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Modified VIM for the Solutions of Gas Dynamics

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Advances in Mathematical Modeling and Scientific Computing (ICRDM 2022)

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Abstract

A method for solving nonlinear differential equations, the variational iteration method (VIM), has gotten a lot of interest in recent years. VIM has been slightly tinkered within the modified variational iterative method (MVIM). For a specific type of nonlinear differential equation, the implementation of the MVIM method results in unnecessary calculations and additional time spent on repeated calculations for series solutions. The drawbacks of VIM are addressed by introducing a modified version, and its usefulness is demonstrated through a few examples.

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References

  1. Abassy, T.A., El-Tawil, M., Kamel, H.: The solution of KdV and mKdV equations using Adomian Padé approximation. Int. J. Nonlinear Sci. Numer. Simul. 5(4), 327–339 (2004)

    Article  MathSciNet  Google Scholar 

  2. Wazwaz, A.-M.: A comparison between the variational iteration method and Adomian decomposition method. J. Comput. Appl. Math. 207, 129–136 (2007)

    Article  MathSciNet  Google Scholar 

  3. He, J.H.: A new approach to nonlinear partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 2(4), 230–235 (1997)

    Article  Google Scholar 

  4. He, J.H.: Variational iteration method—a kind of non-linear analytical technique: some examples. Int. J. Non-Linear Mech. 34(4), 699–708 (1999)

    Article  Google Scholar 

  5. Fatima, N., Daniel, S.: Solution of wave equations and heat equations using HPM. Appl. Math. Sci. Comput. Trends Math., 367–374 (2019)

    Google Scholar 

  6. He, J.H.: Approximate analytical solution of Blasius’ equation. Commun. Nonlinear Sci. Numer. Simul. 4(1), 75–78 (1999)

    Article  Google Scholar 

  7. Fatima, N.: © Solution of gas dynamic and wave equations with VIM springer nature Singapore Pte Ltd. In: Advances in Fluid Dynamics Lecture Notes in Mechanical Engineering, pp. 81–91 (2021)

    Chapter  Google Scholar 

  8. He, J.H.: Variational iteration method for autonomous ordinary differential systems. Appl. Math. Comput. 114(2–3), 115–123 (2000)

    Article  MathSciNet  Google Scholar 

  9. He, J.H.: Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B. 20(10), 1141–1199 (2006)

    Article  MathSciNet  Google Scholar 

  10. Tamer, A., et al.: Toward a modified variational iteration method. J. Comput. Appl. Math. 207, 137–147 (2007)

    Article  MathSciNet  Google Scholar 

  11. Tari, H.: Modified variational iteration method. Phys. Lett. A. 369, 290–293 (2007)

    Article  Google Scholar 

  12. He, J.H., Wu, X.H.: Construction of solitary solution and compacton-like solution by variational iteration method. Chaos, Solitons Fractals. 29(1), 108–113 (2006)

    Article  MathSciNet  Google Scholar 

  13. Hopf, E.: The partial differential equation ut + uux = μuxx. Commun. Pure Appl. Math. 3, 201–230 (1950)

    Article  Google Scholar 

  14. Soliman, A.A.: A numerical simulation and explicit solutions of KdV-Burgers’ and Lax’s seventh-order KdV equations. Chaos, Solitons Fractals. 29(2), 294–302 (2006)

    Article  MathSciNet  Google Scholar 

  15. Fatima, N.: The study of heat conduction equation by homotopy perturbation method. SN Comput. Sci. 3, 65 (2022)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Prince Sultan University, Riyadh, Saudi Arabia

    Nahid Fatima

Authors
  1. Nahid Fatima
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Correspondence to Nahid Fatima .

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Editors and Affiliations

  1. Department of Electrical Engineering, Canadian University Dubai, Dubai, United Arab Emirates

    Firuz Kamalov

  2. Department of Mathematics, Dr B R Ambedkar National Institute of Technology, Jalandhar, Punjab, India

    R. Sivaraj

  3. Department of Mathematical Sciences, United Arab Emirates University, Al Ain, United Arab Emirates

    Ho-Hon Leung

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Fatima, N. (2024). Modified VIM for the Solutions of Gas Dynamics. In: Kamalov, F., Sivaraj, R., Leung, HH. (eds) Advances in Mathematical Modeling and Scientific Computing. ICRDM 2022. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-41420-6_12

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