Abstract and Applied Analysis

Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations

A. H. Bhrawy and M. A. Alghamdi

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Abstract

A shifted Jacobi Galerkin method is introduced to get a direct solution technique for solving the third- and fifth-order differential equations with constant coefficients subject to initial conditions. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. A quadrature Galerkin method is introduced for the numerical solution of these problems with variable coefficients. A new shifted Jacobi collocation method based on basis functions satisfying the initial conditions is presented for solving nonlinear initial value problems. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms. The algorithms are easy to implement and yield very accurate results.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 364360, 25 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495862

Digital Object Identifier
doi:10.1155/2012/364360

Mathematical Reviews number (MathSciNet)
MR2970001

Zentralblatt MATH identifier
1253.65157

Citation

Bhrawy, A. H.; Alghamdi, M. A. Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations. Abstr. Appl. Anal. 2012 (2012), Article ID 364360, 25 pages. doi:10.1155/2012/364360. https://projecteuclid.org/euclid.aaa/1355495862


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References

  • J. P. Boyd, Chebyshev and Fourier Spectral Methods, Dover, Mineola, NY, USA, 2nd edition, 2001.
  • C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, New York, NY, USA, 1989.
  • B.-Y. Guo, Spectral Methods and Their Applications, World Scientific, River Edge, NJ, USA, 1998.
  • A. H. Bhrawy, A. S. Alofi, and S. I. El-Soubhy, “An extension of the legendre-galerkin method for solving sixth-order differential equations with variable polynomial coefficients,” Mathematical Problems in Engineering, vol. 2012, Article ID 896575, 13 pages, 2012.
  • E. H. Doha, W. M. Abd-Elhameed, and A. H. Bhrawy, “Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of $2n$th-order linear differential equations,” Applied Mathematical Mod-elling, vol. 33, no. 4, pp. 1982–1996, 2009.
  • E. H. Doha and A. H. Bhrawy, “Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials,” Numerical Algorithms, vol. 42, no. 2, pp. 137–164, 2006.
  • E. H. Doha and A. H. Bhrawy, “Efficient spectral-Galerkin algorithms for direct solution of the integrated forms of second-order equations using ultraspherical polynomials,” The ANZIAM Journal, vol. 48, no. 3, pp. 361–386, 2007.
  • E. H. Doha and A. H. Bhrawy, “A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations,” Numerical Methods for Partial Differential Equations, vol. 25, no. 3, pp. 712–739, 2009.
  • E. H. Doha, A. H. Bhrawy, and W. M. Abd-Elhameed, “Jacobi spectral Galerkin method for elliptic Neumann problems,” Numerical Algorithms, vol. 50, no. 1, pp. 67–91, 2009.
  • E. H. Doha and W. M. Abd-Elhameed, “Efficient solutions of multidimensional sixth-order boundary value problems using symmetric generalized Jacobi-Galerkin method,” Abstract and Applied Analysis, Abstract and Applied Analysis, vol. 2012, Article ID 749370, 19 pages, 2012.
  • E. H. Doha and A. H. Bhrawy, “An efficient direct solver for multidimensional elliptic Robin boundary value problems using a Legendre spectral-Galerkin method,” Computers and Mathematics with Applications, vol. 64, no. 4, pp. 558–571, 2012.
  • C. Lanczos, Applied Analysis, Pitman, London, UK, 1957.
  • A. H. Bhrawy, A. S. Alofi, and S. I. El-Soubhy, “Spectral shifted Jacobi tau and collocation methods for solving fifth-order boundary value problems,” Abstract and Applied Analysis, vol. 2011, Article ID 823273, 14 pages, 2011.
  • E. H. Doha, A. H. Bhrawy, and R. M. Hafez, “On shifted Jacobi spectral method for high-order multipoint boundary value problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, pp. 3802–3810, 2012.
  • A. H. Bhrawy and W. M. Abd-Elhameed, “New algorithm for the numerical solutions of nonlinear third-order differential equations using Jacobi-Gauss collocation method,” Mathematical Problems in Engineering, vol. 2011, Article ID 837218, 14 pages, 2011.
  • A. H. Bhrawy and A. S. Alofi, “A Jacobi-Gauss collocation method for solving nonlinear Lane-Emden type equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 62–70, 2012.
  • E. H. Doha and A. H. Bhrawy, “Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials,” Applied Numerical Mathematics., vol. 58, no. 8, pp. 1224–1244, 2008.
  • E. H. Doha, A. H. Bhrawy, and R. M. Hafez, “A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1820–1832, 2011.
  • E. H. Doha, A. H. Bhrawy, and R. M. Hafez, “A Jacobi dual-Petrov-Galerkin method for solving some odd-order ordinary differential equations,” Abstract and Applied Analysis, vol. 2011, Article ID 947230, 21 pages, 2011.
  • E. H. Doha, “On the coefficients of differentiated expansions and derivatives of Jacobi polynomials,” Journal of Physics A, vol. 35, no. 15, pp. 3467–3478, 2002.
  • E. H. Doha, “On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials,” Journal of Physics A, vol. 37, no. 3, pp. 657–675, 2004.
  • Y. Luke, The Special Functions and Their Approximations, vol. 2, Academic Press, New York, NY, USA, 1969.
  • I. H. Abdel-Halim Hassan, “Differential transformation technique for solving higher-order initial value problems,” Applied Mathematics and Computation, vol. 154, no. 2, pp. 299–311, 2004.
  • D. K. Salkuyeh, “Convergence of the variational iteration method for solving linear systems of ODEs with constant coefficients,” Computers & Mathematics with Applications, vol. 56, no. 8, pp. 2027–2033, 2008.