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NUMERICAL SIMULATION OF GENERALIZED OSKOLKOV EQUATION VIA THE SEPTIC B-SPLINE COLLOCATION METHOD

Year 2022, Volume: 5 Issue: 2, 108 - 116, 31.07.2022
https://doi.org/10.33773/jum.1134983

Abstract

In this paper, one of the nonlinear evolution equation (NLEE) namely generalised Oskolkov equation which defines the dynamics of an incompressible visco-elastic Kelvin-Voigt fluid is investigated. We discuss numerical
solutions of the equation for two test problems including single solitary wave and Gaussian initial condition, applying the collocation finite element method. The algorithm, based upon Crank Nicolson approach in time, is unconditionally stable. To demonstrate the proficiency and accuracy of the numerical algorithm, error norms L2, L∞ and invariant I are calculated and the obtained results are indicated both in tabular and graphical form. The obtained numerical results provide the method is more suitable and systematically handle the solution procedures of nonlinear equations arising in mathematical physics.

References

  • Referans1 M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, in: London Mathematical Society Lecture Notes, Cambridge University Press, Cambridge (1991).
  • Referans2 T. Ak, S. B. G. Karakoc, A. Biswas, Numerical scheme to dispersive shallow water waves, Journal of Computational and Theoretical Nanoscience, Vol.13,N.10, pp.7084–7092 (2016).
  • Referans3 S. K. Bhowmik, S. B. G. Karakoc, Numerical approximation of the generalized regularized long wave equation using Petrov Galerkin finite element method, Numerical Methods for Partial Differential Equation, Vol.35,N.6, pp.2236–2257 (2019).
  • Referans4 T. Ak, S. B. G. Karakoc, T. Houria, Numerical simulation for treatment of dispersive shallow water waves with Rosenau KdV equation, The European Physical Journal Plus, Vol.131, pp.1–15 (2016).
  • Referans5 S. B. G. Karakoc, T. Geyikli, A. Bashan, A numerical solution of the Modified Regularized Long Wave MRLW equation using quartic B splines, TWMS J. App. Eng. Math., Vol.3,N.2, pp.231-244 (2013).
  • Referans6 T. Ak, S. B. G. Karakoc, A numerical technique based on collocation method for solving modified Kawahara equation, Journal of Ocean Engineering and Science, Vol.3,N.1, pp.67–75 (2018).
  • Referans7 H. Zeybek and S. B. G. Karakoc, Application of the Collocation Method With B-Splines to the GEW Equation, Electronic Transactions on Numerical Analysis, Vol.46, pp.71–88 (2017).
  • Referans8 E. Pindza, E. Mar´e, Solving the Generalized Regularized Long Wave Equation Using a Distributed Approximating Functional Method, International Journal of Computational Mathematics, Vol.2014, 12 pages (2014).
  • Referans9 M. M. Roshid, T. Bairagi, Harun-Or-Roshid, M.M. Rahman, Lump, interaction of lump and kink and solitonic solution of nonlinear evolution equation which describe incompressible viscoelastic Kelvin–Voigt fluid, Partial Differential Equations in Applied Mathematics, Vol.5 100354, (2022).
  • Referans10 A. O. Kondyukov, T. G. Sukacheva, Phase space of the initial-boundary value problem for the Oskolkov system of nonzero order, Computational Mathematics and Mathematical Physics, Vol.55,N.5, pp.823-828 (2015).
  • Referans11 M. Alquran,Bright and dark soliton solutions to the Ostrovsky-Benjamin-Bona-Mahony (OSBBM) equation, J. Math. Comput. Sci., Vol.2,N.1, pp.15-22 (2012).
  • Referans12 Khan, K., Akbar, M. A., Alam, M. N.: Traveling wave solutions of the nonlinear Drinfel’d– Sokolov–Wilson equation and modified Benjamin–Bona–Mahony equations, Journal of the Egyptian Mathematical Society, Vol.21,N.3, pp.233-240 (2013).
  • Referans13 G. A. Sviridyuk, A. S. Shipilov, On the stability of solutions of the Oskolkov equations on a graph, Differential Equations, Vol.46,N.5, pp.742-747 (2010).
  • Referans14 S. Akcagil, T. Aydemir, O. F. Gozukizil, Exact travelling wave solutions of nonlinear pseudoparabolic equations by using the G’/G Expansion Method, New Trends in Mathematical Sciences, Vol.4,N.4, pp.51-66 (2016).
  • Referans15 M. M. Roshid, O. H. Roshid, Exact and explicit traveling wave solutions to two nonlinear evolution equations which describe incompressible viscoelastic Kelvin-Voigt fluid, Heliyon, Vol.4, pp.1-21 (2018).
  • Referans16 O. F. Gozukizil, S. Akcagil, The tanh-cothmethod for some nonlinear pseudoparabolic equations with exact solutions, Advances in Difference Equations, Vol.143, pp.1-19 (2013).
  • Referans17 M. Roshid, H. Bashar, Breather wave and kinky periodic wave solutions of one-dimensional Oskolkov equation. Mathematical Modelling of Engineering Problems, Vol.6,N.3, pp.460-466 (2019).
  • Referans18 M. N. Alam and C. Tunc, An analytical method for solving exact solutions of the nonlinear Bogoyavlenskii equation and the nonlinear diffusive predator-prey system, Alexandria Engineering Journal, Vol.55, pp.1855-1865 (2016).
  • Referans19 S. B. G. Karakoc, S.K. Bhowmik, D. Y. Sucu, A Novel Scheme Based on Collocation Finite Element Methodto Generalised Oskolkov Equation, journal of Science and Arts, Vol.4,N.47, pp.895–908 (2021).
  • Referans20 M. P. Prenter, Splines and variational methods, Courier Corporation, New York, (2008).
  • Referans21 A. Esen, A lumped Galerkin method for the numerical solution of the modified equal-width wave equation using quadratic B-splines, International Journal of Computer Mathematics, Vol.83, pp.449-459 (2011).
  • Referans122 G. S. Rubin, A. Randolph, J. Graves, A cubic spline approximation for problems in fluid mechanics, NASA STI/Recon Technical Report N., Vol.75, 33345, (1975).
  • Referans23 S. B. G. Karakoc, A Quartic Subdomain Finite Element Method for the Modified KdV Equation, Statistic, Optimization and Information Computing, Vol.6,N.4, pp.609–618 (2018).
  • Referans24 T. Ak, T. Aydemir, A. Saha, A. Kara, Propagation of nonlinear shock waves for the generalised Oskolkov equation and its dynamic motions in thepresence of an external periodic perturbation, Pramana. J. Phys, Vol.90,N.78, pp.1-16 (2018).
Year 2022, Volume: 5 Issue: 2, 108 - 116, 31.07.2022
https://doi.org/10.33773/jum.1134983

Abstract

References

  • Referans1 M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, in: London Mathematical Society Lecture Notes, Cambridge University Press, Cambridge (1991).
  • Referans2 T. Ak, S. B. G. Karakoc, A. Biswas, Numerical scheme to dispersive shallow water waves, Journal of Computational and Theoretical Nanoscience, Vol.13,N.10, pp.7084–7092 (2016).
  • Referans3 S. K. Bhowmik, S. B. G. Karakoc, Numerical approximation of the generalized regularized long wave equation using Petrov Galerkin finite element method, Numerical Methods for Partial Differential Equation, Vol.35,N.6, pp.2236–2257 (2019).
  • Referans4 T. Ak, S. B. G. Karakoc, T. Houria, Numerical simulation for treatment of dispersive shallow water waves with Rosenau KdV equation, The European Physical Journal Plus, Vol.131, pp.1–15 (2016).
  • Referans5 S. B. G. Karakoc, T. Geyikli, A. Bashan, A numerical solution of the Modified Regularized Long Wave MRLW equation using quartic B splines, TWMS J. App. Eng. Math., Vol.3,N.2, pp.231-244 (2013).
  • Referans6 T. Ak, S. B. G. Karakoc, A numerical technique based on collocation method for solving modified Kawahara equation, Journal of Ocean Engineering and Science, Vol.3,N.1, pp.67–75 (2018).
  • Referans7 H. Zeybek and S. B. G. Karakoc, Application of the Collocation Method With B-Splines to the GEW Equation, Electronic Transactions on Numerical Analysis, Vol.46, pp.71–88 (2017).
  • Referans8 E. Pindza, E. Mar´e, Solving the Generalized Regularized Long Wave Equation Using a Distributed Approximating Functional Method, International Journal of Computational Mathematics, Vol.2014, 12 pages (2014).
  • Referans9 M. M. Roshid, T. Bairagi, Harun-Or-Roshid, M.M. Rahman, Lump, interaction of lump and kink and solitonic solution of nonlinear evolution equation which describe incompressible viscoelastic Kelvin–Voigt fluid, Partial Differential Equations in Applied Mathematics, Vol.5 100354, (2022).
  • Referans10 A. O. Kondyukov, T. G. Sukacheva, Phase space of the initial-boundary value problem for the Oskolkov system of nonzero order, Computational Mathematics and Mathematical Physics, Vol.55,N.5, pp.823-828 (2015).
  • Referans11 M. Alquran,Bright and dark soliton solutions to the Ostrovsky-Benjamin-Bona-Mahony (OSBBM) equation, J. Math. Comput. Sci., Vol.2,N.1, pp.15-22 (2012).
  • Referans12 Khan, K., Akbar, M. A., Alam, M. N.: Traveling wave solutions of the nonlinear Drinfel’d– Sokolov–Wilson equation and modified Benjamin–Bona–Mahony equations, Journal of the Egyptian Mathematical Society, Vol.21,N.3, pp.233-240 (2013).
  • Referans13 G. A. Sviridyuk, A. S. Shipilov, On the stability of solutions of the Oskolkov equations on a graph, Differential Equations, Vol.46,N.5, pp.742-747 (2010).
  • Referans14 S. Akcagil, T. Aydemir, O. F. Gozukizil, Exact travelling wave solutions of nonlinear pseudoparabolic equations by using the G’/G Expansion Method, New Trends in Mathematical Sciences, Vol.4,N.4, pp.51-66 (2016).
  • Referans15 M. M. Roshid, O. H. Roshid, Exact and explicit traveling wave solutions to two nonlinear evolution equations which describe incompressible viscoelastic Kelvin-Voigt fluid, Heliyon, Vol.4, pp.1-21 (2018).
  • Referans16 O. F. Gozukizil, S. Akcagil, The tanh-cothmethod for some nonlinear pseudoparabolic equations with exact solutions, Advances in Difference Equations, Vol.143, pp.1-19 (2013).
  • Referans17 M. Roshid, H. Bashar, Breather wave and kinky periodic wave solutions of one-dimensional Oskolkov equation. Mathematical Modelling of Engineering Problems, Vol.6,N.3, pp.460-466 (2019).
  • Referans18 M. N. Alam and C. Tunc, An analytical method for solving exact solutions of the nonlinear Bogoyavlenskii equation and the nonlinear diffusive predator-prey system, Alexandria Engineering Journal, Vol.55, pp.1855-1865 (2016).
  • Referans19 S. B. G. Karakoc, S.K. Bhowmik, D. Y. Sucu, A Novel Scheme Based on Collocation Finite Element Methodto Generalised Oskolkov Equation, journal of Science and Arts, Vol.4,N.47, pp.895–908 (2021).
  • Referans20 M. P. Prenter, Splines and variational methods, Courier Corporation, New York, (2008).
  • Referans21 A. Esen, A lumped Galerkin method for the numerical solution of the modified equal-width wave equation using quadratic B-splines, International Journal of Computer Mathematics, Vol.83, pp.449-459 (2011).
  • Referans122 G. S. Rubin, A. Randolph, J. Graves, A cubic spline approximation for problems in fluid mechanics, NASA STI/Recon Technical Report N., Vol.75, 33345, (1975).
  • Referans23 S. B. G. Karakoc, A Quartic Subdomain Finite Element Method for the Modified KdV Equation, Statistic, Optimization and Information Computing, Vol.6,N.4, pp.609–618 (2018).
  • Referans24 T. Ak, T. Aydemir, A. Saha, A. Kara, Propagation of nonlinear shock waves for the generalised Oskolkov equation and its dynamic motions in thepresence of an external periodic perturbation, Pramana. J. Phys, Vol.90,N.78, pp.1-16 (2018).
There are 24 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Seydi Battal Gazi Karakoç 0000-0002-2348-4170

Derya Sucu 0000-0001-8396-8081

Mohamad Arif Taghachi 0000-0001-5760-6538

Publication Date July 31, 2022
Submission Date June 23, 2022
Acceptance Date July 27, 2022
Published in Issue Year 2022 Volume: 5 Issue: 2

Cite

APA Karakoç, S. B. G., Sucu, D., & Taghachi, M. A. (2022). NUMERICAL SIMULATION OF GENERALIZED OSKOLKOV EQUATION VIA THE SEPTIC B-SPLINE COLLOCATION METHOD. Journal of Universal Mathematics, 5(2), 108-116. https://doi.org/10.33773/jum.1134983
AMA Karakoç SBG, Sucu D, Taghachi MA. NUMERICAL SIMULATION OF GENERALIZED OSKOLKOV EQUATION VIA THE SEPTIC B-SPLINE COLLOCATION METHOD. JUM. July 2022;5(2):108-116. doi:10.33773/jum.1134983
Chicago Karakoç, Seydi Battal Gazi, Derya Sucu, and Mohamad Arif Taghachi. “NUMERICAL SIMULATION OF GENERALIZED OSKOLKOV EQUATION VIA THE SEPTIC B-SPLINE COLLOCATION METHOD”. Journal of Universal Mathematics 5, no. 2 (July 2022): 108-16. https://doi.org/10.33773/jum.1134983.
EndNote Karakoç SBG, Sucu D, Taghachi MA (July 1, 2022) NUMERICAL SIMULATION OF GENERALIZED OSKOLKOV EQUATION VIA THE SEPTIC B-SPLINE COLLOCATION METHOD. Journal of Universal Mathematics 5 2 108–116.
IEEE S. B. G. Karakoç, D. Sucu, and M. A. Taghachi, “NUMERICAL SIMULATION OF GENERALIZED OSKOLKOV EQUATION VIA THE SEPTIC B-SPLINE COLLOCATION METHOD”, JUM, vol. 5, no. 2, pp. 108–116, 2022, doi: 10.33773/jum.1134983.
ISNAD Karakoç, Seydi Battal Gazi et al. “NUMERICAL SIMULATION OF GENERALIZED OSKOLKOV EQUATION VIA THE SEPTIC B-SPLINE COLLOCATION METHOD”. Journal of Universal Mathematics 5/2 (July 2022), 108-116. https://doi.org/10.33773/jum.1134983.
JAMA Karakoç SBG, Sucu D, Taghachi MA. NUMERICAL SIMULATION OF GENERALIZED OSKOLKOV EQUATION VIA THE SEPTIC B-SPLINE COLLOCATION METHOD. JUM. 2022;5:108–116.
MLA Karakoç, Seydi Battal Gazi et al. “NUMERICAL SIMULATION OF GENERALIZED OSKOLKOV EQUATION VIA THE SEPTIC B-SPLINE COLLOCATION METHOD”. Journal of Universal Mathematics, vol. 5, no. 2, 2022, pp. 108-16, doi:10.33773/jum.1134983.
Vancouver Karakoç SBG, Sucu D, Taghachi MA. NUMERICAL SIMULATION OF GENERALIZED OSKOLKOV EQUATION VIA THE SEPTIC B-SPLINE COLLOCATION METHOD. JUM. 2022;5(2):108-16.

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