Investigation of Some Explicit Exact Solution of the Damped Forced KDV Burger Equation by Modified Exp(-φ(ξ))-Expansion Method

Authors

Ranjan Barman

Department of Mathematics, Dinhata College, Cooch Behar, 736135 (India)

Article Information

DOI: 10.51244/IJRSI.2025.1208004125

Subject Category: Mathematics

Volume/Issue: 12/9 | Page No: 4688-4694

Publication Timeline

Submitted: 2025-10-04

Accepted: 2025-10-11

Published: 2025-10-24

Abstract

This work presented the some explicit exact solution of the damped forced KdV-Burger equation with variable coefficients. We have successfully applied the exp(-φ(ξ))-expansion method with modification to obtain the generalized explicit exact solution of the damped forced KdV-Burger’s equation. The obtained solution contains the hyperbolic function and trigonometric function. The dynamic behavior of the solution is demonstrated graphically in three dimensional and two dimensional space.

Keywords

Exp(-φ(ξ))-expansion method, variable of separation, KdV-Burger’s equation,forcing term, variable coefficients etc.

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References

1. Johnson, R.S., Shallow water waves on a viscous fluid—the undular bore. Physics of Fluids, 15(1972) 1693–1699. [Google Scholar] [Crossref]

2. van Wijngaarden, L., On the motion of gas bubbles in a perfect fluid. Annual Review of Fluid Mechanics, 4(1972) 369–373. [Google Scholar] [Crossref]

3. Johnson, R.S., A nonlinear equation incorporating damping and dispersion. Journal of Fluid Mechanics, 42(1970) 49–60. [Google Scholar] [Crossref]

4. Grad, H. and Hu, P.N., Unified shock profile in a plasma. Physics of Fluids, 10(1967) 2596–2602. [Google Scholar] [Crossref]

5. Hu, P.N. Collisional theory of shock and nonlinear waves in a plasma. Physics of Fluids, 15(1972) 854–864. [Google Scholar] [Crossref]

6. Ruderman, M.S. Method of derivation of the Korteweg–de Vries–Burgers equation. Journal of Applied Mathematics and Mechanics, 39(1975) 656–664. [Google Scholar] [Crossref]

7. Karahara, T. Weak nonlinear magneto-acoustic waves in a cold plasma in the presence of effective electron–ion collisions. Journal of the Physical Society of Japan, 27(1970) 1321–1329. [Google Scholar] [Crossref]

8. Su, C.H. and Gardner, C.S. Korteweg–de Vries equation and generalizations–Part III: derivation of the Korteweg–de Vries equation and Burgers equation. Journal of Mathematical Physics, 10(1969) 536–539. [Google Scholar] [Crossref]

9. Gao, G. A theory of interaction between dissipation and dispersion of turbulence. Scienceentia Sinica (Series A), 28(1985) 616–627. [Google Scholar] [Crossref]

10. [10] Liu, S.D. and Liu, S.K. KdV–Burgers equation modelling of turbulence. Scienceentia Sinica (Series A), 35(1992) 576–586. [Google Scholar] [Crossref]

11. Wadati, M. Wave propagation in nonlinear lattice. Journal of the Physical Society of Japan, 38(1975) 673–680. [Google Scholar] [Crossref]

12. Zayko, Y.N. Polarization waves in nonlinear dielectric. Zhurnal Teknicheskoi Fiziki, 59(1989) 172–173 (in Russian). [Google Scholar] [Crossref]

13. Burgers, J.M. Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. Transactions of the Royal Netherlands Academy of Science (Amsterdam), 17(1939) 1–53. [Google Scholar] [Crossref]

14. Korteweg, D.J. and de Vries, G. On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary wave. Philosophical Magazine, 39(1895) 422–443. [Google Scholar] [Crossref]

15. Johnson, R.S. A Modern Introduction to the Mathematical Theory of Water Waves (Cambridge,UK:Cambridge University Press 1997). [Google Scholar] [Crossref]

16. Aljahdaly NH, El-Tantawy SA. Novel analytical solution to the damped Kawahara equation and its application for modeling the dissipative nonlinear structures in a fluid medium. Journal of Ocean Engineering and Science. 2022;7(5):492-7. [Google Scholar] [Crossref]

17. Raut S, Barman R, Sarkar T. Integrability, breather, lump and quasi-periodic waves of non-autonomous Kadomtsev–Petviashvili equation based on Bell-polynomial approach. Wave Motion. 2023;119:103125. [Google Scholar] [Crossref]

19. Grimshaw RHJ, Chan KH, Chow KW. Transcritical flow of a stratified fluid: the forced extended Korteweg-de Vries model. Phys Fluids. 2002;14:755–774. [Google Scholar] [Crossref]

20. Li M, Xiao JH, Wang M, et al. Solitons for a forced extended Korteweg-de Vries equation with variable coefficients in atmospheric dynamics. Z Naturforsch A. 2013;68:235–244. [Google Scholar] [Crossref]

21. A. M. Wazwaz, The tanh method for travelling wave solutions of nonlinear equations, Applied Mathematics and Computation 154 (2004) 713–723. [Google Scholar] [Crossref]

22. F. Tascan, A. Yakut, Conservation laws and exact solutions with symmetry reduction of nonlinear reaction diffusion equations, International Journal of Nonlinear Sciences and Numerical Simulation 16 (2015) 191–196. [Google Scholar] [Crossref]

23. E. Zayed, H.A. Zedan, K.A. Gepreel, Group analysis and modified extended tanh-function to find the invariant solutions and soliton solutions for nonlinear euler equations, International Journal of Nonlinear Sciences and Numerical Simulation 5 (2011) 221–234. [Google Scholar] [Crossref]

24. M. Mirzazadeh, M. Eslami, E. Zerrad, M.F. Mahmood, A. Biswas, M. Belic,Optical solitons in nonlinear directional couplers by sine-cosine function method and bernoulli’s equation approach, Nonlinear Dynamics 81 (2015)1933–1949. [Google Scholar] [Crossref]

25. E. Fan, H. Zhang, A note on the homogeneous balance method, Physcis Letters A 246 (1998) 403–406. [Google Scholar] [Crossref]

26. M. A. Abdou, The extended f-expansion method and its application for a class of nonlinear evolution equations, Chaos, Solitons and Fractals 31(2007) 95–104. [Google Scholar] [Crossref]

27. A. Bekir, A. Boz, Exact solutions for nonlinear evolution equations using exp-function method, Physics Letters A 372 (2008) 1619–1625. [Google Scholar] [Crossref]

28. E. Misirli, Y. Gurefe, Exact solutions of the drinfel’d-sokolov-wilson equations by using exp-function method, Applied Mathematics and Computation 216 (2010) 2623–2627. [Google Scholar] [Crossref]

29. M. Mirzazadeh, Modified simple equation method and its applications to nonlinear partial differential equations, Inf. Sci. Lett. 3 (2014) 1–9. [Google Scholar] [Crossref]

30. A. Bekir, M. Kaplan, O. G¨uner, A novel modified simple equation method and its application to some nonlinear evolution equation systems, AIP Conference Proceedings 1611 (2014) 30–36. [Google Scholar] [Crossref]

31. N. Taghizadeh, M. Mirzazadeh, The first integral method to some complex non-linear partial differential equations, Applications and Applied Mathematics 7 (2012) 117–132. [Google Scholar] [Crossref]

32. Y. Gurefe, E. Misirli, Y. Pandir, A. Sonmezoglu, M. Ekici, New exact solutions of the davey-stewartson equation with power-law nonlinearity, Bulletin of the Malaysian Mathematical Sciences Society 38 (2015) 1223–115 1234. [Google Scholar] [Crossref]

33. Y. Gurefe, E. Misirli, New variable separation solutions of two-dimensional burgers system, Applied Mathematics and Computation 217 (2011) 9189–9197. [Google Scholar] [Crossref]

34. M. Kaplan, A. Bekir, M. N.Ozer , Solving nonlinear evolution equation system using two different methods, Open Physics 13 (2015) 383–388. [Google Scholar] [Crossref]

35. E. M. E. Zayed, K.A.E Alurrfi, The (g’/g,1/g)-expansion method and its applications to two nonlinear schrodinger equations describing the propagation of femtosecond pulses in nonlinear optical fibers, Optik 127 (2016)1581–1589. [Google Scholar] [Crossref]

36. M. Mirzazadeh, M. Eslami, A. Biswas, 1-soliton solution of kdv6 equation, Nonlinear Dynamics 80 (2015) 387–396. [Google Scholar] [Crossref]

37. M. Kaplan, A. Akbulut, A. Bekir, Exact travelling wave solutions of the nonlinear evolution equations by auxiliary equation method, Zeitschrift furnaturforschung A 70(2015) 969-974. [Google Scholar] [Crossref]

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