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Pramana

, 92:8 | Cite as

A dynamical study of certain nonlinear diffusion–reaction equations with a nonlinear convective flux term

  • Anand Malik
  • Hitender KumarEmail author
  • Rishi Pal Chahal
  • Fakir Chand
Article
  • 37 Downloads

Abstract

We explore the dynamics of quadratic and quartic nonlinear diffusion–reaction equations with nonlinear convective flux term, which arise in well-known physical and biological problems such as population dynamics of the species. Three integration techniques, namely the \(({G^\prime }/{G})\)-expansion method, its generalised version and Kudryashov method, are adopted to solve these equations. We attain new travelling and solitary wave solutions in the form of Jacobi elliptic functions, hyperbolic functions, trigonometric functions and rational solutions with some constraint relations that naturally appear from the structure of these solutions. The travelling population fronts, which are the general solutions of nonlinear diffusion–reaction equations, describe the species invasion if higher population density corresponds to the species invasion. This effort highlights the significant features of the employed algebraic approaches and shows the diversity in the constructed solutions.

Keywords

Nonlinear diffusion–reaction equation Jacobi elliptic functions kink–antikink soliton 

PACS Nos

02.30.Ik 02.30.Jr 02.70.Wz 05.45.Yv 

Notes

Acknowledgements

The authors would like to thank the anonymous referees for many useful suggestions and detailed comments that helped them to improve this paper.

References

  1. 1.
    J D Murray, Mathematical biology: An introduction (Springer-Verlag, New York, 1993)Google Scholar
  2. 2.
    P G Drazin and R S Johnson, Solitons: An introduction (Cambridge University Press, Cambridge, 1989)CrossRefGoogle Scholar
  3. 3.
    R Hirota, Direct method of finding exact solutions of nonlinear evoluton equations (Springer, Berlin, 1976)zbMATHGoogle Scholar
  4. 4.
    F Cariello and M Tabor, Physica D 39, 77 (1989)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    W Hereman and M Takaoka, J. Phys. A 23, 4805 (1990)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    E V Krishnan, S Kumar and A Biswas, Nonlinear Dyn. 70, 1213 (2012)CrossRefGoogle Scholar
  7. 7.
    A L Fabian, R Kohl and A Biswas, Commun. Nonlinear Sci. Numer. Simul. 14, 1227 (2009)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    H Triki, S Crutcher, A Yildirim, T Hayat, O M Aldossary and A Biswas, Rom. Rep. Phys. 64, 367 (2012)Google Scholar
  9. 9.
    H Kumar and F Chand, J. Nonlinear Opt. Phys. Mater. 22, 1350001 (2013)ADSCrossRefGoogle Scholar
  10. 10.
    H Kumar and F Chand, Opt. Laser Technol. 54, 265 (2013)ADSCrossRefGoogle Scholar
  11. 11.
    H Kumar, A Malik, M S Gautam and F Chand, Acta Phys. Pol. A 131, 275 (2017)CrossRefGoogle Scholar
  12. 12.
    M Wang, Phys. Lett. A 199, 169 (1995)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    H Kumar, A Malik and F Chand, J. Math. Phys. 53, 103704 (2012)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    A M Wazwaz, Appl. Math. Comput. 154, 713 (2004)MathSciNetGoogle Scholar
  15. 15.
    E Fan and H Zhang, Phys. Lett. A 246, 403 (1998)ADSCrossRefGoogle Scholar
  16. 16.
    E Fan, Phys Lett. A 277, 212 (2000)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Q Zhou, Q Zhu, Y Liu, H Yu, P Yao and A Biswas, Laser Phys. 25, 015402 (2015)ADSCrossRefGoogle Scholar
  18. 18.
    M Ekici, A Sonmezoglu, Q Zhou, S P Moshokoa, M Z Ullah, A H Arnous, A Biswas and M Belic, Opt. Quant. Electr. 50, 75 (2018)CrossRefGoogle Scholar
  19. 19.
    A Biswas, M Ekici, A Sonmezoglu, H Triki, Q Zhou, S P Moshokoa and M Belic, Optik 158, 790 (2018)ADSCrossRefGoogle Scholar
  20. 20.
    A R Seadawy and K El-Rashidy, Pramana – J. Phys. 87: 20 (2016)ADSCrossRefGoogle Scholar
  21. 21.
    M A Khater, A R Seadawy and D Lu, Pramana – J. Phys. 90: 59 (2018)ADSCrossRefGoogle Scholar
  22. 22.
    A H Khater, D K Callebaut, W Malfliet and A R Seadawy, Phys. Scr. 64, 533 (2001)ADSCrossRefGoogle Scholar
  23. 23.
    A H Khater, D K Callebaut and A R Seadawy, Phys. Scr. 67, 340 (2003)ADSCrossRefGoogle Scholar
  24. 24.
    A H Khater, D K Callebaut, M A Helal and A R Seadawy, Phys. Scr. 74, 384 (2006)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    A H Khater, D K Callebaut, M A Helal and A R Seadawy, Eur. Phys. J. D 39, 237 (2006)ADSCrossRefGoogle Scholar
  26. 26.
    M A Helal and A R Seadawy, Z. Angew. Math. Phys. 62, 839 (2011)MathSciNetCrossRefGoogle Scholar
  27. 27.
    M A Helal and A R Seadawy, Comput. Math. Appl. 62, 3741 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    A H Khater, M A Helal and A R Seadawy, Il Nuovo Cimento B 115, 1303 (2000)ADSGoogle Scholar
  29. 29.
    A R Seadawy, Appl. Math. Lett. 25, 687 (2012)MathSciNetCrossRefGoogle Scholar
  30. 30.
    A R Seadawy, Appl. Math. Sci. 6, 4081 (2012)MathSciNetGoogle Scholar
  31. 31.
    A M Wazwaz, Math. Comput. Modell. 40, 499 (2004)CrossRefGoogle Scholar
  32. 32.
    E Fan and Y C Hon, Appl. Math. Comput. 141, 351 (2003)MathSciNetGoogle Scholar
  33. 33.
    Z Fu and Q Zhao, Phys. Lett. A 289, 69 (2001)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    J H He and M A Abdou, Chaos Solitons Fractals 34, 1421 (2007)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    H Kumar, A Malik, F Chand and S C Mishra, Indian J. Phys. 86, 819 (2012)ADSCrossRefGoogle Scholar
  36. 36.
    H Kumar and F Chand, AIP Adv. 3, 032128 (2013)ADSCrossRefGoogle Scholar
  37. 37.
    H Kumar and F Chand, J. Theor. Appl. Phys. 8, 114 (2014)ADSCrossRefGoogle Scholar
  38. 38.
    H Kumar and F Chand, Optik 125, 2938 (2014)ADSCrossRefGoogle Scholar
  39. 39.
    A R Seadaway, Pramana – J. Phys. 89: 49 (2017)ADSCrossRefGoogle Scholar
  40. 40.
    H Kumar, A Malik and F Chand, Pramana – J. Phys. 80, 361 (2013)ADSCrossRefGoogle Scholar
  41. 41.
    H Kumar and P Saravanan, Sci. Iran. B 24(5), 2429 (2017)Google Scholar
  42. 42.
    M Wang, X Li and J Zhang, Phys. Lett. A 372, 417 (2008)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    J Zhang, X Wei and Y Lu, Phys. Lett. A 372, 3653 (2008)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    S Zhang, L Tong and W Wang, Phys. Lett. A 372, 2254 (2008)ADSCrossRefGoogle Scholar
  45. 45.
    E M E Zayed and K A Gepreel, J. Math. Phys. 50, 013502 (2008)ADSCrossRefGoogle Scholar
  46. 46.
    D D Ganji and M Abdollahzadeh, J. Math. Phys. 50, 013519 (2009)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    T Ozis and I Aslan, Commun. Theor. Phys. 51, 577 (2009)CrossRefGoogle Scholar
  48. 48.
    E M E Zayed and K A Gepreel, Int. J. Nonlinear Sci. 7, 501 (2009)MathSciNetGoogle Scholar
  49. 49.
    A Malik, F Chand and S C Mishra, Appl. Math. Comput. 216, 2596 (2010)MathSciNetGoogle Scholar
  50. 50.
    M Mirzazadeh, M Eslami, D Milovic and A Biswas, Optik 125, 5480 (2014)ADSCrossRefGoogle Scholar
  51. 51.
    A Malik, F Chand, H Kumar and S C Mishra, Pramana – J. Phys. 78, 513 (2012)ADSCrossRefGoogle Scholar
  52. 52.
    F Chand and A Malik, Int. J. Nonlinear Sci. 14, 416 (2012)MathSciNetGoogle Scholar
  53. 53.
    E M E Zayed, J. Phys. A 42, 195202 (2009)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    A Malik, F Chand, H Kumar and S C Mishra, Comput. Math. Appl. 64, 2850 (2012)MathSciNetCrossRefGoogle Scholar
  55. 55.
    S Guo and Y Zhou, Appl. Math. Comput. 215, 3214 (2010)MathSciNetGoogle Scholar
  56. 56.
    N A Kudryashov, Commun. Nonlinear Sci. Numer. Simul. 17, 2248 (2012)ADSMathSciNetCrossRefGoogle Scholar
  57. 57.
    H Triki, H Leblond and D Mihalache, Nonlinear Dyn. 86, 2115 (2016)CrossRefGoogle Scholar
  58. 58.
    S B Bhardwaj, R M Singh, K Sharma and S C Mishra, Pramana – J. Phys. 86, 1253 (2016)ADSCrossRefGoogle Scholar
  59. 59.
    A Malik, F Chand, H Kumar and S C Mishra, Indian J. Phys. 86, 129 (2012)ADSCrossRefGoogle Scholar
  60. 60.
    M R Meyers and C Krebs, Sci. Am. 230, 38 (1974)CrossRefGoogle Scholar
  61. 61.
    A Mishra and R Kumar, Phys. Lett. A 374, 2921 (2010)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  • Anand Malik
    • 1
  • Hitender Kumar
    • 2
    Email author
  • Rishi Pal Chahal
    • 1
  • Fakir Chand
    • 3
  1. 1.Department of PhysicsChaudhary Bansi Lal UniversityBhiwaniIndia
  2. 2.Department of PhysicsPt. Neki Ram Sharma Government CollegeRohtakIndia
  3. 3.Department of PhysicsKurukshetra UniversityKurukshetraIndia

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