- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- 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
Advances in Mathematical Physics
Volume 2013 (2013), Article ID 821327, 14 pages
Spectral-Collocation Methods for Fractional Pantograph Delay-Integrodifferential Equations
1College of Civil Engineering and Mechanics, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan 411105, China
2Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan 411105, China
Received 15 May 2013; Accepted 15 September 2013
Academic Editor: Varsha Daftardar-Gejji
Copyright © 2013 Yin Yang and Yunqing Huang. 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.
- J. H. He, “Nonlinear oscillation with fractional derivative and its Cahpinpalications,” in Proceedings of the International Conference on Vibrating Engineering, pp. 288–291, Dalian, China, 1998.
- J. H. He, “Some applications of nonlinear fractional differential equations and therir approximations,” Bulletin of Science, Technology and Society, vol. 15, pp. 86–90, 1999.
- I. Podlubny, Fractional Differential Equations, Academic Press, NewYork, NY, USA, 1999.
- F. Mainardi, Fractional Calculus Continuum Mechanics, Springer, Berlin, Germany, 1997.
- W. M. Ahmad and R. El-Khazali, “Fractional-order dynamical models of love,” Chaos, Solitons and Fractals, vol. 33, no. 4, pp. 1367–1375, 2007.
- F. Huang and F. Liu, “The time fractional diffusion equation and the advection-dispersion equation,” The ANZIAM Journal, vol. 46, no. 3, pp. 317–330, 2005.
- Y. Luchko and R. Gorenflo, The Initial Value Problem for Some Fractional Differential Equations with the Caputo Derivatives, Preprint series A08-98, Fachbreich Mathematik und Informatik, Freic Universitat, Berlin, Germany, 1998.
- N. T. Shawagfeh, “Analytical approximate solutions for nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 131, no. 2-3, pp. 517–529, 2002.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993.
- O. P. Agrawal and P. Kumar, “Comparison of five numerical schemes for fractional differential equations,” in Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, J. Sabatier, et al., Ed., pp. 43–60, Springer, Berlin, Germany, 2007.
- M. M. Khader and A. S. Hendy, “The approximate and exact solutions of the fractional-order delay differential equations using Legendre seudospectral method,” International Journal of Pure and Applied Mathematics, vol. 74, no. 3, pp. 287–297, 2012.
- A. Saadatmandi and M. Dehghan, “A Legendre collocation method for fractional integro-differential equations,” Journal of Vibration and Control, vol. 17, no. 13, pp. 2050–2058, 2011.
- E. A. Rawashdeh, “Legendre wavelets method for fractional integro-differential equations,” Applied Mathematical Sciences, vol. 5, no. 2, pp. 2467–2474, 2011.
- M. Rehman and R. A. Khan, “The Legendre wavelet method for solving fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 11, pp. 4163–4173, 2011.
- A. H. Bhrawy and M. A. Alghamdi, “A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals,” Boundary Value Problems, vol. 1, no. 62, pp. 1–13, 2012.
- E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations,” Applied Mathematical Modelling, vol. 35, no. 12, pp. 5662–5672, 2011.
- E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order,” Computers & Mathematics with Applications, vol. 62, no. 5, pp. 2364–2373, 2011.
- N. H. Sweilam and M. M. Khader, “A Chebyshev pseudo-spectral method for solving fractional-order integro-differential equations,” The ANZIAM Journal, vol. 51, no. 4, pp. 464–475, 2010.
- A. H. Bhrawy, A. S. Alofi, and S. S. Ezz-Eldien, “A quadrature tau method for fractional differential equations with variable coefficients,” Applied Mathematics Letters, vol. 24, no. 12, pp. 2146–2152, 2011.
- A. H. Bhrawy and M. Alshomrani, “A shifted Legendre spectral method for fractional-order multi-point boundary value problems,” Advances in Difference Equations, vol. 2012, article 8, 2012.
- A. Saadatmandi and M. Dehghan, “A new operational matrix for solving fractional-order differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1326–1336, 2010.
- A. H. Bhrawy, M. M. Tharwat, and A. Yildirim, “A new formula for fractional integrals of Chebyshev polynomials: application for solving multi-term fractional differential equations,” Applied Mathematical Modelling, vol. 37, no. 6, pp. 4245–4252, 2013.
- A. H. Bhrawy and A. S. Alofi, “The operational matrix of fractional integration for shifted Chebyshev polynomials,” Applied Mathematics Letters, vol. 26, no. 1, pp. 25–31, 2013.
- E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “A new Jacobi operational matrix: an application for solving fractional differential equations,” Applied Mathematical Modelling, vol. 36, no. 10, pp. 4931–4943, 2012.
- Y. Chen and T. Tang, “Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel,” Mathematics of Computation, vol. 79, no. 269, pp. 147–167, 2010.
- Y. Wei and Y. Chen, “Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions,” Advances in Applied Mathematics and Mechanics, vol. 4, no. 1, pp. 1–20, 2012.
- C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods, Fundamentals in Single Domains, Springer, Berlin, Germany, 2006.
- G. Mastroianni and D. Occorsio, “Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey,” Journal of Computational and Applied Mathematics, vol. 134, no. 1-2, pp. 325–341, 2001.
- D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, Germany, 1989.
- D. L. Ragozin, “Polynomial approximation on compact manifolds and homogeneous spaces,” Transactions of the American Mathematical Society, vol. 150, pp. 41–53, 1970.
- D. L. Ragozin, “Constructive polynomial approximation on spheres and projective spaces,” Transactions of the American Mathematical Society, vol. 162, pp. 157–170, 1971.
- D. Colton and R. Kress, Inverse Coustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, Springer, Berlin, Germany, 2nd edition, 1998.
- P. Nevai, “Mean convergence of Lagrange interpolation. III,” Transactions of the American Mathematical Society, vol. 282, no. 2, pp. 669–698, 1984.
- A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type, World Scientific, New York, NY, USA, 2003.
- D. Baleanu, A. H. Bhrawy, and T. M. Taha, “Two efficient generalized Laguerre spectral algorithms for fractional initial value problems,” Abstract and Applied Analysis, vol. 2013, Article ID 546502, 10 pages, 2013.