Skip to main content
SpringerLink
Log in

On the solitary wave solutions and physical characterization of gas diffusion in a homogeneous medium via some efficient techniques

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

This paper aims to determine some novel solitary wave solutions of the Chaffee–Infante equation, which have not yet been presented for this equation. This equation arises in various branches of science and technology, such as plasma physics, coastal engineering, fluid dynamics, signal processing through optical fibers, ion-acoustic waves in plasma, the sound waves, and the electromagnetic waves field. The \((2+1)\)-dimensional Chaffee–Infante equation describes the dynamical behavior of gas diffusion in a homogeneous medium. Soliton solutions are obtained for this equation using several computational schemes. Many physical significances are explained by sketching some two-dimensional and three-dimensional diagrams for the acquired solutions in three different types. These figures give us a better understanding of the behavior of these solutions. Moreover, the stability property is investigated based on the Hamiltonian system’s characterizations. The methods provide efficient way for the solving other equations that occur in other branches of science.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. T.T. Guy, J.R. Bogning, Modeling nonlinear partial differential equations and construction of solitary wave solutions in an inductive electrical line. J. Adv. Math. Comput. Sci. (2019). https://doi.org/10.9734/jamcs/2019/v33i230174

    Article  Google Scholar 

  2. K. Hosseini, P. Mayeli, R. Ansari, Modified Kudryashov method for solving the conformable time-fractional Klein–Gordon equations with quadratic and cubic nonlinearities. Optik 130, 737–742 (2017)

    Article  ADS  Google Scholar 

  3. X. Zhang, T. Sagiya, Shear strain concentration mechanism in the lower crust below an intraplate strike-slip fault based on rheological laws of rocks. Earth Planets Space 69(1), 82 (2017)

    Article  ADS  Google Scholar 

  4. M.A. Abdou, An analytical method for space-time fractional nonlinear differential equations arising in plasma physics. J. Ocean Eng. Sci. 2(4), 288–292 (2017)

    Article  Google Scholar 

  5. M. Delkhosh, K. Parand, A hybrid numerical method to solve nonlinear parabolic partial differential equations of time-arbitrary order. Comput. Appl. Math. 38(2), 76 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. M.M.A. Khater, A.R. Seadawy, D. Lu, Dispersive optical soliton solutions for higher order nonlinear Sasa–Satsuma equation in mono mode fibers via new auxiliary equation method. Superlattices Microstruct. 113, 346–358 (2018)

    Article  ADS  Google Scholar 

  7. F. Alabau-Boussouira, F. Ancona, A. Porretta, C. Sinestrari, Trends in Control Theory and Partial Differential Equations, vol. 32 (Springer, Switzerland, 2019)

    Book  MATH  Google Scholar 

  8. H. Lin, Electronic structure from equivalent differential equations of Hartree–Fock equations. Chin. Phys. B 28(8), 087101 (2019)

    Article  ADS  Google Scholar 

  9. F. Grassi, A. Loukas, N. Perraudin, B. Ricaud, A time-vertex signal processing framework: scalable processing and meaningful representations for time-series on graphs. IEEE Trans. Signal Process. 66(3), 817–829 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. A. Pels, J. Gyselinck, R.V. Sabariego, S. Schöps, Solving nonlinear circuits with pulsed excitation by multirate partial differential equations. IEEE Trans. Magn. 54(3), 1–4 (2017)

    Article  Google Scholar 

  11. A.C. Cevikel, New exact solutions of the space-time fractional KdV-Burgers and nonlinear fractional foam Drainage equation. Thermal Sci. 22(Suppl. 1), 15–24 (2018)

    Article  Google Scholar 

  12. M. Raissi, Deep hidden physics models: deep learning of nonlinear partial differential equations. J. Mach. Learn. Res. 19(1), 932–955 (2018)

    MathSciNet  MATH  Google Scholar 

  13. C.L. Fefferman, J.C. Robinson, J.L. Rodrigo, J.L. Rodrigo Diez, Partial Differential Equations in Fluid Mechanics, vol. 452 (Cambridge University Press, Cambridge, 2018)

    Book  Google Scholar 

  14. A. Lopatiev, O. Ivashchenko, O. Khudolii, Y. Pjanylo, S. Chernenko, T. Yermakova, Systemic approach and mathematical modeling in physical education and sports. J. Phys. Educ. Sport 17(1), 146–155 (2017)

    Google Scholar 

  15. G.-Q. Sun, Mathematical modeling of population dynamics with Allee effect. Nonlinear Dyn. 85(1), 1–12 (2016)

    Article  MathSciNet  Google Scholar 

  16. P. Zhang, X. Xiao, Z.W. Ma, A review of the composite phase change materials: fabrication, characterization, mathematical modeling and application to performance enhancement. Appl. Energy 165, 472–510 (2016)

    Article  Google Scholar 

  17. B. Ghanbari. On novel non-differentiable exact solutions to local fractional Gardner’s equation using an effective technique. Math. Methods Appl. Sci. 44(6), 4673–4685 (2021)

  18. B. Ghanbari, K.S. Nisar, M. Aldhaifallah, Abundant solitary wave solutions to an extended nonlinear Schrödinger’s equation with conformable derivative using an efficient integration method. Adv. Differ. Equ. 2020(1), 1–25 (2020)

  19. B. Ghanbari, A. Yusuf, D. Baleanu et al., The new exact solitary wave solutions and stability analysis for the \((2+ 1) \)-dimensional Zakharov–Kuznetsov equation. Adv. Differ. Equ. 2019(1), 1–15 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Inc, M. Miah, S.A. Akher Chowdhury, H. Rezazadeh, M.A. Akinlar, Y.-M. Chu, New exact solutions for the Kaup–Kupershmidt equation. AIMS Math. 5(6), 6726–6738 (2020)

    Article  MathSciNet  Google Scholar 

  21. H.M. Srivastava, H. Günerhan, B. Ghanbari, Exact traveling wave solutions for resonance nonlinear Schrödinger equation with intermodal dispersions and the Kerr law nonlinearity. Math. Methods Appl. Sci. 42(18), 7210–7221 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. X. Fu-Ding, L. Xiao-Dan, S. Xiao-Peng, T. Di, Application of computer algebra in solving Chaffee-Infante equation. Commun. Theor. Phys. 49(4), 825 (2008)

    Article  ADS  MATH  Google Scholar 

  23. Y. Mao, Exact solutions to \((2+1)\)-dimensional Chaffee–Infante equation. Pramana 91(1), 9 (2018)

    Article  ADS  Google Scholar 

  24. M.A. Akbar, N.H. Mohd Ali, J. Hussain, Optical soliton solutions to the \((2+1)\)-dimensional Chaffee–Infante equation and the dimensionless form of the zakharov equation. Adv. Differ. Equ. 2019(1), 446 (2019)

    Article  MathSciNet  Google Scholar 

  25. R. Sakthivel, C. Chun, New soliton solutions of Chaffee–Infante equations using the Exp-function method. Z. Naturforschung A 65(3), 197–202 (2010)

    Article  ADS  Google Scholar 

  26. H. Allcott, M. Gentzkow, Yu. Chuan, Trends in the diffusion of misinformation on social media. Res. Politics 6(2), 2053168019848554 (2019)

    Article  Google Scholar 

  27. K. Gawrisch, H.C. Gaede, O. Soubias, W.E. Teague, Phospholipid structural features influence lateral diffusion. Biophys. J. 118(3), 166a (2020)

    Article  ADS  Google Scholar 

  28. A. Tabernero, L. Baldino, S. Cardea, E.M. del Valle, E. Reverchon, A phenomenological approach to study mechanical properties of polymeric porous structures processed using supercritical CO2. Polymers 11(3), 485 (2019)

    Article  Google Scholar 

  29. C.K.W. Tam, A phenomenological approach to jet noise: the two-source model. Philos. Trans. R. Soc. A 377(2159), 20190078 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  30. M. Balonis, G. Sant, O.B. Isgor, Mitigating steel corrosion in reinforced concrete using functional coatings, corrosion inhibitors, and atomistic simulations. Cem. Concr. Compos. 101, 15–23 (2019)

    Article  Google Scholar 

  31. Y. Ji, P.M. Kowalski, P. Kegler, N.M. Huittinen, N. Marks, V. Vinograd, Y. Arinicheva, S. Neumeier, D. Bosbach, Rare-Earth orthophosphates from atomistic simulations. Front. Chem. 7, 197 (2019)

    Article  ADS  Google Scholar 

  32. N. Chafee, E.F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type. Appl. Anal. 4(1), 17–37 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  33. M. Tahir, S. Kumar, H. Rehman, M. Ramzan, A. Hasan, M.S. Osman, Exact traveling wave solutions of Chaffee–Infante equation in (2+1)-dimensions and dimensionless Zakharov equation. Math. Methods Appl. Sci. 44(2), 1500–1513 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  34. L. Qiang, Z. Yun, W. Yuanzheng, Qualitative analysis and travelling wave solutions for the Chaffee–Infante equation. Rep. Math. Phys. 71(2), 177–193 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. G. Zhang, Z. Li, Y. Duan, Exact solitary wave solutions of nonlinear wave equations. Sci. China Ser. A Math. 44(3), 396–401 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. C.-Q. Dai, X. Yun-Jie, Exact solutions for a wick-type stochastic reaction duffing equation. Appl. Math. Modell. 39(23–24), 7420–7426 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. A.T. Ali, M.M.A. Khater, R.A.M. Attia, A.-H. Abdel-Aty, D. Lu, Abundant numerical and analytical solutions of the generalized formula of Hirota–Satsuma coupled KdV system. Chaos Solitons Fractals 131, 109473 (2020)

    Article  MathSciNet  Google Scholar 

  38. M. Eslami, H. Rezazadeh, The first integral method for Wu–Zhang system with conformable time-fractional derivative. Calcolo 53(3), 475–485 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. B. Ghanbari, On the nondifferentiable exact solutions to Schamel’s equation with local fractional derivative on cantor sets. Numer. Methods Partial Differ. Equ. (2020). https://doi.org/10.1002/num.22740

  40. B. Ghanbari, A. Atangana, A new application of fractional Atangana–Baleanu derivatives: designing ABC-fractional masks in image processing. Phys. A Stat. Mech. Appl. 542, 123516 (2020)

    Article  MathSciNet  Google Scholar 

  41. B. Ghanbari, D. Baleanu, A novel technique to construct exact solutions for nonlinear partial differential equations. Eur. Phys. J. Plus 134(10), 1–21 (2019)

    Article  Google Scholar 

  42. B. Ghanbari, M. Inc, A new generalized exponential rational function method to find exact special solutions for the resonance nonlinear Schrödinger equation. Eur. Phys. J. Plus 133(4), 142 (2018)

    Article  Google Scholar 

  43. M. Inc, H. Rezazadeh, J. Vahidi, M. Eslami, M.A. Akinlar, M.N. Ali, Y.-M. Chu, New solitary wave solutions for the conformable Klein–Gordon equation with quantic nonlinearity. AIMS Math. 5(6), 6972–6984 (2020)

    Article  MathSciNet  Google Scholar 

  44. R.M. Jena, S. Chakraverty, H. Rezazadeh, D. Domiri Ganji, On the solution of time-fractional dynamical model of brusselator reaction–diffusion system arising in chemical reactions. Math. Methods Appl. Sci. 43(7), 3903–3913 (2020)

    MathSciNet  MATH  Google Scholar 

  45. M.M.A. Khater, J.F. Alzaidi, R.A.M. Attia, D. Lu et al., Analytical and numerical solutions for the current and voltage model on an electrical transmission line with time and distance. Phys. Scripta 95(5), 055206 (2020)

    Article  ADS  Google Scholar 

  46. M.M.A. Khater, C. Park, D. Lu, R.A.M. Attia, Analytical, semi-analytical, and numerical solutions for the Cahn–Allen equation. Adv. Differ. Equ. 2020(1), 1–12 (2020)

    Article  MathSciNet  Google Scholar 

  47. S. Akhtet, H. Roshid, M.N. Alam, N. Rahman, K. Khan, M.A. Akbar, Application of exp (– \(\varphi \)( \(\eta \)))-expansion method to find the exact solutions of nonlinear evolution equations. IOSR-JM 9(6), 106–13 (2014)

    Google Scholar 

  48. D. Lu, A.R. Seadawy, J. Wang, M. Arshad, U. Farooq, Soliton solutions of the generalised third-order nonlinear Schrödinger equation by two mathematical methods and their stability. Pramana 93(3), 1–9 (2019)

    Article  Google Scholar 

  49. M.A. Abdou, On the fractional order space-time nonlinear equations arising in plasma physics. Indian J. Phys. 93(4), 537–541 (2019)

    Article  ADS  Google Scholar 

  50. M. Eslami, A. Neirameh, New exact solutions for higher order nonlinear Schrödinger equation in optical fibers. Opt. Quantum Electron. 50(1), 1–8 (2018)

    Article  Google Scholar 

  51. H.-O. Roshid, M.R. Kabir, R.C. Bhowmik, B.K. Datta, Investigation of solitary wave solutions for Vakhnenko–Parkes equation via exp-function and exp (- \(\phi \) ())-expansion method. SpringerPlus 3(1), 1–10 (2014)

    Article  Google Scholar 

  52. M.M.A. Khater, E.H.M. Zahran, Soliton soltuions of nonlinear evolutions equation by using the extended exp (- \(\phi \) ()) expansion method. Int. J. Comput. Appl. 975, 8887 (2016)

    Google Scholar 

  53. M. Wang, X. Li, J. Zhang, The (g’/g)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372(4), 417–423 (2008)

  54. E.M.E. Zayed, The (g’/g)-expansion method and its applications to some nonlinear evolution equations in the mathematical physics. J. Appl. Math. Comput. 30(1), 89–103 (2009)

  55. S. Zhang, L. Dong, J.-M. Ba, Y.-N. Sun, The (g’/g)-expansion method for nonlinear differential-difference equations. Phys. Lett. A 373(10), 905–910 (2009)

  56. M. Shakeel, S.T. Mohyud-Din, A novel (g’/g)-expansion method and its application to the (3+ 1)-dimensional burger’s equations. Int. J. Appl. Comput. Math. 2(1), 13–24 (2016)

  57. N.A. Kudryashov, A note on the g’/g-expansion method. Appl. Math. Comput. 217(4), 1755–1758 (2010)

  58. I. Aslan, A note on the (g’/g)-expansion method again. Appl. Math. Comput. 217(2), 937–938 (2010)

  59. N.A. Kudryashov, N.B. Loguinova, Extended simplest equation method for nonlinear differential equations. Appl. Math. Comput. 205(1), 396–402 (2008)

    MathSciNet  MATH  Google Scholar 

  60. S. Bilige, T. Chaolu, An extended simplest equation method and its application to several forms of the fifth-order kdv equation. Appl. Math. Comput. 216(11), 3146–3153 (2010)

    MathSciNet  MATH  Google Scholar 

  61. C. Zhang, Z. Zhang, Application of the enhanced modified simple equation method for Burger–Fisher and modified Volterra equations. Adv. Differ. Equ. 2017(1), 1–8 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  62. A.K.M.K.S. Hossain, M.A. Akbar, A.-M. Wazwaz, Closed form solutions of complex wave equations via the modified simple equation method. Cogent Phys. 4(1), 1312751 (2017)

    Article  Google Scholar 

  63. M.A. Akbar, N.H. Mohd Ali, S.T. Mohyud-Din, Assessment of the further improved (g’/g)-expansion method and the extended tanh-method in probing exact solutions of nonlinear pdes. SpringerPlus 2(1), 1–9 (2013)

  64. S.A. El-Wakil, M.A. Abdou, E.K. El-Shewy, A. Hendi, H.G. Abdelwahed, (g’/g)-expansion method equivalent to the extended tanh-function method. Phys. Scripta 81(3), 035011 (2010)

  65. Ş. Akçaği, T. Aydemir, Comparison between the (g’/g)-expansion method and the modified extended tanh method. Open Phys. 14(1), 88–94 (2016)

  66. H.A. Zedan, New approach for tanh and extended-tanh methods with applications on Hirota–Satsuma equations. Comput. Appl. Math. 28(1), 1–14 (2009)

    MathSciNet  MATH  Google Scholar 

  67. S. Bibi, S.T. Mohyud-Din, U. Khan, N. Ahmed, Khater method for nonlinear Sharma Tasso-Olever (sto) equation of fractional order. Results Phys. 7, 4440–4450 (2017)

    Article  ADS  Google Scholar 

  68. H. Rezazadeh, A. Korkmaz, M.M.A. Khater, M. Eslami, D. Lu, R.A.M. Attia, New exact traveling wave solutions of biological population model via the extended rational sinh-cosh method and the modified Khater method. Mod. Phys. Lett. B 33(28), 1950338 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  69. C. Yue, D. Lu, M.M.A. Khater, A.-H. Abdel-Aty, W. Alharbi, R.A.M. Attia. On explicit wave solutions of the fractional nonlinear DSW system via the modified Khater method. Fractals 28(8), 2040034 (2020)

  70. M.M.A. Khater, M.S. Mohamed, C. Park, R.A.M. Attia, Effective computational schemes for a mathematical model of relativistic electrons arising in the laser thermonuclear fusion. Results Phys. 19, 103701 (2020)

    Article  Google Scholar 

  71. J. Li, R.A.M. Attia, M.M.A. Khater, D. Lu, The new structure of analytical and semi-analytical solutions of the longitudinal plasma wave equation in a magneto-electro-elastic circular rod. Mod. Phys. Lett. B 34(12), 2050123 (2020)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to extend our special thanks to Prof. Piergiulio Tempesta, the handling editor of the paper, and three anonymous referees for their valuable suggestions and advisory comments that increased the quality of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Jiangsu University, Zhenjiang, China

    Mostafa M. A. Khater

  2. Department of Mathematics, Obour High Institute for Engineering and Technology, 11828, Cairo, Egypt

    Mostafa M. A. Khater

  3. Department of Basic Science, Kermanshah University of Technology, Kermanshah, Iran

    Behzad Ghanbari

  4. Department of Mathematics, Faculty of Engineering and Natural Sciences, Bahçeşehir University, 34349, Istanbul, Turkey

    Behzad Ghanbari

Authors
  1. Mostafa M. A. Khater
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Behzad Ghanbari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Behzad Ghanbari.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khater, M.M.A., Ghanbari, B. On the solitary wave solutions and physical characterization of gas diffusion in a homogeneous medium via some efficient techniques. Eur. Phys. J. Plus 136, 447 (2021). https://doi.org/10.1140/epjp/s13360-021-01457-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-021-01457-1

Search

Navigation