- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 934060, 8 pages
Homotopy Perturbation Method for Fractional Gas Dynamics Equation Using Sumudu Transform
1Department of Mathematics, Jagannath University, Rampura, Chaksu, Jaipur, Rajasthan 303901, India
2Department of Mathematics, Jagannath Gupta Institute of Engineering & Technology, Jaipur, Rajasthan 302022, India
3Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia
Received 12 September 2012; Revised 19 November 2012; Accepted 6 December 2012
Academic Editor: Lan Xu
Copyright © 2013 Jagdev Singh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- H. Beyer and S. Kempfle, “Definition of physically consistent damping laws with fractional derivatives,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 75, no. 8, pp. 623–635, 1995.
- J. H. He, “Some applications of nonlinear fractional differential equations and their approximations,” Bulletin of Science, Technology & Society, vol. 15, no. 2, pp. 86–90, 1999.
- J.-H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57–68, 1998.
- R. Hilfer, “Fractional time evolution,” in Applications of Fractional Calculus in Physics, pp. 87–130, World Scientific, River Edge, NJ, USA, 2000.
- I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
- F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the space-time fractional diffusion equation,” Fractional Calculus & Applied Analysis, vol. 4, no. 2, pp. 153–192, 2001.
- S. Z. Rida, A. M. A. El-Sayed, and A. A. M. Arafa, “On the solutions of time-fractional reaction-diffusion equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 3847–3854, 2010.
- A. Yıldırım, “He's homotopy perturbation method for solving the space- and time-fractional telegraph equations,” International Journal of Computer Mathematics, vol. 87, no. 13, pp. 2998–3006, 2010.
- L. Debnath, “Fractional integral and fractional differential equations in fluid mechanics,” Fractional Calculus & Applied Analysis, vol. 6, no. 2, pp. 119–155, 2003.
- M. Caputo, Elasticita E Dissipazione, Zani-Chelli, Bologna, Italy, 1969.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
- K. B. Oldham and J. Spanier, 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, vol. 111, Academic Press, New York, NY, USA, 1974.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
- X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, NY, USA, 2012.
- X. J. Yang, “Local fractional integral transforms,” Progress in Nonlinear Science, vol. 4, pp. 1–225, 2011.
- X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic, Hong Kong, China, 2011.
- X. J. Yang, “Heat transfer in discontinuous media,” Advances in Mechanical Engineering and Its Applications, vol. 1, no. 3, pp. 47–53, 2012.
- X. J. Yang, “Local fractional partial differential equations with fractal boundary problems,” Advances in Computational Mathematics and Its Applications, vol. 1, no. 1, pp. 60–63, 2012.
- D. Q. Zeng and Y. M. Qin, “The Laplace-Adomian-Pade technique for the seepage flows with the Riemann-Liouville derivatives,” Communications in Fractional Calculus, vol. 3, no. 1, pp. 26–29, 2012.
- J. H. He, “Asymptotic methods for solitary solutions and compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012.
- J.-H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999.
- J.-H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003.
- J.-H. He, “New interpretation of homotopy perturbation method. Addendum: ‘some asymptotic methods for strongly nonlinear equations’,” International Journal of Modern Physics B, vol. 20, no. 18, pp. 2561–2568, 2006.
- D. D. Ganji, “The application of He's homotopy perturbation method to nonlinear equations arising in heat transfer,” Physics Letters A, vol. 355, no. 4-5, pp. 337–341, 2006.
- A. Yildirim, “An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 4, pp. 445–450, 2009.
- D. D. Ganji and M. Rafei, “Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation by homotopy perturbation method,” Physics Letters A, vol. 356, no. 2, pp. 131–137, 2006.
- M. M. Rashidi, D. D. Ganji, and S. Dinarvand, “Explicit analytical solutions of the generalized Burger and Burger-Fisher equations by homotopy perturbation method,” Numerical Methods for Partial Differential Equations, vol. 25, no. 2, pp. 409–417, 2009.
- H. Aminikhah and M. Hemmatnezhad, “An efficient method for quadratic Riccati differential equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 4, pp. 835–839, 2010.
- S. H. Kachapi and D. D. Ganji, Nonlinear Equations: Analytical Methods and Applications, Springer, 2012.
- H. Jafari, A. M. Wazwaz, and C. M. Khalique, “Homotopy perturbation and variational iteration methods for solving fuzzy differential equations,” Communications in Fractional Calculus, vol. 3, no. 1, pp. 38–48, 2012.
- Y. M. Qin and D. Q. Zeng, “Homotopy perturbation method for the q-diffusion equation with a source term,” Communications in Fractional Calculus, vol. 3, no. 1, pp. 34–37, 2012.
- M. Javidi and M. A. Raji, “Combination of Laplace transform and homotopy perturbation method to solve the parabolic partial differential equations,” Communications in Fractional Calculus, vol. 3, no. 1, pp. 10–19, 2012.
- J. S. Duan, R. Rach, D. Buleanu, and A. M. Wazwaz, “A review of the Adomian decomposition method and its applications to fractional differential equations,” Communications in Fractional Calculus, vol. 3, no. 2, pp. 73–99, 2012.
- D. D. Ganji, “A semi-Analytical technique for non-linear settling particle equation of motion,” Journal of Hydro-Environment Research, vol. 6, no. 4, pp. 323–327, 2012.
- J. Singh, D. Kumar, and Sushila, “Homotopy perturbation Sumudu transform method for nonlinear equations,” Advances in Applied Mathematics and Mechanics, vol. 4, pp. 165–175, 2011.
- A. Ghorbani and J. Saberi-Nadjafi, “He's homotopy perturbation method for calculating adomian polynomials,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 2, pp. 229–232, 2007.
- A. Ghorbani, “Beyond Adomian polynomials: he polynomials,” Chaos, Solitons and Fractals, vol. 39, no. 3, pp. 1486–1492, 2009.
- S. Das and R. Kumar, “Approximate analytical solutions of fractional gas dynamic equations,” Applied Mathematics and Computation, vol. 217, no. 24, pp. 9905–9915, 2011.
- G. K. Watugala, “Sumudu transform—a new integral transform to solve differential equations and control engineering problems,” Mathematical Engineering in Industry, vol. 6, no. 4, pp. 319–329, 1998.
- S. Weerakoon, “Application of Sumudu transform to partial differential equations,” International Journal of Mathematical Education in Science and Technology, vol. 25, no. 2, pp. 277–283, 1994.
- S. Weerakoon, “Complex inversion formula for Sumudu transform,” International Journal of Mathematical Education in Science and Technology, vol. 29, no. 4, pp. 618–621, 1998.
- M. A. Aşiru, “Further properties of the Sumudu transform and its applications,” International Journal of Mathematical Education in Science and Technology, vol. 33, no. 3, pp. 441–449, 2002.
- A. Kadem, “Solving the one-dimensional neutron transport equation using Chebyshev polynomials and the Sumudu transform,” Analele Universitatii din Oradea, vol. 12, pp. 153–171, 2005.
- A. Kılıçman, H. Eltayeb, and K. A. M. Atan, “A note on the comparison between Laplace and Sumudu transforms,” Iranian Mathematical Society, vol. 37, no. 1, pp. 131–141, 2011.
- A. Kılıçman and H. E. Gadain, “On the applications of Laplace and Sumudu transforms,” Journal of the Franklin Institute, vol. 347, no. 5, pp. 848–862, 2010.
- H. Eltayeb, A. Kılıçman, and B. Fisher, “A new integral transform and associated distributions,” Integral Transforms and Special Functions, vol. 21, no. 5-6, pp. 367–379, 2010.
- A. Kılıçman and H. Eltayeb, “A note on integral transforms and partial differential equations,” Applied Mathematical Sciences, vol. 4, no. 1–4, pp. 109–118, 2010.
- A. Kılıçman, H. Eltayeb, and R. P. Agarwal, “On Sumudu transform and system of differential equations,” Abstract and Applied Analysis, Article ID 598702, 11 pages, 2010.
- J. Zhang, “A Sumudu based algorithm for solving differential equations,” Academy of Sciences of Moldova, vol. 15, no. 3, pp. 303–313, 2007.
- V. B. L. Chaurasia and J. Singh, “Application of Sumudu transform in Schödinger equation occurring in quantum mechanics,” Applied Mathematical Sciences, vol. 4, no. 57–60, pp. 2843–2850, 2010.
- G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60 of Fundamental Theories of Physics, Kluwer Academic, Dordrecht, The Netherlands, 1994.
- Z. Odibat and S. Momani, “Numerical methods for nonlinear partial differential equations of fractional order,” Applied Mathematical Modelling, vol. 32, no. 1, pp. 28–39, 2008.