- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 608943, 8 pages
An Efficient Approach for Fractional Harry Dym Equation by Using Sumudu Transform
1Department of Mathematics, JaganNath Gupta Institute of Engineering and Technology, Jaipur, Rajasthan 302022, India
2Department of Mathematics, JaganNath University, Village-Rampura, Tehsil-Chaksu, Jaipur, Rajasthan 303901, India
3Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
Received 13 March 2013; Accepted 22 April 2013
Academic Editor: Mustafa Bayram
Copyright © 2013 Devendra Kumar 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.
- G. O. Young, “Definition of physical consistent damping laws with fractional derivatives,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 75, pp. 623–635, 1995.
- R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
- I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
- F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the space-time fractional diffusion equation,” Fractional Calculus and Applied Analysis, vol. 4, pp. 153–192, 2001.
- L. Debnath, “Fractional integrals and fractional differential equations in fluid mechanics,” Fractional Calculus and Applied Analysis, vol. 6, 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, Wiley, New York, NY, USA, 1993.
- K. B. Oldham and J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, NY, USA, 1974.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
- R. Mokhtari, “Exact solutions of the Harry-Dym equation,” Communications in Theoretical Physics, vol. 55, no. 2, pp. 204–208, 2011.
- M. D. Kruskal and J. Moser, Dynamical Systems, Theory and Applications, Lecturer Notes Physics, Springer, Berlin, Germany, 1975.
- G. L. Vasconcelos and L. P. Kadanoff, “Stationary solutions for the Saffman-Taylor problem with surface tension,” Physical Review A, vol. 44, no. 10, pp. 6490–6495, 1991.
- F. Gesztesy and K. Unterkofler, “Isospectral deformations for Strum-Liouville and Dirac-type operators and associated nonlinear evolution equations,” Reports on Mathematical Physics, vol. 31, no. 2, pp. 113–137, 1992.
- S. Kumar, M. P. Tripathi, and O. P. Singh, “A fractional model of Harry Dym equation and its approximate solution,” Ain Shams Engineering Journal, vol. 4, no. 1, pp. 111–115, 2013.
- 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, “Asymptotic methods for solitary solutions and compactions,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012.
- D. D. Ganji, “The applications of He’s homotopy perturbation method to nonlinear equation arising in heat transfer,” Physics Letters A, vol. 335, pp. 337–341, 2006.
- 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.
- 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.
- N. H. Sweilam and M. M. Khader, “Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method,” Computers and Mathematics with Applications, vol. 58, no. 11-12, pp. 2134–2141, 2009.
- 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.
- A. Yildirim, “He's homotopy perturbation method for nonlinear differential-difference equations,” International Journal of Computer Mathematics, vol. 87, no. 5, pp. 992–996, 2010.
- 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.
- 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. A. Khuri, “A Laplace decomposition algorithm applied to a class of nonlinear differential equations,” Journal of Applied Mathematics, vol. 1, no. 4, pp. 141–155, 2001.
- M. Khan and M. Hussain, “Application of Laplace decomposition method on semi-infinite domain,” Numerical Algorithms, vol. 56, no. 2, pp. 211–218, 2011.
- M. Khan, M. A. Gondal, and S. Kumar, “A new analytical solution procedure for nonlinear integral equations,” Mathematical and Computer Modelling, vol. 55, pp. 1892–1897, 2012.
- M. A. Gondal and M. Khan, “Homotopy perturbation method for nonlinear exponential boundary layer equation using Laplace transformation, He's polynomials and pade technology,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 12, pp. 1145–1153, 2010.
- J. Singh, D. Kumar, and Sushila, “Homotopy perturbation sumudu transform method for nonlinear equations,” Advances in Theoretical and Applied Mechanics, vol. 4, pp. 165–175, 2011.
- D. Kumar, J. Singh, and S. Rathore, “Sumudu decomposition method for nonlinear equations,” International Mathematical Forum, vol. 7, no. 11, pp. 515–521, 2012.
- 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, “Applications 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 transforms,” International Journal of Mathematical Education in Science and Technology, vol. 29, no. 4, pp. 618–621, 1998.
- M. A. Asiru, “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,” Bulletin of the 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, 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, vol. 2010, Article ID 598702, 11 pages, 2010.
- J. Zhang, “A sumudu based algorithm for solving differential equations,” Computer Science Journal of Moldova, vol. 15, 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.
- S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “Traveling wave solutions of seventh-order generalized KdV equations using he's polynomials,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 2, pp. 227–233, 2009.
- K. Abbaoui and Y. Cherruault, “New ideas for proving convergence of decomposition methods,” Computers and Mathematics with Applications, vol. 29, no. 7, pp. 103–108, 1995.
- G. Adomian, Solving Frontier Problems of Physics: the Decomposition Method, Kluwer Academic Publishers, Boston, Mass, USA, 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.