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Generalized Lagrangian Jacobi Gauss collocation method for solving unsteady isothermal gas through a micro-nano porous medium

The European Physical Journal Plus volume 133, Article number: 28 (2018) Cite this article

Abstract.

In the present paper, a new method based on the Generalized Lagrangian Jacobi Gauss (GLJG) collocation method is proposed. The nonlinear Kidder equation, which explains unsteady isothermal gas through a micro-nano porous medium, is a second-order two-point boundary value ordinary differential equation on the unbounded interval \( [0, \infty)\). Firstly, using the quasilinearization method, the equation is converted to a sequence of linear ordinary differential equations. Then, by using the GLJG collocation method, the problem is reduced to solving a system of algebraic equations. It must be mentioned that this equation is solved without domain truncation and variable changing. A comparison with some numerical solutions made and the obtained results indicate that the presented solution is highly accurate. The important value of the initial slope, \( y'(0)\), is obtained as \( -1.191790649719421734122828603800159364\) for \(\eta = 0.5\). Comparing to the best result obtained so far, it is accurate up to 36 decimal places.

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References

  1. R.E. Kidder, J. Appl. Mech. 27, 329 (1957)

    MathSciNet  Google Scholar 

  2. S. Kazem, J.A. Rad, K. Parand, M. Shaban, H. Saberi, Int. J. Comput. Math. 16, 2240 (2012)

    Article  Google Scholar 

  3. R.P. Agrawal, D. O’Regan, Stud. Appl. Math. 108, 245 (2002)

    Article  MathSciNet  Google Scholar 

  4. A.M. Wazwaz, Appl. Math. Comput. 118, 123 (2001)

    MathSciNet  Google Scholar 

  5. A. Taghavi, H. Fani, Int. J. Math. Comput. Phys. Electr. Comput. Eng. 3, 991 (2009)

    Google Scholar 

  6. K. Parand, M. Shahini, A. Taghavi, Int. J. Contemp. Math. Sci. 4, 1005 (2009)

    MathSciNet  Google Scholar 

  7. K. Parand, M. Nikarya, J. Comput. Theor. NanoSci. 11, 131 (2014)

    Article  Google Scholar 

  8. M.A. Noor, S.T. Mohyud-Din, Comput. Math. Appl. 58, 2182 (2009)

    Article  MathSciNet  Google Scholar 

  9. Y. Khan, N. Faraz, A. Yildirim, World. Ap. Sci. J. 9, 1818 (2010)

    Google Scholar 

  10. S. Upadhyay, K.N. Rai, Comput. Math. Appl. Res. 3, 251 (2014)

    Google Scholar 

  11. J.V. Baxley, J. Math. Anal. Appl. 147, 122 (1990)

    Article  MathSciNet  Google Scholar 

  12. D.E. Panayotounakos, A.B. Sotiropoulou, N.B. Sotiropoulos, M. Manios, Int. J. Non-Linear Mech. 42, 157 (2007)

    Article  ADS  Google Scholar 

  13. M. Tatari, M. Dehghan, M. Razzaghi, Math. Comput. Model. 45, 639 (2007)

    Article  Google Scholar 

  14. F. Shakeri, M. Dehghan, Math. Comput. Model. 48, 486 (2008)

    Article  Google Scholar 

  15. M.A. Abdou, A.A. Soliman, J. Comput. Appl. Math. 181, 245 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  16. W.X. Ma, T. Huang, Y. Zhang, Phys. Scr. 82, 065003 (2010)

    Article  ADS  Google Scholar 

  17. A. Mohebbi, M. Abbaszadeh, Numer. Algor. 63, 431 (2013)

    Article  Google Scholar 

  18. M. Dehghan, M. Abbaszadeh, A. Mohebbi, Eng. Anal. Bound. Elem. 64, 205 (2016)

    Article  MathSciNet  Google Scholar 

  19. M. Dehghan, M. Abbaszadeh, A. Mohebb, Eng. Anal. Bound. Elem. 50, 412 (2015)

    Article  MathSciNet  Google Scholar 

  20. H. Saeedi, Iran. J. Sci. Tech. Trans. A: Sci. 41, 723 (2017)

    Article  MathSciNet  Google Scholar 

  21. H. Saeedi, G.N. Chuev, J. Appl. Math. Comput. 49, 213 (2015)

    Article  MathSciNet  Google Scholar 

  22. J. Shen, SIAM. J. Numer. Anal. 38, 1113 (2000)

    Article  MathSciNet  Google Scholar 

  23. B.Y. Guo, X.Y. Zhang, Appl. Numer. Math. 57, 455 (2007)

    Article  MathSciNet  Google Scholar 

  24. B.Y. Guo, J. Shen, C.L. Xu, Adv. Comput. Math. 37, 35 (2003)

    Article  Google Scholar 

  25. H. Ma, T. Zhao, Numer. Methods Part D E 23, 968 (2007)

    Article  Google Scholar 

  26. J.P. Boyd, J. Comput. Phys. 69, 112 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  27. B.Y. Guo, Y.G. Yi, J. Sci. Comput. 43, 201 (2010)

    Article  MathSciNet  Google Scholar 

  28. J.P. Boyd, J. Sci. Comput. 3, 109 (1988)

    Article  MathSciNet  Google Scholar 

  29. A. Saadatmandi, F. Mashhadi-Fini, Math. Methods Appl. Sci. 38, 1265 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  30. M. Shamsi, M. Dehghan, Numer. Methods Part D E 23, 196 (2007)

    Article  Google Scholar 

  31. L.N. Trefethen, Spectral methods in MATLAB, doi.org/10.1137/1.9780898719598.bm (2000)

  32. D. Garg, M. Patterson, W.W. Hager, A.V. Rao, D.A. Benson, G.T. Huntington, Automatica. 46, 11 (2010)

    Article  Google Scholar 

  33. F. Fahroo, I.M. Ross, J. Guid. Control Dyn. 24, 2 (2001)

    Google Scholar 

  34. F. Fahroo, I.M. Ross, J. Guid. Control Dyn. 25, 1 (2002)

    Article  Google Scholar 

  35. I.M. Ross, F. Fahroo, Lecture Notes in Control and Information Sciences, Vol. 295 (Springer, Berlin, Heidelberg, 2003)

  36. M. Dehghan, M. Shamsi, Numer. Methods Part D E 22, 6 (2006)

    Google Scholar 

  37. P. Williams, J. Guid. Control Dyn. 27, 2 (2004)

    Google Scholar 

  38. A. Bhrawy, M. Zaky, Math. Methods Appl. Sci. 39, 1765 (2015)

    Article  Google Scholar 

  39. A. Bhrawy, J.F. Alzaidy, M.A. Abdelkawy, A. Biswas, Non-Linear Dyn. 84, 1553 (2016)

    Article  Google Scholar 

  40. A. Bhrawy, E.H. Doha, S.S. Ezz-Eldien, M.A. Abdelkawy, Comput. Model. Eng. Sci. 104, 185 (2015)

    Google Scholar 

  41. A. Bhrawy, E.H. Doha, D. Baleanu, R.M. Hafez, Math. Methods Appl. Sci. 38, 3022 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  42. E.H. Doha, A. Bhrawy, M.A. Abdelkawy, J. Comput. Non-Linear Dyn. 10, 021016 (2015)

    Article  Google Scholar 

  43. J. Shen, T. Tang, L.L. Wang, Spectral Methods: Algorithms, Analysis and Applications (Springer-Verlag, Berlin Heidelberg, 2011)

  44. Z. Sabeh, M. Shamsi, M. Dehghan, Math. Methods Appl. Sci. 39, 3350 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  45. M. Delkhosh, K. Parand, Generalized Pseudospectral Method: Theory and Application, submitted to J. Comput. Sci

  46. K. Parand, P. Mazaheri, M. Delkhosh, A. Ghaderi, SeMA J. 74, 569 (2017)

    Article  MathSciNet  Google Scholar 

  47. K. Parand, M. Delkhosh, J. Comput. Nonlinear Dyn. 13, 011007 (2018)

    Article  Google Scholar 

  48. K. Parand, P. Mazaheri, H. Yousefi, M. Delkhosh, E. Phys. J. Plus 132, 77 (2017)

    Article  Google Scholar 

  49. R.A. Khan, Math. Comput. Model. 43, 727 (2006)

    Article  Google Scholar 

  50. M. El-Gebeily, D. O’Regan, J. Comput. Appl. Math. 192, 270 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  51. A. Vatsala, L. Wang, J. Math. Anal. Appl. 237, 644 (1999)

    Article  MathSciNet  Google Scholar 

  52. M. El-Gebeily, D. O’Regan, Non-Linear Anal: Real. World. Appl. 8, 174 (2007)

    Google Scholar 

  53. V.B. Mandelzweig, J. Math. Phys. 40, 6266 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  54. R. Kalaba, J. Math. Mech. 8, 519 (1959)

    MathSciNet  Google Scholar 

  55. R.E. Bellman, R.E. Kalaba, Quasilinearization and Nonlinear Boundary-Value Problems (Rand Publication, 1965)

  56. K. Parand, M. Delkhosh, J. Comput. Appl. Math. 317, 624 (2017)

    Article  MathSciNet  Google Scholar 

  57. V. Mandelzweig, F. Tabakin, Comput. Phys. Commun. 141, 268 (2001)

    Article  ADS  Google Scholar 

  58. M.A. Noor, S.T. Mohyud-Din, Comput. Math. Appl. 58, 2182 (2009)

    Article  MathSciNet  Google Scholar 

  59. K. Parand, A. Taghavi, M. Shahini, Acta. Phys. Pol. B 40, 1749 (2009)

    ADS  Google Scholar 

  60. A. Taghavi, K. Parand, A. Shams, H.G. Sofloo, J. Comput. Theor. NanoSci. 7, 542 (2010)

    Article  Google Scholar 

  61. S.T. Mohyud-Din, A. Yildirim, M.M. Hosseini, Int. J. Differ. Equ. 2010, 426213 (2010)

    Google Scholar 

  62. A.R. Rezaei, K. Parand, A. Pirkhedri, J. Comput. Theor. NanoSci. 8, 282 (2011)

    Article  Google Scholar 

  63. J.A. Rad, S.M. Ghaderi, K. Parand, J. Comput. Theor. NanoSci. 8, 2033 (2011)

    Article  Google Scholar 

  64. S. Abbasbandy, Acta. Phys. Pol. A 121, 581 (2012)

    Article  Google Scholar 

  65. A.M. Wazwaz, Cent. Eur. J. Eng. 4, 64 (2014)

    Google Scholar 

  66. R. Iacono, J.P. Boyd, Stud. Appl. Math. 135, 63 (2015)

    Article  MathSciNet  Google Scholar 

  67. C.I. Gheorghiu, Rom. J. 11, 69 (2015)

    Google Scholar 

  68. K. Parand, M. Hemami, Iranian J. Sci. Tech. Trans. A: Sci. 41, 677 (2017)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Sciences, Shahid Beheshti University, G.C. Tehran, Iran

    Kourosh Parand, Sobhan Latifi, Mehdi Delkhosh & Mohammad M. Moayeri

  2. Department of Cognitive Modeling, Institute for Cognitive and Brain Sciences, Shahid Beheshti University, G.C. Tehran, Iran

    Kourosh Parand

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  1. Kourosh Parand
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  2. Sobhan Latifi
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  3. Mehdi Delkhosh
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  4. Mohammad M. Moayeri
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Correspondence to Kourosh Parand.

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Parand, K., Latifi, S., Delkhosh, M. et al. Generalized Lagrangian Jacobi Gauss collocation method for solving unsteady isothermal gas through a micro-nano porous medium. Eur. Phys. J. Plus 133, 28 (2018). https://doi.org/10.1140/epjp/i2018-11859-5

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  • DOI: https://doi.org/10.1140/epjp/i2018-11859-5