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The approximate solutions of fractional gas dynamics equations by means of fractional natural decomposition method

Year 2017, Volume: 5 Issue: 4, 271 - 279, 01.10.2017

Abstract

 In this work, a new practical method, which is named by the Fractional Natural Decomposition Method (FNDM), is proposed to obtain the approximate analytical solutions of fractional gas dynamics equations. The FNDM is a mixture of the Natural Transform Method and the Adomian Decomposition Method. In this method, the fractional derivatives are considered as Caputo sense and the nonlinear terms are determined by virtue of Adomian polynomials. Some test examples are given to demonstrate the efficiency and accuracy of the FNDM.

References

  • K.B. Oldham and J. Spanier, Fractional Calculus: Theory and Applications, Differentiation and Integration to Arbitrary Order, New York: Academic Press, 1974.
  • I. Podlubny, Fractional Differential Equations, an Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications, Academic Press, 1999.
  • A. A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Application of Fractional Differential Equations. North Holland Mathematics Studies, 204, 2006.
  • F. Biagini F., Y. Hu , B. Oksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer-Verlag, London, 2008.
  • F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, An Introduction to Mathematical Models, Imperial College Press, London, 2010.
  • J. L. Steger and R.F. Warming; Flux vector splitting of the inviscid gas dynamic equations with application to finite-difference methods, Journal of Computational Physics., 40, 263293, 1981. doi:10.1016/0021-9991(81)90210-2
  • D. J. Evans and H. Bulut, A new approach to the gas dynamics equation: an application of the decomposition method, International Journal of Computer Mathematics, 79(7),817–822, 2002.
  • H. Jafari, C. Chun, S. Seifi, and M. Saeidy, Analytical solution for nonlinear gas dynamic equation by homotopy analysis method, Applications and Applied Mathematics, 4(1), 149–154, 2009.
  • S. Das, R. Kumar, Approximate analytical solutions of fractional gas Dynamics, Applied Mathematics and Computation, 217(24), 9905–9915, 2011.
  • S. Kumar, H. Kocak and A. Yildirim, A Fractional Model of Gas Dynamics Equations and its Analytical Approximate Solution Using Laplace Transform. Zeitschrift fur Naturforschung A, 67(6-7), 389–396,2012.
  • J. Singh, D. Kumar, A. Kilicman, Homotopy perturbation method for fractional gas dynamics equation using Sumudu transform, Abstract and Applied Analysis, vol. 2013, Article ID 934060, 8 pages, 2013. doi:10.1155/2013/934060
  • H. Aminikhah and A. Jamalian, Numerical approximation for nonlinear gas dynamic equation, International Journal of Partial Differential Equations, vol. 2013, Article ID 846749, 7 pages, 2013.
  • S. Kumar and M. Rashidi, New analytical method for gas dynamics equation arising in shock fronts, Computer Physics Communications, 185, 1947–1954,2014.
  • A. Esen and O. Tasbozan, An approach to time fractional gas dynamics equation: quadratic B-spline Galerkin method, Applied Mathematicsand Computation, 261, 330–336,2015.
  • O.S. Iyiola, On the solutions of nonlinear time fractional gas dynamics equations an analytical approach, International Journal of Pure and Applied Mathematics (IJPAM), 98(4), 491–502,2015.
  • A. Esen , B. Karaagac and O. Tasbozan, Finite Difference Methods for Fractional Gas Dynamics Equation, Applied Mathematics and Information Sciences Letters, 4(1), 1–4, 2016.
  • M. Tamsir and V.K. Srivastava, Revisiting the approximate analytical solution of fractional-order gas dynamics equation, Alexandria Engineering Journal, 55 (2), 867–874,2016.
  • Z. H. Khan and W. A. Khan, N-transform properties and applications, NUST Journal of Engineering Sciences, 1, 127–133,2008.
  • R. Silambarasn and F.B.M. Belgacem, Applications of the natural transform to Maxwell’s quations , Progress In Electromagnetics Research Symposium Proceedings, Suzhou, China, Sept. 12–16, 2011.
  • S.K.Q. Al-Omari, On the application of natural transforms, International Journal of Pure and Applied Mathematics 85(4), 729–744, 2013.
  • M.S. Rawashdeh and S. Maitama, Solving nonlinear ordinary differential equations using the NDM, Journal of Applied Analysis and Computation, 5(1), 77–88, 2015.
  • M. Omran, A. Kilicman and X. Zhang, Natural transform of fractional order and some properties, Cogent Mathematics, 3(1), 2016.
  • S. Maitama, A new analytical approach to linear and nonlinear partial differential equations Nonlinear Studies, 23(4), 675–684, 2016.
  • M.S. Rawashdeh and H. Al-Jammal, New approximate solutions to fractional nonlinear systems of partial differential equations ung the FNDM, Advance Differential Equations, 2016(1), 235, 2016.
  • S. Rida, A. Arafa, A. Abedl-Rady and H. Abdl-Rahaim, Fractional physical differential equations via natural transform, Chinese Journal of Physics, 55(4), 1569–1575, 2017.
  • M. Rawashdeh and S. Maitama, Finding exact solutions of nonlinear PDEs using the natural decomposition method, Mathematical Methods in the Applied Sciences,40(1), 223–236, 2017.
  • M.S. Rawashdeh, The fractional natural decomposition method: theories and applications, Mathematical Methods in the Applied Sciences,40(7), 2362–2376, 2017.
  • F.B.M. Belgacem and R.Silambarasan, Theory of Natural Transform, Mathematics in Engineering, Science and Aerospace (MESA),3(1), 99–124, 2012.
Year 2017, Volume: 5 Issue: 4, 271 - 279, 01.10.2017

Abstract

References

  • K.B. Oldham and J. Spanier, Fractional Calculus: Theory and Applications, Differentiation and Integration to Arbitrary Order, New York: Academic Press, 1974.
  • I. Podlubny, Fractional Differential Equations, an Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications, Academic Press, 1999.
  • A. A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Application of Fractional Differential Equations. North Holland Mathematics Studies, 204, 2006.
  • F. Biagini F., Y. Hu , B. Oksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer-Verlag, London, 2008.
  • F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, An Introduction to Mathematical Models, Imperial College Press, London, 2010.
  • J. L. Steger and R.F. Warming; Flux vector splitting of the inviscid gas dynamic equations with application to finite-difference methods, Journal of Computational Physics., 40, 263293, 1981. doi:10.1016/0021-9991(81)90210-2
  • D. J. Evans and H. Bulut, A new approach to the gas dynamics equation: an application of the decomposition method, International Journal of Computer Mathematics, 79(7),817–822, 2002.
  • H. Jafari, C. Chun, S. Seifi, and M. Saeidy, Analytical solution for nonlinear gas dynamic equation by homotopy analysis method, Applications and Applied Mathematics, 4(1), 149–154, 2009.
  • S. Das, R. Kumar, Approximate analytical solutions of fractional gas Dynamics, Applied Mathematics and Computation, 217(24), 9905–9915, 2011.
  • S. Kumar, H. Kocak and A. Yildirim, A Fractional Model of Gas Dynamics Equations and its Analytical Approximate Solution Using Laplace Transform. Zeitschrift fur Naturforschung A, 67(6-7), 389–396,2012.
  • J. Singh, D. Kumar, A. Kilicman, Homotopy perturbation method for fractional gas dynamics equation using Sumudu transform, Abstract and Applied Analysis, vol. 2013, Article ID 934060, 8 pages, 2013. doi:10.1155/2013/934060
  • H. Aminikhah and A. Jamalian, Numerical approximation for nonlinear gas dynamic equation, International Journal of Partial Differential Equations, vol. 2013, Article ID 846749, 7 pages, 2013.
  • S. Kumar and M. Rashidi, New analytical method for gas dynamics equation arising in shock fronts, Computer Physics Communications, 185, 1947–1954,2014.
  • A. Esen and O. Tasbozan, An approach to time fractional gas dynamics equation: quadratic B-spline Galerkin method, Applied Mathematicsand Computation, 261, 330–336,2015.
  • O.S. Iyiola, On the solutions of nonlinear time fractional gas dynamics equations an analytical approach, International Journal of Pure and Applied Mathematics (IJPAM), 98(4), 491–502,2015.
  • A. Esen , B. Karaagac and O. Tasbozan, Finite Difference Methods for Fractional Gas Dynamics Equation, Applied Mathematics and Information Sciences Letters, 4(1), 1–4, 2016.
  • M. Tamsir and V.K. Srivastava, Revisiting the approximate analytical solution of fractional-order gas dynamics equation, Alexandria Engineering Journal, 55 (2), 867–874,2016.
  • Z. H. Khan and W. A. Khan, N-transform properties and applications, NUST Journal of Engineering Sciences, 1, 127–133,2008.
  • R. Silambarasn and F.B.M. Belgacem, Applications of the natural transform to Maxwell’s quations , Progress In Electromagnetics Research Symposium Proceedings, Suzhou, China, Sept. 12–16, 2011.
  • S.K.Q. Al-Omari, On the application of natural transforms, International Journal of Pure and Applied Mathematics 85(4), 729–744, 2013.
  • M.S. Rawashdeh and S. Maitama, Solving nonlinear ordinary differential equations using the NDM, Journal of Applied Analysis and Computation, 5(1), 77–88, 2015.
  • M. Omran, A. Kilicman and X. Zhang, Natural transform of fractional order and some properties, Cogent Mathematics, 3(1), 2016.
  • S. Maitama, A new analytical approach to linear and nonlinear partial differential equations Nonlinear Studies, 23(4), 675–684, 2016.
  • M.S. Rawashdeh and H. Al-Jammal, New approximate solutions to fractional nonlinear systems of partial differential equations ung the FNDM, Advance Differential Equations, 2016(1), 235, 2016.
  • S. Rida, A. Arafa, A. Abedl-Rady and H. Abdl-Rahaim, Fractional physical differential equations via natural transform, Chinese Journal of Physics, 55(4), 1569–1575, 2017.
  • M. Rawashdeh and S. Maitama, Finding exact solutions of nonlinear PDEs using the natural decomposition method, Mathematical Methods in the Applied Sciences,40(1), 223–236, 2017.
  • M.S. Rawashdeh, The fractional natural decomposition method: theories and applications, Mathematical Methods in the Applied Sciences,40(7), 2362–2376, 2017.
  • F.B.M. Belgacem and R.Silambarasan, Theory of Natural Transform, Mathematics in Engineering, Science and Aerospace (MESA),3(1), 99–124, 2012.
There are 28 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Birol Ibis This is me

Publication Date October 1, 2017
Published in Issue Year 2017 Volume: 5 Issue: 4

Cite

APA Ibis, B. (2017). The approximate solutions of fractional gas dynamics equations by means of fractional natural decomposition method. New Trends in Mathematical Sciences, 5(4), 271-279.
AMA Ibis B. The approximate solutions of fractional gas dynamics equations by means of fractional natural decomposition method. New Trends in Mathematical Sciences. October 2017;5(4):271-279.
Chicago Ibis, Birol. “The Approximate Solutions of Fractional Gas Dynamics Equations by Means of Fractional Natural Decomposition Method”. New Trends in Mathematical Sciences 5, no. 4 (October 2017): 271-79.
EndNote Ibis B (October 1, 2017) The approximate solutions of fractional gas dynamics equations by means of fractional natural decomposition method. New Trends in Mathematical Sciences 5 4 271–279.
IEEE B. Ibis, “The approximate solutions of fractional gas dynamics equations by means of fractional natural decomposition method”, New Trends in Mathematical Sciences, vol. 5, no. 4, pp. 271–279, 2017.
ISNAD Ibis, Birol. “The Approximate Solutions of Fractional Gas Dynamics Equations by Means of Fractional Natural Decomposition Method”. New Trends in Mathematical Sciences 5/4 (October 2017), 271-279.
JAMA Ibis B. The approximate solutions of fractional gas dynamics equations by means of fractional natural decomposition method. New Trends in Mathematical Sciences. 2017;5:271–279.
MLA Ibis, Birol. “The Approximate Solutions of Fractional Gas Dynamics Equations by Means of Fractional Natural Decomposition Method”. New Trends in Mathematical Sciences, vol. 5, no. 4, 2017, pp. 271-9.
Vancouver Ibis B. The approximate solutions of fractional gas dynamics equations by means of fractional natural decomposition method. New Trends in Mathematical Sciences. 2017;5(4):271-9.