×

Homotopy perturbation method for fractional gas dynamics equation using Sumudu transform. (English) Zbl 1262.76082

Summary: A user friendly algorithm based on new homotopy perturbation Sumudu transform method (HPSTM) is proposed to solve nonlinear fractional gas dynamics equation. The fractional derivative is considered in the Caputo sense. Further, the same problem is solved by Adomian decomposition method (ADM). The results obtained by the two methods are in agreement and hence this technique may be considered an alternative and efficient method for finding approximate solutions of both linear and nonlinear fractional differential equations. The HPSTM is a combined form of Sumudu transform, homotopy perturbation method, and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The numerical solutions obtained by the proposed method show that the approach is easy to implement and computationally very attractive.

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76N15 Gas dynamics (general theory)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Beyer, H.; Kempfle, S., Definition of physically consistent damping laws with fractional derivatives, Zeitschrift für Angewandte Mathematik und Mechanik, 75, 8, 623-635 (1995) · Zbl 0850.65069 · doi:10.1002/zamm.19950750820
[2] He, J. H., Some applications of nonlinear fractional differential equations and their approximations, Bulletin of Science, Technology & Society, 15, 2, 86-90 (1999)
[3] He, J.-H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering, 167, 1-2, 57-68 (1998) · Zbl 0942.76077 · doi:10.1016/S0045-7825(98)00108-X
[4] Hilfer, R., Fractional time evolution, Applications of Fractional Calculus in Physics, 87-130 (2000), River Edge, NJ, USA: World Scientific, River Edge, NJ, USA · Zbl 0994.34050 · doi:10.1142/9789812817747_0002
[5] Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, Mathematics in Science and Engineering, 198 (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0924.34008
[6] Mainardi, F.; Luchko, Y.; Pagnini, G., The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus & Applied Analysis, 4, 2, 153-192 (2001) · Zbl 1054.35156
[7] Rida, S. Z.; El-Sayed, A. M. A.; Arafa, A. A. M., On the solutions of time-fractional reaction-diffusion equations, Communications in Nonlinear Science and Numerical Simulation, 15, 12, 3847-3854 (2010) · Zbl 1222.65115 · doi:10.1016/j.cnsns.2010.02.007
[8] Yıldırım, A., He’s homotopy perturbation method for solving the space- and time-fractional telegraph equations, International Journal of Computer Mathematics, 87, 13, 2998-3006 (2010) · Zbl 1206.65239 · doi:10.1080/00207160902874653
[9] Debnath, L., Fractional integral and fractional differential equations in fluid mechanics, Fractional Calculus & Applied Analysis, 6, 2, 119-155 (2003) · Zbl 1076.35095
[10] Caputo, M., Elasticita E Dissipazione (1969), Bologna, Italy: Zani-Chelli, Bologna, Italy
[11] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0789.26002
[12] Oldham, K. B.; Spanier, J., The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, With an Annotated Chronological Bibliography by Bertram Ross, Mathematics in Science and Engineering, 111 (1974), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0292.26011
[13] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204 (2006), Amsterdam, The Netherlands: Elsevier Science B.V., Amsterdam, The Netherlands · Zbl 1092.45003
[14] Yang, X. J., Advanced Local Fractional Calculus and Its Applications (2012), New York, NY, USA: World Science Publisher, New York, NY, USA
[15] Yang, X. J., Local fractional integral transforms, Progress in Nonlinear Science, 4, 1-225 (2011)
[16] Yang, X. J., Local Fractional Functional Analysis and Its Applications (2011), Hong Kong, China: Asian Academic, Hong Kong, China
[17] Yang, X. J., Heat transfer in discontinuous media, Advances in Mechanical Engineering and Its Applications, 1, 3, 47-53 (2012)
[18] Yang, X. J., Local fractional partial differential equations with fractal boundary problems, Advances in Computational Mathematics and Its Applications, 1, 1, 60-63 (2012)
[19] Zeng, D. Q.; Qin, Y. M., The Laplace-Adomian-Pade technique for the seepage flows with the Riemann-Liouville derivatives, Communications in Fractional Calculus, 3, 1, 26-29 (2012)
[20] He, J. H., Asymptotic methods for solitary solutions and compactons, Abstract and Applied Analysis, 2012 (2012) · Zbl 1257.35158 · doi:10.1155/2012/916793
[21] He, J.-H., Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 3-4, 257-262 (1999) · Zbl 0956.70017 · doi:10.1016/S0045-7825(99)00018-3
[22] He, J.-H., Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation, 135, 1, 73-79 (2003) · Zbl 1030.34013 · doi:10.1016/S0096-3003(01)00312-5
[23] He, J.-H., New interpretation of homotopy perturbation method. Addendum: ‘some asymptotic methods for strongly nonlinear equations’, International Journal of Modern Physics B, 20, 18, 2561-2568 (2006) · doi:10.1142/S0217979206034819
[24] Ganji, D. D., The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer, Physics Letters A, 355, 4-5, 337-341 (2006) · Zbl 1255.80026 · doi:10.1016/j.physleta.2006.02.056
[25] Yildirim, A., An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 10, 4, 445-450 (2009)
[26] Ganji, D. D.; Rafei, M., Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation by homotopy perturbation method, Physics Letters A, 356, 2, 131-137 (2006) · Zbl 1160.35517 · doi:10.1016/j.physleta.2006.03.039
[27] Rashidi, M. M.; Ganji, D. D.; Dinarvand, S., Explicit analytical solutions of the generalized Burger and Burger-Fisher equations by homotopy perturbation method, Numerical Methods for Partial Differential Equations, 25, 2, 409-417 (2009) · Zbl 1159.65085 · doi:10.1002/num.20350
[28] Aminikhah, H.; Hemmatnezhad, M., An efficient method for quadratic Riccati differential equation, Communications in Nonlinear Science and Numerical Simulation, 15, 4, 835-839 (2010) · Zbl 1221.65193 · doi:10.1016/j.cnsns.2009.05.009
[29] Kachapi, S. H.; Ganji, D. D., Nonlinear Equations: Analytical Methods and Applications (2012), Springer · Zbl 1342.65001
[30] Jafari, H.; Wazwaz, A. M.; Khalique, C. M., Homotopy perturbation and variational iteration methods for solving fuzzy differential equations, Communications in Fractional Calculus, 3, 1, 38-48 (2012)
[31] Qin, Y. M.; Zeng, D. Q., Homotopy perturbation method for the q-diffusion equation with a source term, Communications in Fractional Calculus, 3, 1, 34-37 (2012)
[32] Javidi, M.; Raji, M. A., Combination of Laplace transform and homotopy perturbation method to solve the parabolic partial differential equations, Communications in Fractional Calculus, 3, 1, 10-19 (2012)
[33] Duan, J. S.; Rach, R.; Buleanu, D.; Wazwaz, A. M., A review of the Adomian decomposition method and its applications to fractional differential equations, Communications in Fractional Calculus, 3, 2, 73-99 (2012)
[34] Ganji, D. D., A semi-Analytical technique for non-linear settling particle equation of motion, Journal of Hydro-Environment Research, 6, 4, 323-327 (2012) · doi:10.1016/j.jher.2012.04.002
[35] Singh, J.; Kumar, D.; Sushila, Homotopy perturbation Sumudu transform method for nonlinear equations, Advances in Applied Mathematics and Mechanics, 4, 165-175 (2011) · Zbl 1247.76062
[36] Ghorbani, A.; Saberi-Nadjafi, J., He’s homotopy perturbation method for calculating adomian polynomials, International Journal of Nonlinear Sciences and Numerical Simulation, 8, 2, 229-232 (2007) · Zbl 1401.65056
[37] Ghorbani, A., Beyond Adomian polynomials: he polynomials, Chaos, Solitons and Fractals, 39, 3, 1486-1492 (2009) · Zbl 1197.65061 · doi:10.1016/j.chaos.2007.06.034
[38] Das, S.; Kumar, R., Approximate analytical solutions of fractional gas dynamic equations, Applied Mathematics and Computation, 217, 24, 9905-9915 (2011) · Zbl 1387.35606 · doi:10.1016/j.amc.2011.03.144
[39] Watugala, G. K., Sumudu transform—a new integral transform to solve differential equations and control engineering problems, Mathematical Engineering in Industry, 6, 4, 319-329 (1998) · Zbl 0916.44002
[40] Weerakoon, S., Application of Sumudu transform to partial differential equations, International Journal of Mathematical Education in Science and Technology, 25, 2, 277-283 (1994) · Zbl 0812.35004 · doi:10.1080/0020739940250214
[41] Weerakoon, S., Complex inversion formula for Sumudu transform, International Journal of Mathematical Education in Science and Technology, 29, 4, 618-621 (1998) · Zbl 1018.44004
[42] Aşiru, M. A., Further properties of the Sumudu transform and its applications, International Journal of Mathematical Education in Science and Technology, 33, 3, 441-449 (2002) · Zbl 1013.44001 · doi:10.1080/002073902760047940
[43] Kadem, A., Solving the one-dimensional neutron transport equation using Chebyshev polynomials and the Sumudu transform, Analele Universitatii din Oradea, 12, 153-171 (2005) · Zbl 1164.82331
[44] Kılıçman, A.; Eltayeb, H.; Atan, K. A. M., A note on the comparison between Laplace and Sumudu transforms, Iranian Mathematical Society, 37, 1, 131-141 (2011) · Zbl 1242.44001
[45] Kılıçman, A.; Gadain, H. E., On the applications of Laplace and Sumudu transforms, Journal of the Franklin Institute, 347, 5, 848-862 (2010) · Zbl 1286.35185 · doi:10.1016/j.jfranklin.2010.03.008
[46] Eltayeb, H.; Kılıçman, A.; Fisher, B., A new integral transform and associated distributions, Integral Transforms and Special Functions, 21, 5-6, 367-379 (2010) · Zbl 1191.35017 · doi:10.1080/10652460903335061
[47] Kılıçman, A.; Eltayeb, H., A note on integral transforms and partial differential equations, Applied Mathematical Sciences, 4, 1-4, 109-118 (2010) · Zbl 1194.35017
[48] Kılıçman, A.; Eltayeb, H.; Agarwal, R. P., On Sumudu transform and system of differential equations, Abstract and Applied Analysis (2010) · Zbl 1197.34001 · doi:10.1155/2010/598702
[49] Zhang, J., A Sumudu based algorithm for solving differential equations, Academy of Sciences of Moldova, 15, 3, 303-313 (2007) · Zbl 1187.34015
[50] Chaurasia, V. B. L.; Singh, J., Application of Sumudu transform in Schödinger equation occurring in quantum mechanics, Applied Mathematical Sciences, 4, 57-60, 2843-2850 (2010) · Zbl 1218.33005
[51] Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method. Solving Frontier Problems of Physics: The Decomposition Method, Fundamental Theories of Physics, 60 (1994), Dordrecht, The Netherlands: Kluwer Academic, Dordrecht, The Netherlands · Zbl 0802.65122
[52] Odibat, Z.; Momani, S., Numerical methods for nonlinear partial differential equations of fractional order, Applied Mathematical Modelling, 32, 1, 28-39 (2008) · Zbl 1133.65116 · doi:10.1016/j.apm.2006.10.025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.