Research Article
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Year 2022, Volume: 35 Issue: 4, 1597 - 1612, 01.12.2022
https://doi.org/10.35378/gujs.892116

Abstract

References

  • [1] Benjamin, T.B., Bona, J.L., Mahony, J.J., “Model Equations for Long Waves in Nonlinear Dispersive Systems”, Philosophical Transactions of the Royal Society A, 272(1220): 47-78, (1972). DOI: 10.1098/rsta.1972.0032
  • [2] Wazwaz, A., “Partial differential equations and solitary waves theory”, Springer-Verlag Berlin Heidelberg, (2009).
  • [3] Wazwaz, A., “Analytic study on nonlinear variants of the RLW and the PHI-four equations”, Communications in Nonlinear Science and Numerical Simulation, 12(3): 314-327, (2007).
  • [4] Lou, Y., “Bifurcation of travelling wave solutions in a nonlinear variant of the RLW equation”, Communications in Nonlinear Science and Numerical Simulation, 12(8): 1488–1503, (2007).
  • [5] Soliman, A.A., “Exact traveling wave solution of nonlinear variants of the RLW and the PHI-four equations”, Physics Letters A, 368(5): 383–390, (2007).
  • [6] Nuruddeen, R.I., Aboodh, K.S., Ali, K.K., “Investigating the tangent dispersive solitary wave solutions to the Equal Width and Regularized Long Wave equations”, Journal of King Saud University – Science, 32(1): 677–681, (2020).
  • [7] Feng, D., Li, J., Lu, J., He, T., “New explicit and exact solutions for a system of variant RLW equations”, Applied Mathematics and Computation, 198(2): 715–720, (2008).
  • [8] Roshid, H., Roshid, Md.M. Rahman, N., Pervin, Mst.R., “New solitary wave in shallow water, plasma and ion acoustic plasma via the GZK-BBM equation and the RLW equation”, Propulsion and Power Research, 6(1): 49–57, (2017).
  • [9] Kutluay, S., Esen, A., “A finite difference solution of the regularized long-wave equation”, Mathematical Problems in Engineering, 2006: 1–14, (2006).
  • [10] Inan, B., Bahadır, A.R., “A Fully Implicit Finite Difference Scheme for the Regularized Long Wave Equation”, General Mathematics Notes, 33(2):40-59, (2016).
  • [11] Jain, P.C. Shankar, R. Singh, T.V., “Numerical solution of regularized long wave equation”, Communications in Numerical Methods in Engineering, 9: 579-586, (1993).
  • [12] Dag, I. Irk, D., Kaçmaz, O., Adar, N., “Trigonometric B-spline collocation algorithm for solving the RLW equation”, Applied and Computational Mathematics, 15(1): 96-105, (2016).
  • [13] Irk, D., Dag, I. Doğan, A., “Numerical integration of the RLW equation using cubic splines”, The Australian & New Zealand Industrial and Applied Mathematics Journal, 47(1): 131-142, (2005).
  • [14] Islam, S., Haq, S., Ali, A., “A meshfree method for the numerical solution of the RLW equation”, Journal of Computational and Applied Mathematics, 223(2): 997–1012, (2009).
  • [15] Saka, B., Dag, I., Irk, D., “Quintic B-spline collocation method for numerical solution of the RLW equation”, The Australian & New Zealand Industrial and Applied Mathematics Journal, 49(3): 389–410, (2008). DOI: 10.1017/S1446181108000072
  • [16] Mokhtari, R., Mohammadi, M., “Numerical solution of GRLW equation using Sinc-collocation method”, Computer Physics Communications, 181(7): 1266–1274, (2010).
  • [17] Raslan, K.R., “A computational method for the regularized long wave (RLW) equation”, Applied Mathematics and Computation, 167(2): 1101–1118, (2005).
  • [18] Dag, I., Saka, B., Irk, D., “Application of cubic B-splines for numerical solution of the RLW equation”, Applied Mathematics and Computation, 159(2): 373–389, (2004).
  • [19] Saka, B., Dag, I., “Quartic B-Spline Collocation Algorithms for Numerical Solution of the RLW Equation”, Numerical Methods for Partial Differential Equations, 23(3): 731–751, (2007).
  • [20] Esen, A., Kutluay, S., “Application of a lumped Galerkin method to the regularized long wave equation”, Applied Mathematics and Computation, 174(2): 833–845, (2006).
  • [21] Mei, L., Chen, Y., “Explicit multistep method for the numerical solution of RLW equation”, Applied Mathematics and Computation, 218(18): 9547–9554, (2012).
  • [22] Mei, L., Chen, Y., “Numerical solutions of RLW equation using Galerkin method with extrapolation techniques”, Computer Physics Communications, 183(8): 1609–1616, (2012).
  • [23] Gardner, L.R.T., Dag, I., “The Boundary-Forced Regularised Long-Wave Equation”, Il Nuovo Cimento, 110 B(12): 1487-1496, (1995).
  • [24] Dag, I., Saka, B., Irk, D., “Galerkin method for the numerical solution of the RLW equation using quintic B-splines”, Journal of Computational and Applied Mathematics, 190(1-2): 532–547, (2006).
  • [25] Saka, B., Dag, I., Doğan, A., “Galerkin method for the numerical solution of the RLW equation using quadratic B-splines”, International Journal of Computer Mathematics, 81(6): 727–739, (2004).
  • [26] Doğan, A., “Numerical solution of RLW equation using linear finite elements within Galerkin’s method”, Applied Mathematical Modelling, 26(7): 771–783, (2002).
  • [27] Gardner, L.R.T., Gardner, G.A., Dag, I., “A B-spline finite element method for the regularized long wave equation”, Communications in Numerical Methods in Engineering, 11: 59-68, (1995).
  • [28] Görgülü, M.Z., Dag, I., Irk, D., “Simulations of solitary waves of RLW equation by exponential B-spline Galerkin method”, Chinese Physics B, 26(8): 080202, (2017).
  • [29] Zaki, S.I., “Solitary waves of the splitted RLW equation”, Computer Physics Communications, 138(1): 80–91, (2001).
  • [30] Gardner, L.R.T., Gardner, G.A., Doğan, A., “A least-squares finite element scheme for the RLW equation”, Communications in Numerical Methods in Engineering, 12: 795-804, (1996).
  • [31] Dag, I., “Least-squares quadratic B-spline finite element method for the regularized long wave equation”, Computer Methods in Applied Mechanics and Engineering, 182(1-2): 205-215, (2000).
  • [32] Dag, I., Korkmaz, A., Saka, B., “Cosine Expansion-Based Differential Quadrature Algorithm for Numerical Solution of the RLW Equation”, Numerical Methods for Partial Differential Equations, 26(3): 544–560, (2010).
  • [33] Sloan, D.M., “Fourier pseudospectral solution of the regularised long wave equation”, Journal of Computational and Applied Mathematics, 36(2): 159-179, (1991).
  • [34] Hosseini, M.M., Ghaneai, H., Mohyud-Din, S.T., Usman, M., “Tri-prong scheme for regularized long wave equation”, Journal of the Association of Arab Universities for Basic and Applied Sciences, 20(1): 68–77, (2016).
  • [35] El-Danaf, T.S., Ramadan, M.A., Abd Alaal, F.E.I., “The use of adomian decomposition method for solving the regularized long-wave equation”, Chaos, Solitons and Fractals, 26(3): 747–757, (2005).
  • [36] Çelik, E., Bayram, M., “On the numerical solution of differential-algebraic equations by Pade series”, Applied Mathematics and Computation, 137: 151–160, (2003).
  • [37] Çelik, E., Karaduman, E., Bayram, M., “Numerical Method to Solve Chemical Differential-Algebraic Equations”, International Journal of Quantum Chemistry, 89: 447–451, (2002).
  • [38] Çelik, E., Bayram, M., Yeloğlu, T., “Solution of Differential-Algebraic Equations (DAEs) by Adomian Decomposition Method”, International Journal Pure & Applied Mathematical Sciences, 3(1): 93-100, (2006).
  • [39] Bellman, R., Kashef, B.G., Casti, J., “Differential quadrature: a tecnique for the rapid solution of nonlinear differential equations”, Journal of Computational Physics, 10(1): 40-52, (1972).
  • [40] Başhan, A., “Highly efficient approach to numerical solutions of two different forms of the modified Kawahara equation via contribution of two effective methods”, Mathematics and Computers in Simulation, 179: 111-125, (2021).
  • [41] Başhan, A., “An effective approximation to the dispersive soliton solutions of the coupled KdV equation via combination of two efficient methods”, Computational and Applied Mathematics, 39(2): 80, (2020).
  • [42] Başhan, A., Yağmurlu, N.M., Uçar, Y., Esen, A., “A new perspective for the numerical solutions of the cmKdV equation via modified cubic B-spline differential quadrature method”, International Journal of Modern Physics C, 29(6): 1850043, (2018).
  • [43] Başhan, A., Yağmurlu, N.M., Uçar, Y., Esen, A., “Finite difference method combined with differential quadrature method for numerical computation of the modified equal width wave equation”, Numerical Methods for Partial Differential Equations, 37(1): 690–706, (2021).
  • [44] Shu, C., “Differential Quadrature and its application in engineering”, Springer-Verlag, London, (2000).
  • [45] Uçar, Y., Yağmurlu, N.M., Başhan, A., “Numerical Solutions and Stability Analysis of Modified Burgers Equation via Modified Cubic B-spline Differential Quadrature Methods”, Sigma Journal of Engineering and Natural Sciences, 37 (1): 129-142, (2019).
  • [46] Başhan, A., Uçar, Y., Yağmurlu, N.M., Esen, A., “Numerical solutions for the fourth order extended Fisher-Kolmogorov equation with high accuracy by differential quadrature method”, Sigma Journal of Engineering and Natural Sciences, 9(3): 273-284, (2018).
  • [47] Başhan, A., “A numerical treatment of the coupled viscous Burgers’ equation in the presence of very large Reynolds number”, Physica A, 545: 123755, (2020).
  • [48] Başhan, A., “Quartic B-spline Differential Quadrature Method for Solving the Extended Fisher-Kolmogorov Equation”, Erzincan University, Journal of Science and Technology, 12(1): 56-62, (2019).
  • [49] Crank, J., Nicolson, P., “A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type”, Mathematical Proceedings of the Cambridge Philosophical Society, 43(1): 50-67, (1947). DOI: 10.1017/S0305004100023197
  • [50] Rubin, S.G., Graves, R.A., “A cubic spline approximation for problems in fluid mechanics”, National Aeronautics and Space Administration, Technical Report, Washington, (1975).
  • [51] Chegini, N.G., Salaripanah, A., Mokhtari, R., Isvand, D., “Numerical solution of the regularized long wave equation using nonpolynomial splines”, Nonlinear Dynamics, 69: 459–471, (2012).
  • [52] Olver, P.J., “Euler operators and conservation laws of the BBM equation”, Mathematical Proceedings of the Cambridge Philosophical Society, 85(01): 143-160, (1979).

Single Solitary Wave and Wave Generation Solutions of the Regularised Long Wave (RLW) Equation

Year 2022, Volume: 35 Issue: 4, 1597 - 1612, 01.12.2022
https://doi.org/10.35378/gujs.892116

Abstract

In this paper, high accurate numerical solutions of the regularised long-wave (RLW) equation is going to be obtained by using effective algorithm including finite difference method, differential quadrature and Rubin-Graves type linearization technique. Solitary wave solutions and Maxwellian initial condition based wave generation solutions are obtained successfully. To observe the development of the present algorithm, the present numerical results are compared with many earlier works. The present results are seen as superior among the given ones. The rates of the convergence are also given.

References

  • [1] Benjamin, T.B., Bona, J.L., Mahony, J.J., “Model Equations for Long Waves in Nonlinear Dispersive Systems”, Philosophical Transactions of the Royal Society A, 272(1220): 47-78, (1972). DOI: 10.1098/rsta.1972.0032
  • [2] Wazwaz, A., “Partial differential equations and solitary waves theory”, Springer-Verlag Berlin Heidelberg, (2009).
  • [3] Wazwaz, A., “Analytic study on nonlinear variants of the RLW and the PHI-four equations”, Communications in Nonlinear Science and Numerical Simulation, 12(3): 314-327, (2007).
  • [4] Lou, Y., “Bifurcation of travelling wave solutions in a nonlinear variant of the RLW equation”, Communications in Nonlinear Science and Numerical Simulation, 12(8): 1488–1503, (2007).
  • [5] Soliman, A.A., “Exact traveling wave solution of nonlinear variants of the RLW and the PHI-four equations”, Physics Letters A, 368(5): 383–390, (2007).
  • [6] Nuruddeen, R.I., Aboodh, K.S., Ali, K.K., “Investigating the tangent dispersive solitary wave solutions to the Equal Width and Regularized Long Wave equations”, Journal of King Saud University – Science, 32(1): 677–681, (2020).
  • [7] Feng, D., Li, J., Lu, J., He, T., “New explicit and exact solutions for a system of variant RLW equations”, Applied Mathematics and Computation, 198(2): 715–720, (2008).
  • [8] Roshid, H., Roshid, Md.M. Rahman, N., Pervin, Mst.R., “New solitary wave in shallow water, plasma and ion acoustic plasma via the GZK-BBM equation and the RLW equation”, Propulsion and Power Research, 6(1): 49–57, (2017).
  • [9] Kutluay, S., Esen, A., “A finite difference solution of the regularized long-wave equation”, Mathematical Problems in Engineering, 2006: 1–14, (2006).
  • [10] Inan, B., Bahadır, A.R., “A Fully Implicit Finite Difference Scheme for the Regularized Long Wave Equation”, General Mathematics Notes, 33(2):40-59, (2016).
  • [11] Jain, P.C. Shankar, R. Singh, T.V., “Numerical solution of regularized long wave equation”, Communications in Numerical Methods in Engineering, 9: 579-586, (1993).
  • [12] Dag, I. Irk, D., Kaçmaz, O., Adar, N., “Trigonometric B-spline collocation algorithm for solving the RLW equation”, Applied and Computational Mathematics, 15(1): 96-105, (2016).
  • [13] Irk, D., Dag, I. Doğan, A., “Numerical integration of the RLW equation using cubic splines”, The Australian & New Zealand Industrial and Applied Mathematics Journal, 47(1): 131-142, (2005).
  • [14] Islam, S., Haq, S., Ali, A., “A meshfree method for the numerical solution of the RLW equation”, Journal of Computational and Applied Mathematics, 223(2): 997–1012, (2009).
  • [15] Saka, B., Dag, I., Irk, D., “Quintic B-spline collocation method for numerical solution of the RLW equation”, The Australian & New Zealand Industrial and Applied Mathematics Journal, 49(3): 389–410, (2008). DOI: 10.1017/S1446181108000072
  • [16] Mokhtari, R., Mohammadi, M., “Numerical solution of GRLW equation using Sinc-collocation method”, Computer Physics Communications, 181(7): 1266–1274, (2010).
  • [17] Raslan, K.R., “A computational method for the regularized long wave (RLW) equation”, Applied Mathematics and Computation, 167(2): 1101–1118, (2005).
  • [18] Dag, I., Saka, B., Irk, D., “Application of cubic B-splines for numerical solution of the RLW equation”, Applied Mathematics and Computation, 159(2): 373–389, (2004).
  • [19] Saka, B., Dag, I., “Quartic B-Spline Collocation Algorithms for Numerical Solution of the RLW Equation”, Numerical Methods for Partial Differential Equations, 23(3): 731–751, (2007).
  • [20] Esen, A., Kutluay, S., “Application of a lumped Galerkin method to the regularized long wave equation”, Applied Mathematics and Computation, 174(2): 833–845, (2006).
  • [21] Mei, L., Chen, Y., “Explicit multistep method for the numerical solution of RLW equation”, Applied Mathematics and Computation, 218(18): 9547–9554, (2012).
  • [22] Mei, L., Chen, Y., “Numerical solutions of RLW equation using Galerkin method with extrapolation techniques”, Computer Physics Communications, 183(8): 1609–1616, (2012).
  • [23] Gardner, L.R.T., Dag, I., “The Boundary-Forced Regularised Long-Wave Equation”, Il Nuovo Cimento, 110 B(12): 1487-1496, (1995).
  • [24] Dag, I., Saka, B., Irk, D., “Galerkin method for the numerical solution of the RLW equation using quintic B-splines”, Journal of Computational and Applied Mathematics, 190(1-2): 532–547, (2006).
  • [25] Saka, B., Dag, I., Doğan, A., “Galerkin method for the numerical solution of the RLW equation using quadratic B-splines”, International Journal of Computer Mathematics, 81(6): 727–739, (2004).
  • [26] Doğan, A., “Numerical solution of RLW equation using linear finite elements within Galerkin’s method”, Applied Mathematical Modelling, 26(7): 771–783, (2002).
  • [27] Gardner, L.R.T., Gardner, G.A., Dag, I., “A B-spline finite element method for the regularized long wave equation”, Communications in Numerical Methods in Engineering, 11: 59-68, (1995).
  • [28] Görgülü, M.Z., Dag, I., Irk, D., “Simulations of solitary waves of RLW equation by exponential B-spline Galerkin method”, Chinese Physics B, 26(8): 080202, (2017).
  • [29] Zaki, S.I., “Solitary waves of the splitted RLW equation”, Computer Physics Communications, 138(1): 80–91, (2001).
  • [30] Gardner, L.R.T., Gardner, G.A., Doğan, A., “A least-squares finite element scheme for the RLW equation”, Communications in Numerical Methods in Engineering, 12: 795-804, (1996).
  • [31] Dag, I., “Least-squares quadratic B-spline finite element method for the regularized long wave equation”, Computer Methods in Applied Mechanics and Engineering, 182(1-2): 205-215, (2000).
  • [32] Dag, I., Korkmaz, A., Saka, B., “Cosine Expansion-Based Differential Quadrature Algorithm for Numerical Solution of the RLW Equation”, Numerical Methods for Partial Differential Equations, 26(3): 544–560, (2010).
  • [33] Sloan, D.M., “Fourier pseudospectral solution of the regularised long wave equation”, Journal of Computational and Applied Mathematics, 36(2): 159-179, (1991).
  • [34] Hosseini, M.M., Ghaneai, H., Mohyud-Din, S.T., Usman, M., “Tri-prong scheme for regularized long wave equation”, Journal of the Association of Arab Universities for Basic and Applied Sciences, 20(1): 68–77, (2016).
  • [35] El-Danaf, T.S., Ramadan, M.A., Abd Alaal, F.E.I., “The use of adomian decomposition method for solving the regularized long-wave equation”, Chaos, Solitons and Fractals, 26(3): 747–757, (2005).
  • [36] Çelik, E., Bayram, M., “On the numerical solution of differential-algebraic equations by Pade series”, Applied Mathematics and Computation, 137: 151–160, (2003).
  • [37] Çelik, E., Karaduman, E., Bayram, M., “Numerical Method to Solve Chemical Differential-Algebraic Equations”, International Journal of Quantum Chemistry, 89: 447–451, (2002).
  • [38] Çelik, E., Bayram, M., Yeloğlu, T., “Solution of Differential-Algebraic Equations (DAEs) by Adomian Decomposition Method”, International Journal Pure & Applied Mathematical Sciences, 3(1): 93-100, (2006).
  • [39] Bellman, R., Kashef, B.G., Casti, J., “Differential quadrature: a tecnique for the rapid solution of nonlinear differential equations”, Journal of Computational Physics, 10(1): 40-52, (1972).
  • [40] Başhan, A., “Highly efficient approach to numerical solutions of two different forms of the modified Kawahara equation via contribution of two effective methods”, Mathematics and Computers in Simulation, 179: 111-125, (2021).
  • [41] Başhan, A., “An effective approximation to the dispersive soliton solutions of the coupled KdV equation via combination of two efficient methods”, Computational and Applied Mathematics, 39(2): 80, (2020).
  • [42] Başhan, A., Yağmurlu, N.M., Uçar, Y., Esen, A., “A new perspective for the numerical solutions of the cmKdV equation via modified cubic B-spline differential quadrature method”, International Journal of Modern Physics C, 29(6): 1850043, (2018).
  • [43] Başhan, A., Yağmurlu, N.M., Uçar, Y., Esen, A., “Finite difference method combined with differential quadrature method for numerical computation of the modified equal width wave equation”, Numerical Methods for Partial Differential Equations, 37(1): 690–706, (2021).
  • [44] Shu, C., “Differential Quadrature and its application in engineering”, Springer-Verlag, London, (2000).
  • [45] Uçar, Y., Yağmurlu, N.M., Başhan, A., “Numerical Solutions and Stability Analysis of Modified Burgers Equation via Modified Cubic B-spline Differential Quadrature Methods”, Sigma Journal of Engineering and Natural Sciences, 37 (1): 129-142, (2019).
  • [46] Başhan, A., Uçar, Y., Yağmurlu, N.M., Esen, A., “Numerical solutions for the fourth order extended Fisher-Kolmogorov equation with high accuracy by differential quadrature method”, Sigma Journal of Engineering and Natural Sciences, 9(3): 273-284, (2018).
  • [47] Başhan, A., “A numerical treatment of the coupled viscous Burgers’ equation in the presence of very large Reynolds number”, Physica A, 545: 123755, (2020).
  • [48] Başhan, A., “Quartic B-spline Differential Quadrature Method for Solving the Extended Fisher-Kolmogorov Equation”, Erzincan University, Journal of Science and Technology, 12(1): 56-62, (2019).
  • [49] Crank, J., Nicolson, P., “A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type”, Mathematical Proceedings of the Cambridge Philosophical Society, 43(1): 50-67, (1947). DOI: 10.1017/S0305004100023197
  • [50] Rubin, S.G., Graves, R.A., “A cubic spline approximation for problems in fluid mechanics”, National Aeronautics and Space Administration, Technical Report, Washington, (1975).
  • [51] Chegini, N.G., Salaripanah, A., Mokhtari, R., Isvand, D., “Numerical solution of the regularized long wave equation using nonpolynomial splines”, Nonlinear Dynamics, 69: 459–471, (2012).
  • [52] Olver, P.J., “Euler operators and conservation laws of the BBM equation”, Mathematical Proceedings of the Cambridge Philosophical Society, 85(01): 143-160, (1979).
There are 52 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Mathematics
Authors

Ali Başhan 0000-0001-8500-493X

Publication Date December 1, 2022
Published in Issue Year 2022 Volume: 35 Issue: 4

Cite

APA Başhan, A. (2022). Single Solitary Wave and Wave Generation Solutions of the Regularised Long Wave (RLW) Equation. Gazi University Journal of Science, 35(4), 1597-1612. https://doi.org/10.35378/gujs.892116
AMA Başhan A. Single Solitary Wave and Wave Generation Solutions of the Regularised Long Wave (RLW) Equation. Gazi University Journal of Science. December 2022;35(4):1597-1612. doi:10.35378/gujs.892116
Chicago Başhan, Ali. “Single Solitary Wave and Wave Generation Solutions of the Regularised Long Wave (RLW) Equation”. Gazi University Journal of Science 35, no. 4 (December 2022): 1597-1612. https://doi.org/10.35378/gujs.892116.
EndNote Başhan A (December 1, 2022) Single Solitary Wave and Wave Generation Solutions of the Regularised Long Wave (RLW) Equation. Gazi University Journal of Science 35 4 1597–1612.
IEEE A. Başhan, “Single Solitary Wave and Wave Generation Solutions of the Regularised Long Wave (RLW) Equation”, Gazi University Journal of Science, vol. 35, no. 4, pp. 1597–1612, 2022, doi: 10.35378/gujs.892116.
ISNAD Başhan, Ali. “Single Solitary Wave and Wave Generation Solutions of the Regularised Long Wave (RLW) Equation”. Gazi University Journal of Science 35/4 (December 2022), 1597-1612. https://doi.org/10.35378/gujs.892116.
JAMA Başhan A. Single Solitary Wave and Wave Generation Solutions of the Regularised Long Wave (RLW) Equation. Gazi University Journal of Science. 2022;35:1597–1612.
MLA Başhan, Ali. “Single Solitary Wave and Wave Generation Solutions of the Regularised Long Wave (RLW) Equation”. Gazi University Journal of Science, vol. 35, no. 4, 2022, pp. 1597-12, doi:10.35378/gujs.892116.
Vancouver Başhan A. Single Solitary Wave and Wave Generation Solutions of the Regularised Long Wave (RLW) Equation. Gazi University Journal of Science. 2022;35(4):1597-612.