About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 542839, 9 pages
http://dx.doi.org/10.1155/2013/542839
Research Article

New Wavelets Collocation Method for Solving Second-Order Multipoint Boundary Value Problems Using Chebyshev Polynomials of Third and Fourth Kinds

1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
2Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt

Received 7 August 2013; Revised 13 September 2013; Accepted 13 September 2013

Academic Editor: Soheil Salahshour

Copyright © 2013 W. M. Abd-Elhameed 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.

Linked References

  1. E. H. Doha, W. M. Abd-Elhameed, and Y. H. Youssri, “Efficient spectral-Petrov-Galerkin methods for the integrated forms of third- and fifth-order elliptic differential equations using general parameters generalized Jacobi polynomials,” Applied Mathematics and Computation, vol. 218, no. 15, pp. 7727–7740, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. W. M. Abd-Elhameed, E. H. Doha, and Y. H. Youssri, “Efficient spectral-Petrov-Galerkin methods for third- and fifth-order differential equations using general parameters generalized Jacobi polynomials,” Quaestiones Mathematicae, vol. 36, no. 1, pp. 15–38, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  3. E. H. Doha, W. M. Abd-Elhameed, and A. H. Bhrawy, “New spectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobi polynomials,” Collectanea Mathematica, vol. 64, no. 3, pp. 373–394, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  4. E. H. Doha, A. H. Bhrawy, and R. M. Hafez, “On shifted Jacobi spectral method for high-order multi-point boundary value problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 10, pp. 3802–3810, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods, Springer, Berlin, Germany, 2006. View at MathSciNet
  6. Siraj-ul-Islam, I. Aziz, and B. Šarler, “The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1577–1590, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. 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, vol. 2012, Article ID 749370, 19 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. W. M. Abd-Elhameed, “Efficient spectral Legendre dual-Petrov-Galerkin algorithms for the direct solution of (2n+1)th-order linear differential equations,” Journal of the Egyptian Mathematical Society, vol. 17, no. 2, pp. 189–211, 2009. View at Zentralblatt MATH · View at MathSciNet
  10. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. M. Moshiinsky, “Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas,” Boletin de La Sociedad Matematica Mexicana, vol. 7, article 125, 1950.
  13. S. P. Timoshenko, Theory of Elastic Stability, McGraw-Hill Book, New York, NY, USA, 2nd edition, 1961. View at MathSciNet
  14. R. P. Agarwal and I. Kiguradze, “On multi-point boundary value problems for linear ordinary differential equations with singularities,” Journal of Mathematical Analysis and Applications, vol. 297, no. 1, pp. 131–151, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Z. Du, “Solvability of functional differential equations with multi-point boundary value problems at resonance,” Computers & Mathematics with Applications, vol. 55, no. 11, pp. 2653–2661, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. W. Feng and J. R. L. Webb, “Solvability of m-point boundary value problems with nonlinear growth,” Journal of Mathematical Analysis and Applications, vol. 212, no. 2, pp. 467–480, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  17. H. B. Thompson and C. Tisdell, “Three-point boundary value problems for second-order, ordinary, differential equations,” Mathematical and Computer Modelling, vol. 34, no. 3-4, pp. 311–318, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. M. R. Scott and H. A. Watts, “SUPORT—a computer code for two-point boundary-value problems via orthonormalization,” Sandia Labs Report 75-0198, Sandia Laboratories, Albuquerque, NM, USA, 1975.
  19. M. R. Scott and H. A. Watts, “Computational solution of linear two-point boundary value problems via orthonormalization,” SIAM Journal on Numerical Analysis, vol. 14, no. 1, pp. 40–70, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. R. Scott and W. H. Vandevender, “A comparison of several invariant imbedding algorithms for the solution of two-point boundary-value problems,” Applied Mathematics and Computation, vol. 1, no. 3, pp. 187–218, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. T. Y. Na, Computational Methods in Engineering Boundary Value Problems, vol. 145, Academic Press, New York, NY, USA, 1979. View at MathSciNet
  22. K. E. Bisshopp and D. C. Drucker, “Large deflection of cantilever beams,” Quarterly of Applied Mathematics, vol. 3, pp. 272–275, 1945. View at Zentralblatt MATH · View at MathSciNet
  23. W. Glabisz, “The use of Walsh-wavelet packets in linear boundary value problems,” Computers & Structures, vol. 82, no. 2-3, pp. 131–141, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  24. Siraj-ul-Islam, M. A. Noor, I. A. Tirmizi, and M. A. Khan, “Quadratic non-polynomial spline approach to the solution of a system of second-order boundary-value problems,” Applied Mathematics and Computation, vol. 179, no. 1, pp. 153–160, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. S. Robert and J. Shipman, “Solution of Troesch's two-point boundary value problems by shooting techniques,” Journal of Computational Physics, vol. 10, pp. 232–241, 1972. View at Publisher · View at Google Scholar
  26. E. Wiebel, Confinement of a Plasma Column by Radiation Pressure in the Plasma in a Magnetic Field, Stanford University Press, Stanfor, Calif, USA, 1958.
  27. H. H. Keller and E. S. Holdrege, “Radiation heat transfer for annular fins of trapezoidal profile,” International Journal of High Performance Computing Applications, vol. 92, pp. 113–116, 1970.
  28. M. Tatari and M. Dehgan, “The use of the Adomian decomposition method for solving multipoint boundary value problems,” Physica Scripta, vol. 73, pp. 672–676, 2006.
  29. M. Lakestani and M. Dehghan, “The solution of a second-order nonlinear differential equation with Neumann boundary conditions using semi-orthogonal B-spline wavelets,” International Journal of Computer Mathematics, vol. 83, no. 8-9, pp. 685–694, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. A. Constantmldes, Applied Numerical Methods with Personal Computers, McGraw-Hill, New York, NY, USA, 1987.
  31. D. E. Newland, An Introduction to Random Vibrations, Spectral and Wavelet Analysis, Longman Scientific and Technical, New York, NY, USA, 1993.
  32. M. Razzaghi and S. Yousefi, “Legendre wavelets method for the solution of nonlinear problems in the calculus of variations,” Mathematical and Computer Modelling, vol. 34, no. 1-2, pp. 45–54, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  33. M. Razzaghi and S. Yousefi, “Legendre wavelets method for constrained optimal control problems,” Mathematical Methods in the Applied Sciences, vol. 25, no. 7, pp. 529–539, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. E. Babolian and F. Fattahzadeh, “Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 417–426, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. L. Zhu and Q. Fan, “Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2333–2341, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. E. H. Doha, W. M. Abd- Elhameed, and Y. H. Youssri, “Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane-Emden type,” New Astronomy, vol. 23-24, pp. 113–117, 2013.
  37. E. H. Doha, W. M. Abd-Elhameed, and M. A. Bassuony, “New algorithms for solving high even-order differential equations using third and fourth Chebyshev-Galerkin methods,” Journal of Computational Physics, vol. 236, pp. 563–579, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  38. J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman & Hall, New York, NY, USA, 2003. View at MathSciNet
  39. Y. Lin, J. Niu, and M. Cui, “A numerical solution to nonlinear second order three-point boundary value problems in the reproducing kernel space,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7362–7368, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. F. Geng, “Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2095–2102, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. F. Z. Geng, “A numerical algorithm for nonlinear multi-point boundary value problems,” Journal of Computational and Applied Mathematics, vol. 236, no. 7, pp. 1789–1794, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. A. Saadatmandi and M. Dehghan, “The use of sinc-collocation method for solving multi-point boundary value problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 2, pp. 593–601, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet