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Approximate Analytical Solutions of the Fractional Sharma-Tasso-Olver Equation Using Homotopy Analysis Method and a Comparison with Other Methods

Yıl 2012, Cilt: 9 Sayı: 2, - , 01.04.2012

Öz

In this paper, the homotopy analysis method (HAM) is successfully applied to
the fractional Sharma-Tasso-Olver equation to obtain its approximate analytical solutions.
Comparison of the obtained results with those of variational iteration method (VIM),
Adomian’s decomposition method (ADM) and homotopy perturbation method (HPM) has
led us to conclude that the method gives significantly important consequences. The HAM
solution includes an auxiliary parameter ~ which provides a convenient way of adjusting
and controlling the convergence region of solution series.

Kaynakça

  • [1] J.-H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering 167 (1998), 57–68.
  • [2] R. Y. Molliq, M. S. M. Noorani and I. Hashim, Variational iteration method for fractional heatand wave-like equations, Nonlinear Analysis: Real World Applications 10 (2009), 1854–1869.
  • [3] N. T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, Applied Mathematics and Computation 131 (2002), 517–529.
  • [4] S. Momani and Z. Obibat, Numerical approach to differential equations of fractional order, Journal of Computational and Applied Mathematics 207 (2007), 96–110.
  • [5] S. Momani and Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Physics Letters A 365 (2007), 345–350.
  • [6] Z. Odibat, Exact solitary solutions for variants of the KdV equations with fractional time derivatives, Chaos, Solitons & Fractals 40 (2009), 1264–1270.
  • [7] H. Xu and J. Cang, Analysis of a time fractional wave-like equation with the homotopy analysis method, Physics Letters A 372 (2008), 1250–1255.
  • [8] L. Song and H. Q. Zhang, Application of homotopy analysis method to fractional KdVBurgers-Kuramoto equation, Physics Letters A 367 (2007), 88–94.
  • [9] H. Jafari and S. Seifi, Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation, Communications in Nonlinear Science and Numerical Simulation 14 (2009), 2006–2012.
  • [10] S. J. Liao, The Proposed Homotopy Analysis Tecnique for the Solution of Nonlinear Problems, Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai 1992.
  • [11] S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton 2003.
  • [12] S. J. Liao, Homotopy analysis method: A new analytical technique for nonlinear problems, Communications in Nonlinear Science and Numerical Simulation 2 (1997), 95–100.
  • [13] S. J. Liao, On the homotopy analysis method for nonlinear problems, Applied Mathematics and Computation 147 (2004), 499–513.
  • [14] L. Podlubny, Fractional Differantial Equations, Academic Press, London 1999.
  • [15] M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophysical Journal International 13 (1967), 529–539.
  • [16] M. Caputo, Elasticit`a e Dissipazione, Zanichelli Publisher, Bologna, 1969.
  • [17] L. Song, Q. Wang and H. Zhang, Rational approximation solution of the fractional SharmaTasso-Olever equation, Journal of Computational and Applied Mathematics 224 (2009), 210–218.
  • [18] S. J. Liao, Notes on the homotopy analysis method: Some definitions and theorems, Communications in Nonlinear Science and Numerical Simulation 14 (2009), 983–997.
Yıl 2012, Cilt: 9 Sayı: 2, - , 01.04.2012

Öz

Kaynakça

  • [1] J.-H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering 167 (1998), 57–68.
  • [2] R. Y. Molliq, M. S. M. Noorani and I. Hashim, Variational iteration method for fractional heatand wave-like equations, Nonlinear Analysis: Real World Applications 10 (2009), 1854–1869.
  • [3] N. T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, Applied Mathematics and Computation 131 (2002), 517–529.
  • [4] S. Momani and Z. Obibat, Numerical approach to differential equations of fractional order, Journal of Computational and Applied Mathematics 207 (2007), 96–110.
  • [5] S. Momani and Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Physics Letters A 365 (2007), 345–350.
  • [6] Z. Odibat, Exact solitary solutions for variants of the KdV equations with fractional time derivatives, Chaos, Solitons & Fractals 40 (2009), 1264–1270.
  • [7] H. Xu and J. Cang, Analysis of a time fractional wave-like equation with the homotopy analysis method, Physics Letters A 372 (2008), 1250–1255.
  • [8] L. Song and H. Q. Zhang, Application of homotopy analysis method to fractional KdVBurgers-Kuramoto equation, Physics Letters A 367 (2007), 88–94.
  • [9] H. Jafari and S. Seifi, Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation, Communications in Nonlinear Science and Numerical Simulation 14 (2009), 2006–2012.
  • [10] S. J. Liao, The Proposed Homotopy Analysis Tecnique for the Solution of Nonlinear Problems, Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai 1992.
  • [11] S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton 2003.
  • [12] S. J. Liao, Homotopy analysis method: A new analytical technique for nonlinear problems, Communications in Nonlinear Science and Numerical Simulation 2 (1997), 95–100.
  • [13] S. J. Liao, On the homotopy analysis method for nonlinear problems, Applied Mathematics and Computation 147 (2004), 499–513.
  • [14] L. Podlubny, Fractional Differantial Equations, Academic Press, London 1999.
  • [15] M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophysical Journal International 13 (1967), 529–539.
  • [16] M. Caputo, Elasticit`a e Dissipazione, Zanichelli Publisher, Bologna, 1969.
  • [17] L. Song, Q. Wang and H. Zhang, Rational approximation solution of the fractional SharmaTasso-Olever equation, Journal of Computational and Applied Mathematics 224 (2009), 210–218.
  • [18] S. J. Liao, Notes on the homotopy analysis method: Some definitions and theorems, Communications in Nonlinear Science and Numerical Simulation 14 (2009), 983–997.
Toplam 18 adet kaynakça vardır.

Ayrıntılar

Bölüm Makaleler
Yazarlar

Alaattin Esen Bu kişi benim

Orkun Taşbozan Bu kişi benim

N. Murat Yağmurlu

Yayımlanma Tarihi 1 Nisan 2012
Yayımlandığı Sayı Yıl 2012 Cilt: 9 Sayı: 2

Kaynak Göster

APA Esen, A., Taşbozan, O., & Yağmurlu, N. M. (2012). Approximate Analytical Solutions of the Fractional Sharma-Tasso-Olver Equation Using Homotopy Analysis Method and a Comparison with Other Methods. Cankaya University Journal of Science and Engineering, 9(2).
AMA Esen A, Taşbozan O, Yağmurlu NM. Approximate Analytical Solutions of the Fractional Sharma-Tasso-Olver Equation Using Homotopy Analysis Method and a Comparison with Other Methods. CUJSE. Nisan 2012;9(2).
Chicago Esen, Alaattin, Orkun Taşbozan, ve N. Murat Yağmurlu. “Approximate Analytical Solutions of the Fractional Sharma-Tasso-Olver Equation Using Homotopy Analysis Method and a Comparison With Other Methods”. Cankaya University Journal of Science and Engineering 9, sy. 2 (Nisan 2012).
EndNote Esen A, Taşbozan O, Yağmurlu NM (01 Nisan 2012) Approximate Analytical Solutions of the Fractional Sharma-Tasso-Olver Equation Using Homotopy Analysis Method and a Comparison with Other Methods. Cankaya University Journal of Science and Engineering 9 2
IEEE A. Esen, O. Taşbozan, ve N. M. Yağmurlu, “Approximate Analytical Solutions of the Fractional Sharma-Tasso-Olver Equation Using Homotopy Analysis Method and a Comparison with Other Methods”, CUJSE, c. 9, sy. 2, 2012.
ISNAD Esen, Alaattin vd. “Approximate Analytical Solutions of the Fractional Sharma-Tasso-Olver Equation Using Homotopy Analysis Method and a Comparison With Other Methods”. Cankaya University Journal of Science and Engineering 9/2 (Nisan 2012).
JAMA Esen A, Taşbozan O, Yağmurlu NM. Approximate Analytical Solutions of the Fractional Sharma-Tasso-Olver Equation Using Homotopy Analysis Method and a Comparison with Other Methods. CUJSE. 2012;9.
MLA Esen, Alaattin vd. “Approximate Analytical Solutions of the Fractional Sharma-Tasso-Olver Equation Using Homotopy Analysis Method and a Comparison With Other Methods”. Cankaya University Journal of Science and Engineering, c. 9, sy. 2, 2012.
Vancouver Esen A, Taşbozan O, Yağmurlu NM. Approximate Analytical Solutions of the Fractional Sharma-Tasso-Olver Equation Using Homotopy Analysis Method and a Comparison with Other Methods. CUJSE. 2012;9(2).