ. Olver, Euler operators and conservation laws of the BBM equation, Math. Proc. Camb. Phil. Soc., 85 (1979) ,143-160.
  • [29] P. M. Prenter, Splines and variational methods, John Wiley, New York, NY, 1975.
  • [30] H. Holden et al., Splitting methods for partial differential equations with rough solutions, European Mathematical Society, Publishing House, Z¨urich, 2010.
  • [31] J.VonNeumann, R. D.Richtmyer, A Method for the Numerical Calculation of Hydrodynamic Shocks, J. Appl. Phys., Vol:21, (1950), 232-237.
  • [32] G. D. Smith, Numerical solutions of partial differential equations: Finite difference methods, Clarendon Press,Oxford, 1985.
  • [33] M.Seydaoglu, U. Erdog˘an, T.O¨ zis¸¸ Numerical solution of Burgers’ equation with high order splitting methods, J. Comput. Appl. Math., Vol: 291, (2016) 410–421.
  • [34] J. Geiser, Iterative Splitting Methods for Differential Equations, CHAPMAN and HALL/CRC., Numerical Analysis and Scientific Computing, Boca Raton, 2011.
There are 34 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Melike Karta

Publication Date October 31, 2022
Submission Date March 17, 2022
Acceptance Date August 6, 2022
Published in Issue Year 2022 Volume: 10 Issue: 2

Cite

APA Karta, M. (2022). Two Effective Numerical Approaches for Equal Width Wave (EW) Equation Using Lie-Trotter Splitting Technique. Konuralp Journal of Mathematics, 10(2), 220-232.
AMA Karta M. Two Effective Numerical Approaches for Equal Width Wave (EW) Equation Using Lie-Trotter Splitting Technique. Konuralp J. Math. October 2022;10(2):220-232.
Chicago Karta, Melike. “Two Effective Numerical Approaches for Equal Width Wave (EW) Equation Using Lie-Trotter Splitting Technique”. Konuralp Journal of Mathematics 10, no. 2 (October 2022): 220-32.
EndNote Karta M (October 1, 2022) Two Effective Numerical Approaches for Equal Width Wave (EW) Equation Using Lie-Trotter Splitting Technique. Konuralp Journal of Mathematics 10 2 220–232.
IEEE M. Karta, “Two Effective Numerical Approaches for Equal Width Wave (EW) Equation Using Lie-Trotter Splitting Technique”, Konuralp J. Math., vol. 10, no. 2, pp. 220–232, 2022.
ISNAD Karta, Melike. “Two Effective Numerical Approaches for Equal Width Wave (EW) Equation Using Lie-Trotter Splitting Technique”. Konuralp Journal of Mathematics 10/2 (October 2022), 220-232.
JAMA Karta M. Two Effective Numerical Approaches for Equal Width Wave (EW) Equation Using Lie-Trotter Splitting Technique. Konuralp J. Math. 2022;10:220–232.
MLA Karta, Melike. “Two Effective Numerical Approaches for Equal Width Wave (EW) Equation Using Lie-Trotter Splitting Technique”. Konuralp Journal of Mathematics, vol. 10, no. 2, 2022, pp. 220-32.
Vancouver Karta M. Two Effective Numerical Approaches for Equal Width Wave (EW) Equation Using Lie-Trotter Splitting Technique. Konuralp J. Math. 2022;10(2):220-32.
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