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Quadratic spline solution of Calculus of Variation Problems

Yıl 2015, Cilt: 5 Sayı: 2, 276 - 285, 01.12.2015

Öz

In this paper, we developed numerical method of order O h 2 and based on quadratic polynomial spline function for the numerical solution of class of two point boundary value problems arising in Calculus of Variation. The present approach gives better approximations over all the existing finite difference methods. Convergence analysis and a bound on the approximate solution are discussed. Numerical examples are also given to demonstrate the higher accuracy and efficiency of our method.

Kaynakça

  • Van Brunt, B., (2004), The Calculus of Variations, Springer-Verlag, New York.
  • Gelfand, I. M., Fomin, S. V., (1963), Calculus of Variations, Prentice-Hall, NJ, (revised English edition translated and edited by R.A. Silverman).
  • Elsgolts, L., (1977), Differential Equations and Calculus of Variations, Mir, Moscow, (translated from the Russian by G. Yankovsky).
  • Chen, C. F. and Hsiao, C. H., (1975), A walsh series direct method for solving variational problems, J. Franklin Inst., 300, pp. 265-280.
  • Chang, R. Y. and Wang, M. L., (1983), Shifted Legendre direct method for variational problems
  • J. Optim. Theory Appl., 39, pp. 299-306. Horng, I. R. and Chou, J. H., (1985), Shifted Chebyshev direct method for solving variational problems, Internat. J. Systems Sci., 16, pp. 855-861.
  • Hwang, C. and Shih, Y. P., (1983), Laguerre series direct method for variational problems, J. Optim. Theory Appl., 39, 1, pp. 143-149.
  • Razzaghi, M. and Marzban, H. R., (2000), Direct method for variational problems via of Block-Pulse and chebyshev functions, Mathematical Problems in Engineering, 6, pp. 85-97.
  • Razzaghi, M. and Yousefi, S., (2000), Legendre wavelets direct method for variational problems
  • Math. Comput. Simulation, 53, pp. 185-192. Razzaghi, M. and Yousefi, S., (2001), Legendre Wavelets Method for the Solution of Nonlinear
  • Problems in the Calculus of Variations, Mathematical and Computer Modellmg, 34, pp. 45-54. Razzaghi, M. and Ordokhani, Y., (2001), An application of rationalized Haar functions for varia- tional problems, Appl. Math. Comput., 122, pp. 353-364.
  • Razzaghi, M. and Ordokhani, Y., (2001), Solution for a classical problem in the calculus of variations via rationalized haar functions, Kybernetika, 37, 5, pp. 575-583.
  • Dehghan, M. and Tatari, M., (2006), The use of Adomian decomposition method for solving prob- lems in calculus of variations, Math. Probl. Eng., 2006, pp. 1-12.
  • Tatari, M. and Dehghan, M., (2007), Solution of problems in calculus of variations via He’s varia- tional iteration method, Phys. Lett. A, 362, pp. 401-406.
  • Saadatmandi, A. and Dehghan, M., (2008), The numerical solution of problems in calculus of variation using Chebyshev finite difference method, Phys. Lett. A, 372, pp. 4037-4040.
  • Abdulaziz, O., Hashim, I. and Chowdhury, M. S. H., (2008), Solving variational problems by homo- topy perturbation method, International Journal of Numerical Methods in Engngineering, 75, pp. 721.
  • Dixit, S., Singh, V. K., Singh, A. K. and Singh, O. P., (2010), Bernstein Direct Method for Solving
  • Variational Problems, International Mathematical Forum, 5, 48, pp. 2351-2370.
  • Yousefi, S. A. and Dehghan, M., (2010), The use of He’s variational iteration method for solving variational problems, Int. J. Comput. Math., 87, 6, pp. 1299-1314.
  • Maleki, M. and Mashali-Firouzi, M., (2010), A numerical solution of problems in calculus of variation using direct method and nonclassical parameterization, Journal of Computational and Applied Mathematics, 234, 1364-1373.
  • Nazemi, A. R., Hesam, S. and Haghbin, A., (2013), A fast numerical method for solving calculus of variation problems, AMO - Advanced Modeling and Optimization, 15, 2, pp. 133-149.
  • Ahlberg, J. H., Nilson, J. H. and Walsh, E. N., (1967), The Theory of Splines and Their Applications.
  • Academic Press, San Diego. De Boor, C., (1978), Practical Guide to Splines. Springer, Berlin.
  • Rashidinia, J., Mohammadi, R. and Jalilian, R., (2008), Cubic spline method for two-point boundary value problems, IUST International Journal of Engineering Science, 19, 5-2, pp. 39-43.
  • Ramadan, M. A., Lashien, I. F. and Zahra, W. K., (2007), Polynomial and nonpolynomial spline ap- proaches to the numerical solution of second order boundary value problems, Appl. Math. Comput., , pp. 476-484.
  • Zahra, W. K. and Elkholy, S. M., (2012), Quadratic spline solution for boundary value problem of fractional order, Numer. Algor., 59, pp. 373-391.
  • Henrici, P., (1962), Discrete variable methods in ordinary differential equations, Wiley, New York.
  • Reza Mohammadi is an Assistant Professor of Applied Mathematics in School of Basic Sciences, University of Neyshabur, Neyshabur, Iran, since
Yıl 2015, Cilt: 5 Sayı: 2, 276 - 285, 01.12.2015

Öz

Kaynakça

  • Van Brunt, B., (2004), The Calculus of Variations, Springer-Verlag, New York.
  • Gelfand, I. M., Fomin, S. V., (1963), Calculus of Variations, Prentice-Hall, NJ, (revised English edition translated and edited by R.A. Silverman).
  • Elsgolts, L., (1977), Differential Equations and Calculus of Variations, Mir, Moscow, (translated from the Russian by G. Yankovsky).
  • Chen, C. F. and Hsiao, C. H., (1975), A walsh series direct method for solving variational problems, J. Franklin Inst., 300, pp. 265-280.
  • Chang, R. Y. and Wang, M. L., (1983), Shifted Legendre direct method for variational problems
  • J. Optim. Theory Appl., 39, pp. 299-306. Horng, I. R. and Chou, J. H., (1985), Shifted Chebyshev direct method for solving variational problems, Internat. J. Systems Sci., 16, pp. 855-861.
  • Hwang, C. and Shih, Y. P., (1983), Laguerre series direct method for variational problems, J. Optim. Theory Appl., 39, 1, pp. 143-149.
  • Razzaghi, M. and Marzban, H. R., (2000), Direct method for variational problems via of Block-Pulse and chebyshev functions, Mathematical Problems in Engineering, 6, pp. 85-97.
  • Razzaghi, M. and Yousefi, S., (2000), Legendre wavelets direct method for variational problems
  • Math. Comput. Simulation, 53, pp. 185-192. Razzaghi, M. and Yousefi, S., (2001), Legendre Wavelets Method for the Solution of Nonlinear
  • Problems in the Calculus of Variations, Mathematical and Computer Modellmg, 34, pp. 45-54. Razzaghi, M. and Ordokhani, Y., (2001), An application of rationalized Haar functions for varia- tional problems, Appl. Math. Comput., 122, pp. 353-364.
  • Razzaghi, M. and Ordokhani, Y., (2001), Solution for a classical problem in the calculus of variations via rationalized haar functions, Kybernetika, 37, 5, pp. 575-583.
  • Dehghan, M. and Tatari, M., (2006), The use of Adomian decomposition method for solving prob- lems in calculus of variations, Math. Probl. Eng., 2006, pp. 1-12.
  • Tatari, M. and Dehghan, M., (2007), Solution of problems in calculus of variations via He’s varia- tional iteration method, Phys. Lett. A, 362, pp. 401-406.
  • Saadatmandi, A. and Dehghan, M., (2008), The numerical solution of problems in calculus of variation using Chebyshev finite difference method, Phys. Lett. A, 372, pp. 4037-4040.
  • Abdulaziz, O., Hashim, I. and Chowdhury, M. S. H., (2008), Solving variational problems by homo- topy perturbation method, International Journal of Numerical Methods in Engngineering, 75, pp. 721.
  • Dixit, S., Singh, V. K., Singh, A. K. and Singh, O. P., (2010), Bernstein Direct Method for Solving
  • Variational Problems, International Mathematical Forum, 5, 48, pp. 2351-2370.
  • Yousefi, S. A. and Dehghan, M., (2010), The use of He’s variational iteration method for solving variational problems, Int. J. Comput. Math., 87, 6, pp. 1299-1314.
  • Maleki, M. and Mashali-Firouzi, M., (2010), A numerical solution of problems in calculus of variation using direct method and nonclassical parameterization, Journal of Computational and Applied Mathematics, 234, 1364-1373.
  • Nazemi, A. R., Hesam, S. and Haghbin, A., (2013), A fast numerical method for solving calculus of variation problems, AMO - Advanced Modeling and Optimization, 15, 2, pp. 133-149.
  • Ahlberg, J. H., Nilson, J. H. and Walsh, E. N., (1967), The Theory of Splines and Their Applications.
  • Academic Press, San Diego. De Boor, C., (1978), Practical Guide to Splines. Springer, Berlin.
  • Rashidinia, J., Mohammadi, R. and Jalilian, R., (2008), Cubic spline method for two-point boundary value problems, IUST International Journal of Engineering Science, 19, 5-2, pp. 39-43.
  • Ramadan, M. A., Lashien, I. F. and Zahra, W. K., (2007), Polynomial and nonpolynomial spline ap- proaches to the numerical solution of second order boundary value problems, Appl. Math. Comput., , pp. 476-484.
  • Zahra, W. K. and Elkholy, S. M., (2012), Quadratic spline solution for boundary value problem of fractional order, Numer. Algor., 59, pp. 373-391.
  • Henrici, P., (1962), Discrete variable methods in ordinary differential equations, Wiley, New York.
  • Reza Mohammadi is an Assistant Professor of Applied Mathematics in School of Basic Sciences, University of Neyshabur, Neyshabur, Iran, since
Toplam 28 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Research Article
Yazarlar

R. Mohammadi Bu kişi benim

A. S. Alavi Bu kişi benim

Yayımlanma Tarihi 1 Aralık 2015
Yayımlandığı Sayı Yıl 2015 Cilt: 5 Sayı: 2

Kaynak Göster