Homotopy Analysis Method for Nonlinear Fractional Gas Dynamics Equation

[1]

**Attia Rani**, Department of Mathematics, University of Wah, Wah Cantt., Pakistan.

[2]

**Munazza Saeed**, Department of Mathematics, University of Wah, Wah Cantt., Pakistan.

[3]

**Qazi Mahmood Ul-Hassan**, Department of Mathematics, University of Wah, Wah Cantt., Pakistan.

[4]

**Muhammad Yaqub Khan**, Department of Mathematics, Riphah International University, Islamabad, Pakistan.

[5]

**Kamran Ayub**, Department of Mathematics, Riphah International University, Islamabad, Pakistan.

Fractional calculus is of vital importance and its significance is increased a lot since last many years. This paper applies Homotopy Analysis Method (HAM) to obtain analytical solutions of nonlinear fractional gas dynamics equation. Numerical results reveal the complete compatibility of proposed algorithm for such problems. Two special cases of the equation has been solved.

Homotopy Analysis Method, Fractional Calculus, Nonlinear Fractional Gas Dynamics Equation

[1]

West BJ, Bolognab M, Grigolini P (2003). Physics of Fractal Operators, Springer, New York.

[2]

Miller KS, Ross B (1993). An introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.

[3]

Liao SJ (1999). An explicit, totally analytic approximation of Blasius viscous flow problems. Int. J. Nonlin. Mech., 34 (4): 759–778.

[4]

Liao SJ (2003). On the analytic solution of magneto hydrodynamic flows of non-Newtonian fluids over a stretching sheet. J. Fluid Mech., 488: 189–212.

[5]

Liao SJ (2004). An analytic approximate approach for free oscillations of self-excited systems. Int. J. Nolin. Mech., 39: 271–280.

[6]

Liao SJ (2004). On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147: 499–513.

[7]

Podlubny I (1999). Fractional differential equations. An introduction to fractional derivatives fractional differential equations, some methods of their solution and some of their applications. SanDiego: Academic Press.

[8]

Caputo M (1967). Linear models of dissipation whose Q is almost frequency independent, Part II. J. Roy. Astr. Soc., 13: 529-535. 13

[9]

Abbasbandy S (2007). The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equation. Phys Lett. A., 361: 478–483.

[10]

Liao SJ. Comparison between the homotopy analysis method and homotopy perturbation method. Appl Math Comput 2005; 169: 1186-94.

[11]

Hayat T, Khan M, Asghar S (2004). Magneto hydrodynamic flow of an Oldroyd 6–constant fluid. Appl. Math. Comput. 155: 417–425.

[12]

Liao SJ (1997). An approximate solution technique which does not depend upon small parameters (II): An application in fluid mechanics. Int. J. Nonlin. Mech., 32: 815–822.

[13]

Ji-Huan He, S. K. Elagan, Z. B. Li, Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Physics Letters A 376 (2012) 257–259.

[14]

El-Shahed M, Gaber M. Two-dimensional q-differential transformation and its application. Appl Math Comput 2011; 217 (22): 9165–72.

[15]

Liu HK. Application of a differential transformation method to strongly nonlinear damped q-difference equations. Comput Math Appl 2011; 61 (9): 2555–61.

[16]

El-Shahed M, Gaber M, Al-Yami M. The fractional q-differential transformation and its application. Commun Nonlinear Sci Numer Simul 2013; 18: 42–55.

[17]

Abdeljawad T, Baleanu D. Caputo q-fractional initial value problems and a q-analogue Mittag–Leffler function. Commun Nonlinear Sci Numer Simul 2011; 16 (12): 4682–8.

[18]

Salahshour S, Ahmadian A, Chan CS. Successive approximation method for Caputo q-fractional IVPs. Commun Nonlinear Sci Numer Simul 2015; 24 (1–3): 153–8.

[19]

Zeng YX, Zeng Y, Wu G-C. Application of the variational iteration method to the initial value problems of q-difference equations-some examples. Commun Numer Anal 2013. http://dx. doi.org/10.5899/2013/cna-00180.

[20]

Wu GC. Variational iteration method for q-difference equations of second order. J Appl Math 2012. http://dx.doi.org/10.1155/ 2012/102850.

[21]

Qin YM, Zeng DQ. Homotopy perturbation method for the qdiffusion equation with a source term. Commun Fract Calculus 2012; 3 (1): 34–7.

[22]

Liao S. Homotopy analysis method in nonlinear differential equations. Heidelberg Dordrecht London., New York: Springer; 2012.

[23]

Liao S. Beyond perturbation: introduction to homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press; 2003.

[24]

Seshadri R, Munjam SR. Mixed convection flow due to a vertical plate in the presence of heat source and chemical reaction. Ain Shams Eng J, in press. doi: http://dx.doi.org/10.1016/j.asej.2015. 05.008.

[25]

Srinivas S, Reddy AS, Ramamohan TR, Shukla AK. Influence of heat transfer on MHD flow in a pipe with expanding or contracting permeable wall. Ain Shams Eng J 2014; 5: 817–30.

[26]

Semary MS, Hassan HN. The homotopy analysis method for strongly nonlinear initial/boundary value problems. Int J Modern Math Sci 2014; 9 (3): 154–72.

[27]

Khader MM, Sweilam NH, EL-Sehrawy ZI, Ghwail SA. Analytical study for the nonlinear vibrations of multiwalled carbon nanotubes using homotopy analysis method. Appl Math Inform Sci 2014; 8 (4): 1675–84.

[28]

Hassan HN, Semary MS. An analytic solution to a parameterized problems arising in heat transfer equations by optimal homotopy analysis method. Walailak J Sci Technol 2014; 11 (8): 659–77.

[29]

Hassan HN, Semary MS. Analytic approximate solution for the Bratu’s problem by optimal homotopy analysis method. Commun Numer Anal 2013: 1–14.

[30]

Karakoc¸ SBG, Eryılmaz A, Basbu¨k M. The approximate solutions of fredholm integrodifferential-difference equations with variable coefficients via homotopy analysis method. Math Problems Eng 2013: 7. http://dx.doi.org/10.1155/2013/248740 261645.