THERMAL SCIENCE

International Scientific Journal

Thermal Science - Online First

online first only

Ficitious time integration method for solving the time fractional gas dynamics equation

ABSTRACT
In this work a powerful approach is presented to solve the time-fractional gas dynamics equation. In fact, we use a fictitious time variable y to convert the dependent variable w(x; t) into a new one with one more dimension. Then by taking a initial guess and implementing the group preserving scheme we solve the problem. Finally four examples are solved to illustrate the power of the offered method.
KEYWORDS
PAPER SUBMITTED: 2019-04-21
PAPER REVISED: 2019-08-02
PAPER ACCEPTED: 2019-08-09
PUBLISHED ONLINE: 2019-10-06
DOI REFERENCE: https://doi.org/10.2298/TSCI190421365P
REFERENCES
  1. F. Mainardi Carpinteri, Fractals and fractional calculus in continuum mechanics, in: A. Carpinteri, F. Mainardi (Eds.),Springer Verlag, Wien, New York, 1997, pp. 277-290.
  2. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
  3. M. Caputo, F. Mainardi, Linear models of dissipation in anelastic solids, Rivista Del Nuovo Cimento 1 (1971) 161-198.
  4. K.B. Oldham, J. Spanier, The Fractional Calculus. Integrations and Differentiations of Arbitrary Order, Academic Press, New York, 1974
  5. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach, New York, 1993.
  6. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  7. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, The Netherlands, 2006.
  8. R. Hilfer, Application of Fractional Calculus in Physics, World Scientific, 2000.
  9. G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, 2005.
  10. R.L. Magin, Fractional Calculus in Bio-engineering, Begell House Publisher, Inc., Connecticut, 2006.
  11. R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus, SpringerVerlag, New York, 1997.
  12. M. S. Hashemi, D. Baleanu and M. Parto-Haghighi, A lie group approach to solve the fractional poisson equation, Rom. J. Phys. 60 , 1289-1297,(2015).
  13. Mir sajjad hashemi, Dumitru Baleanu, Mohammad Parto-hghighi and Elham Darvishi, THERMAL SCIENCE, Year , Vol. 19, Suppl. 1, pp. S77-S83, 2015.
  14. A.S.V. RaviKanth, K. Aruna, Differential transform method for solving the linear and nonlinear Klein-Gordon equation, Comput. Phys. Commun. 180 (5) 708-711, (2009) .
  15. A.S.V. RaviKanth, K. Aruna, Differential transform method for solving linear and non-linear systems of partial differential equations, Phys. Lett. A 372 (46), 6896-6898 (17),(2008) .
  16. A.S.V. RaviKanth, K. Aruna, Two-dimensional differential transform method for solving linear and non-linear Schro dinger equations, Chaos, Solitons Fract. 41 (5), 2277-2281, (2009).
  17. S. Kumar, Om P. Singh, Numerical inversion of the abel integral equation using homotopy perturbation method, Z Naturforsch 65a , 677-682,(2010).
  18. S. Kumar, H. Kocak, A. Yildirim, A fractional model of gas dynamics equation by using Laplace transform, Z Naturforsch 67, 389-396, (2012) .
  19. S. Kumar, A numerical study for solution of time fractional nonlinear shallow-water equation in oceans, Z Naturforsch A 68a , 1-7,(2013).
  20. S. Kumar, Numerical computation of time-fractional FokkerPlanck equation arising in solid state physics and circuit theory, Z Naturforsch 68a 1-8,(2013).
  21. H. Jafari, C. Chun, S. Seifi, M. Saeidy, Analytical solution for nonlinear gas dynamics equation by homotopy analysis method, Appl. Appl. Math. 4 (1) 149-154, (2009) .
  22. A.J.M. Jawad, M.D. Petkovic, A. Biswas, Applications of Hes principles to partial differential equations, Appl. Math. Comput. 217 (2011) 7039-7047.
  23. T.G. Elizarova, Quasi gas dynamics equations, Comput. Fluid Solid Mech., Springer Verlag, 2009, ISBN 978-3-642- 00291-5.
  24. D.J. Evans, H. Bulut, A new approach to the gas dynamics equation: an application of the decomposition method, Int. J. Comput. Math. 79 (7) 817-822, (2002).
  25. J.L. Steger, R.F. Warming, Flux vector splitting of the inviscid gas dynamic equations with application to finite-difference methods, J. Comput. Phys. 40 (2) 263-293, (1981).
  26. A. Aziz, D. Anderson, The use of pocket computer in gas dynamics, Comput. Educat. 9 (1) (1985) 41-56.
  27. M. Rasulov, T. Karaguler, Finite difference scheme for solving system equation of gas dynamics in a class of discontinuous function, Appl. Math. Comput. 143 (1) (2003) 145-164.
  28. T.P. Liu, Nonlinear Waves in Mechanics and Gas Dynamics. Defense Technical Information Center, Accession Number: ADA 238-340, 1990.
  29. Tamsir M, Srivastava VK. Revisiting the approximate analytical solution of fractional-order gas dynamics equation. Alexandria Eng ;55(2):867-74, J 2016.
  30. J. Biazar, M. Eslami, Differential transform method for nonlinear fractional gas dynamics equation, Inter. J. Phys. Sci. 6 (5) 12-03,(2011).
  31. S. Das, R. Kumar, Approximate analytical solutions of fractional gas dynamics, Appl. Math. Comput. 217 (24) 9905-9915, (2011).
  32. S. Kumar, M.M. Rashidi, New analytical method for gas dynamics equation arising in shock fronts, Comput. Phys. Commun. 185 (7) 1947-1954,(2014) .
  33. Podlubny I. Fractional Differential Equations. New York: Academic Press; 1999.
  34. Debnath L. Recent applications of fractional calculus to science and engineering. Int J Math Math Sci 2003;54:3413-42.
  35. Liu, C.-S., Solving an Inverse Sturm-Liouville Problem by a Lie-Group Method, Boundary Value Problems, (2008), 749-865,(2008).