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National Academy Science Letters

, Volume 42, Issue 1, pp 51–57 | Cite as

Analytical Solution for Fractional Gas Dynamics Equation

  • S. Raja Balachandar
  • K. KrishnaveniEmail author
  • K. Kannan
  • S. G. Venkatesh
Short Communication
  • 102 Downloads

Abstract

A new hybrid method based on fractional order shifted Legendre polynomials is constructed in the present study to obtain the analytical solution of a fractional gas dynamics equation. The theoretical analysis such as convergence analysis and error bound for the proposed technique have been demonstrated. The illustrated examples are shown to test the ability and accuracy of the proposed method.

Keywords

Fractional shifted legendre polynomials Caputo derivative Time-fractional PDE Fractional gas dynamics equation 

Mathematics Subject Classification

49K20 26A33 34A08 35R11 

Notes

Acknowledgements

The authors also wish to thank Department of Science and Technology, Government of India for the financial sanction towards this work under FIST Programme SR\(\backslash \) FST\(\backslash \) MSI - 107 \(\backslash \) 2015.

References

  1. 1.
    Jagdev Singh, Devendra Kumar, Kślśman A (2013) Homotopy perturbation method for fractional gas dynamics equation using sumudu transform. Abstr Appl Anal. Article ID 934060Google Scholar
  2. 2.
    Devendra Kumar, Jagdev Singh, Dumitru Baleanu (2016) Numerical computation of a fractional model of differential-difference equation. J Comput Nonlinear Dyn 11(6):061004Google Scholar
  3. 3.
    Kumar Devendra, Singh Jagdev, Kumar Sunil, Sushila BP Singh (2015) Numerical computation of nonlinear shock wave equation of fractional order. Ain Shams Eng J 6(2):605–611CrossRefGoogle Scholar
  4. 4.
    Kumar Devendra, Singh Jagdev, Baleanu Dumitru (2016) A hybrid computational approach for Klein–Gordon equations on Cantor sets. Nonlinear Dyn.  https://doi.org/10.1007/s11071-016-3057-x
  5. 5.
    Momani S (2005) Analytic and approximate solution of the space- and time-fractional telegraph equations. Appl Math Comput 170(2):1126–1134MathSciNetzbMATHGoogle Scholar
  6. 6.
    Al-Khaled K, Momani S (2005) An approximate solution for a fractional diffusion-wave equation using the decomposition method. Appl Math Comput 165:473–483MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hanyga A (2002) Multidimensional solutions of time-fractional diffusion-wave equations. Proc R Soc Lond A 485:933–957ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Debnath L, Bhatta D (2004) Solutions to few linear fractional inhomogeneous partial differential equations in fluid mechanics. Fract Calc Appl Anal 7:21–36MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fix GJ, Roop JP (2004) Least squares finite element solution of a fractional order two-point boundary value problem. Comput Math Appl 48(7–8):1017–1033MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mainardi F (1997) Fractional calculus: some basic problems in continuum and statistical mechanics. Fractals and fractional calculus in continuum mechanics. Springer, New York, pp 291–348Google Scholar
  11. 11.
    Momani S, Shawagfeh NT (2006) Decomposition method for solving fractional Riccati differential equations. Appl Math Comput 182:1083–1092MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ray SS, Chaudhuri KS, Bera RK (2006) Analytical approximate solution of nonlinear dynamic system containing fractional derivative by modiĄed decomposition method. Appl Math Comput 182:544–552MathSciNetzbMATHGoogle Scholar
  13. 13.
    Wang Q (2006) Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method. Appl Math Comput 182:1048–1055MathSciNetzbMATHGoogle Scholar
  14. 14.
    Hashim I, Abdulaziz O, Momani S (2009) Homotopy analysis method for fractional IVPs. Commun Nonlinear Sci Numer Simul 14:674–684ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jafari H, Golbabai A, SeiĄ S, Sayevand K (2010) Homotopy analysis method for solving multi-term linear and nonlinear diffusion-wave equations of fractional order. Comput Math Appl 59:1337–1344MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Yildirim A (2010) He’s homotopy perturbation method for solving the space and time fractional telegraph equations. Int J Comput Math 87:2998–3006MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yildirim A (2010) He’s homotopy perturbation method for solving the space- and time-fractional telegraph equations. Int J Comput Math 87(13):2998–3006MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Momani S, Odibat Z (2007) Numerical comparison of methods for solving linear differential equations of fractional order. Chaos Solitons Fractals 31(5):1248–1255ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Chen W, Ye L, Su H (2010) Fractional diffusion equations by the Kansa method. Comput Math Appl 59:1614–1620MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    He Y, Wei L, Zhang X (2013) Analysis of a local discontinuous Galerkin method for time-fractional advection-diffusion equations. Int J Numer Method Heat Fluid Flow 23(4):634–648MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kazem S (2013) An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations. Appl Math Model 37:1126–1136MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Krishnasamy VS, Razzaghi M (2016) The numerical solution of the BagleyTorvik equation with fractional Taylor method. ASME J Comput Nonlinear Dyn 11(5):051010CrossRefGoogle Scholar
  23. 23.
    Ramswroop Singh J, Kumar D (2014) Numerical study for time-fractional Schrodinger equations arising in quantum mechanics. Nonlinear Eng 3(3):169–177CrossRefGoogle Scholar
  24. 24.
    Singh J, Kumar D, Kilicman A (2014) Numerical solutions of nonlinear fractional partial differential equations arising in spatial diffusion of biological populations. Abst Appl Anal 2014:535793MathSciNetzbMATHGoogle Scholar
  25. 25.
    Rida SZ, El-Sayed AMA, Arafa AAM (2010) On the solutions of time-fractional reaction–diffusion equations. Commun Nonlinear Sci Numer Simul 15(12):3847–3854ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Singh J, Kumar D, Sushila (2011) Homotopy perturbation Sumudu transform method for nonlinear equations. Adv Appl Math Mech 4:165–175zbMATHGoogle Scholar
  27. 27.
    Singh J, Kumar D, Kilicman A (2014) Numerical solutions of nonlinear fractional partial differential equations arising in spatial diffusion of biological populations. Abstr Appl Anal 535–793Google Scholar
  28. 28.
    Singh J, Kumar D, Kumar S (2014) A new fractional model of nonlinear shock wave equation arising in flow of gases. Nonlinear Eng 3(1):43–50CrossRefGoogle Scholar
  29. 29.
    Singh J, Kumar D, Kumar S (2013) New treatment of fractional Fornberg–Whitham equation via Laplace transform. Ain Sham Eng J 4:557–62CrossRefGoogle Scholar
  30. 30.
    Doha EH, Bhrawy AH, Ezz-Eldien SS (2015) An efficient Legendre spectral tau matrix formulation for solving fractional subdiffusion and reaction subdiffusion equations. J Comput Nonlinear Dyn 10(1–8):021019CrossRefGoogle Scholar
  31. 31.
    Bhrawy AH, Taha TM, Machado JAT (2015) A review of operational matrices and spectral techniques for fractional calculus. Nonlinear Dyn 81:1023–1052MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Machado JAT, Mata ME (2015) A fractional perspective to the bond graph modelling of world economies. Nonlinear Dyn 80:1839–1852MathSciNetCrossRefGoogle Scholar
  33. 33.
    Zhou Y, Ionescu C, Machado JAT (2015) Fractional dynamics and its applications. Nonlinear Dyn 80:1661–1664MathSciNetCrossRefGoogle Scholar
  34. 34.
    Yang AM, Zhang YZ, Cattani C, Xie GN, Rashidi MM, Zhou YZ, Yang XJ (2014) Application of local fractional series expansion method to solve KleinGordon equations on Cantor set. Abstr Appl Anal 372–741Google Scholar
  35. 35.
    Yang XJ (2012) Advanced local fractional calculus and its applications. World Science, New YorkGoogle Scholar
  36. 36.
    Razminia K, Razminia A, Machado JAT (2016) Analytical solution of fractional order diffusivity equation with Wellbore storage and skin effects. ASME J Comput Nonlinear Dyn 11(1):011006CrossRefGoogle Scholar
  37. 37.
    Duan JS, Rach R, Buleanu D, Wazwaz AM (2012) A review of the Adomian decomposition method and its applications to fractional differential equations. Commun Fract Calc 3(2):73–99Google Scholar
  38. 38.
    Das S, Kumar R (2011) Approximate analytical solutions of fractional gas dynamic equations. Appl Math Comput 217(24):9905–9915MathSciNetzbMATHGoogle Scholar
  39. 39.
    Kumar S (2013) A numerical study for solution of time fractional nonlinear shallow-water equation in oceans. Zeitschrift fur Naturforschung A 68a:1–7CrossRefGoogle Scholar
  40. 40.
    Kumar S, Kumar A, Baleanu D (2016) Two analytical methods for time-fractional nonlinear coupled Boussinesq–Burger’s equations arise in propagation of shallow water waves. Non-linear Dyn.  https://doi.org/10.1007/s11071-016-2716-2
  41. 41.
    Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, San DiegozbMATHGoogle Scholar
  42. 42.
    Podlubny I (1999) Fractional differential equations. Academic Press, San DiegozbMATHGoogle Scholar
  43. 43.
    Gupta S, Kumar D, Singh J (2015) Numerical study for systems of fractional differential equations via Laplace transform. J Egypt Math Soc 23(2):256–262MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Kumar D, Singh J, Kumar S (2015) Analytical modeling for fractional multi-dimensional diffusion equations by using Laplace transform. Commun Numer Anal 1:16–29MathSciNetCrossRefGoogle Scholar
  45. 45.
    Kazem S, Abbasbandy S, Kumar Sunil (2013) Fractional-order Legendre functions for solving fractional-order differential equations. Appl Math Model 37:5498–5510MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Biazar J, Mostafa E (2011) Differential transform method for nonlinear fractional gas dynamics equation. Int J Phys Sci 6(5):1203–1206Google Scholar
  47. 47.
    Adomian G (1994) Solving frontier problems of physics: the decomposition method. Kluwer Academic Publishers, BostonCrossRefzbMATHGoogle Scholar
  48. 48.
    Evans DJ, Bulut H (2002) A new approach to the gas dynamics equation: an application of the decomposition method. Int J Comput Math 79(7):817–822MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Caputo M (1967) Linear models of dissipation whose Q is almost frequency independent. Part II J R Astr Soc 13 529:383–393ADSGoogle Scholar
  50. 50.
    Shen J, Tang T (2005) High order numerical methods and algorithms. Chinese Science Press, BeijingGoogle Scholar
  51. 51.
    Liu Nanshan, Lin En-Bing (2009) Legendre wavelet method for numerical solutions of partial differential equation. Numer Methods Partial Differ Equ 26(1):81–94MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The National Academy of Sciences, India 2018

Authors and Affiliations

  • S. Raja Balachandar
    • 1
  • K. Krishnaveni
    • 1
    Email author
  • K. Kannan
    • 1
  • S. G. Venkatesh
    • 1
  1. 1.Department of Mathematics, School of Humanities and SciencesSASTRA Deemed to be UniversityThanjavurIndia

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