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Nonlinear Dynamics

, Volume 90, Issue 4, pp 2903–2915 | Cite as

Lie symmetry analysis, soliton and numerical solutions of boundary value problem for variable coefficients coupled KdV–Burgers equation

  • Vikas KumarEmail author
  • Aisha Alqahtani
Original Paper

Abstract

In this article, an initial and boundary value problem for variable coefficients coupled KdV–Burgers equation is considered. With the help of Lie group approach, initial and boundary value problem for variable coefficients coupled KdV–Burgers equation reduced to an initial value problem for nonlinear third-order ordinary differential equations (ODEs). Moreover, the systems of ODEs are solved to obtain soliton solutions. Further, classical fourth-order Runge–Kutta method is applied to systems of ODEs for constructing numerical solutions of coupled KdV–Burgers equation. Numerical solutions are computed, and accuracy of numerical scheme is assessed by applying the scheme half mesh principal to calculate maximum errors.

Keywords

Coupled KdV–Burgers equation Lie symmetry analysis Soliton solution Numerical solution 

Notes

Acknowledgements

The authors would like to thank the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University for supporting this research.

References

  1. 1.
    Su, C.H., Gardner, C.S.: Drivation of the Korteweg–de Vries and Burgers equation. J. Math. Phys. 10, 536–539 (1969)CrossRefzbMATHGoogle Scholar
  2. 2.
    Wijngaarden, L.V.: On the motion of gas bubbles in a perfect fluid. Annu. Rev. Fluid Mech. 4, 369–373 (1972)CrossRefGoogle Scholar
  3. 3.
    Johnson, R.S.: Shallow water waves on a viscous fluid—the undular bore. Phys. Fluids 15, 1693–1699 (1972)CrossRefzbMATHGoogle Scholar
  4. 4.
    Hu, P.N.: Collisional theory of shock and nonlinear waves in a plasma. Phys. Fluids 15, 854–864 (1972)CrossRefGoogle Scholar
  5. 5.
    Korteweg, D.J., Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 39, 422–443 (1895)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Burgers, J.M.: A Mathematical Model Illustrating the Theory of Turbulence. Academic Press, New York (1948)CrossRefGoogle Scholar
  7. 7.
    Feudel, F., Steudel, H.: Nonexistence of prolongation structure for the Korteweg–de Vries–Burgers equation. Phys. Lett. A 107, 5–8 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Johnson, R.S.: A nonlinear equation incorporating damping and dispersion. J. Fluid Mech. 42, 49–60 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Grad, H., Hu, P.N.: Unied shock prole in a plasma. Phys. Fluids 10, 2596–2602 (1967)CrossRefGoogle Scholar
  10. 10.
    Canosa, J., Gazdag, J.: The Korteweg–de Vries–Burgers equation. J. Comput. Phys. 23, 393–403 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Dauletiyarov, K.Z.: Investigation of the dierence method for the Bona–Smith and Burgers–Korteweg–deVries equations. Zh. Vychisl. Mat. i. Mat. Fiz. 24, 383–402 (1984)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Avilov, V.V., Krichever, I.M., Novikov, S.P.: Evolution of the Whiteham zone in the Korteweg–de Vries theory. Soviet. Phys. Dokl. 32, 345–349 (1987)zbMATHGoogle Scholar
  13. 13.
    Bona, J.L., Schonbek, M.E.: Traveling wave solutions to Korteweg–de Vries–Burgers equation. Proc. R. Soc. Edinb. 101, 207–226 (1985)CrossRefzbMATHGoogle Scholar
  14. 14.
    Guan, K.Y., Gao, G.: The qualitative theory of the mixed Korteweg–de Vries–Burgers equation. Sci. Sin. Ser. A 30, 64–73 (1987)Google Scholar
  15. 15.
    Guan, K.Y., Lei, J.Z.: Integrability of second order antonomous system. Ann. Differ. Equ. 10, 117–135 (2002)zbMATHGoogle Scholar
  16. 16.
    Gao, J.X., Lei, J.Z., Guan, K.Y.: Integrable condition on traveling wave solutions of Burgers–KdV equation. J. North. Jiaotong Univ. 27, 38–42 (2003)Google Scholar
  17. 17.
    Shu, J.J.: The proper analytical solution of the Korteweg–de Vries equation. J. Phys. A Math. Gen. 20, 49–56 (1987)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Drazin, P., Johnson, R.: Solitons: An Introduction. Cambridge Univesity Press, New York (1989)CrossRefzbMATHGoogle Scholar
  19. 19.
    Shaojie, Y., Cuncai, H.: Lie symmetry reductions and exact solutions of a coupled KdV–Burgers equation. Appl. Math. Comput. 234, 579–583 (2014)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic Press, New York (1982)zbMATHGoogle Scholar
  21. 21.
    Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York (1989)CrossRefzbMATHGoogle Scholar
  22. 22.
    Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1993)CrossRefzbMATHGoogle Scholar
  23. 23.
    Chou, T.: Lie Group and Its Applications in Differential Equations. Science Press, Beijing (2001)Google Scholar
  24. 24.
    Kaur, L., Gupta, R.K.: Kawahara equation and modified Kawahara equation with time dependent coefficients: symmetry analysis and generalized \(\left( {{G}^{\prime }/G} \right)\)-expansion method. Math. Methods Appl. Sci. 36, 584–601 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Kumar, V., Gupta, R.K., Jiwari, R.: Comparative study of travelling wave and numerical solutions for the coupled short pulse (CSP) equation. Chin. Phys. B 22, 050201 (2013)CrossRefGoogle Scholar
  26. 26.
    Johnpillai, A.G., Kara, A.H., Biswas, A.: Symmetry reduction, exact group-invariant solutions and conservation laws of the Benjamin–Bona–Mahoney equation. Appl. Math. Lett. 26, 376–381 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Kumar, V., Gupta, R.K., Jiwari, R.: Painlevé analysis, Lie symmetries and exact solutions for variable coefficients Benjamin–Bona–Mahony–Burger (BBMB) equation. Commun. Theor. Phys. 60, 175–182 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Gupta, R.K., Kumar, V., Jiwari, R.: Exact and numerical solutions of coupled Short Pulse equation with time-dependent coefficients. Nonlinear Dyn. 79, 455–464 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Singh, M., Gupta, R.K.: Exact solutions for nonlinear evolution equations using novel test function. Nonlinear Dyn. 86, 1171–1182 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Singla, K., Gupta, R.K.: On invariant analysis of some time fractional nonlinear systems of partial differential equations. J. Math. Phys. 57, 101504–14 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Rai, P., Sharma, K.K.: Parameter uniform numerical method for singularly perturbed differential–difference equations with interior layer. Int. J. Comput. Math. 88, 3416–3435 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Rai, P., Sharma, K.K.: Numerical analysis of singularly perturbed delay differential turning point problem. Appl. Math. Comput. 218, 3483–3498 (2011)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of MathematicsD.A.V. College PundariKaithalIndia
  2. 2.Department of Mathematical Sciences, College of SciencesPrincess Nourah bint Abdulrahman UniversityRiyadhSaudi Arabia

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