Analysis of Fractional Gas Dynamic Equations using Atangana-Baleanue Derivative Operator

In this work, iterative Laplace transform method has been successfully applied for nu merical solution of time-fractional homogeneous and non-homogeneous gas dynamic equations. The method gives more realistic series solutions that converge very rapidly in physical problems. The theoretical analysis of solutions is carried out by providing the existence and uniqueness criteria of solution for used technique. It is apparently seen from comparison of absolute error that the proposed method is very efficient and powerful numerical method to find the approximate solution. Also, it is important to point out that even if the problem is nonlinear this method does not require linearization or perturbation. Therefore, ILTM is easy to implement and more convenient than other methods.  

[1] S. Abbasbandy, The application of homotopy analysis method to nonlinear equa tions arising in heat transfer, Physics Letters A, 360(2006), 109-113. [2]  G. Adomian, R. Rach, Modified Adomian polynomials. Math. Comput. Modelling, 24(11)(1996) 39-46. [3]  M. S. Al-luhaibi, New Iterative Method for Fractional gas Dynamics and Coupled Burger’s Equations, The Scientific World Journal, vol. 2015, Article ID 153124, 8 pages, 2015. [4] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non singular kernel: Theory and application to heat transfer model, Thermal Science, 20(2)(2016), 763-769. [5] J. Biazar and M. Eslami, Differential transform method for nonlinear fractional gas dynamic equation, Int. J. Phys. Sci. 6, 1203 (2011). [6]  V. Daftardar-Gejji, H. Jafari, An iterative method for solving non linear functional equations, J. Math. Anal. Appl., 316 (2006),753-763. [7]  S. Das and R. Kumar, Approximate analytical solutions of fractional gas dynamic equations. Applied Mathematics and Computation, 217(24)(2011), 9905-9915. [8]  T. G. Elizarova, Quasi-gas-dynamic equations, in Quasi-Gas Dynamic Equations, pp. 37-62, Springer, Berlin, Heidelberg, 2009. [9] J.H. He, Variational iteration method-a kind of non-linear analytical technique: some examples, Int J Nonlinear Mech, 34(4)(1999), 699-708. [10]  J.H. He, Homotopy perturbation technique, Comp. Meth. Appl. Mech. Eng, 178 (1999), 257-262.