Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Shifted Legendre direct method for variational problems

  • 244 Accesses

  • 99 Citations

Abstract

The shifted Legendre polynomial series is employed to solve variational problems. The solution is carried out by using an operational matrix for integrating the shifted Legendre polynomial vector. Variational problems are reduced to solving algebraic equations. Two illustrative examples are given, and the computational results obtained by Legendre series direct method are compared with the exact solutions.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Schechter, R. S.,The Variational Method in Engineering, McGraw-Hill Book Company, New York, New York, 1967.

  2. 2.

    Chen, C. F., andHsiao, C. H.,A Walsh Series Direct Method for Solving Variational Problems, Journal of Franklin Institute, Vol. 300, No. 4, pp. 265–280, 1975.

  3. 3.

    Villadsen, J., andMichelsen, M. L.,Solution of Differential Equation Models by Polynomial Approximation, Prentice-Hall, Englewood Cliffs, New Jersey, 1978.

  4. 4.

    Ross, B., andFarrell, O. J.,Solved Problems: Gamma and Beta Functions, Legendre Polynomials, Bessel Functions, The Macmillan Company, New York, New York, 1963.

  5. 5.

    Hwang, C.,Study of Operational Matrix Method in Dynamic Systems, National Cheng Kung University, Department of Chemical Engineering, PhD Thesis, 1981.

  6. 6.

    Chen, W. L.,Application of Walsh Functions to Time-Varying and Delay Systems, National Cheng Kung University, Department of Electrical Engineering, PhD Thesis, 1977.

  7. 7.

    Hwang, C., andShih, Y. P.,Laguerre Series Direct Method for Variational Problems, Journal of Optimization Theory and Applications, Vol. 39, No. 1, pp. 143–149, 1983.

  8. 8.

    Kelley, H. J.,Gradient Theory of Optimal Flight Paths, AIAA Journal, Vol. 30, No. 10, pp. 947–954, 1960.

  9. 9.

    Bryson, A. E., Jr., andDenham, W. F.,A Steepest-Ascent Method for Solving Optimum Programming Problems, Journal of Applied Mechanics, Vol. 84, No. 3, pp. 247–257, 1962.

  10. 10.

    Miele, A., Pritchard, R. E., andDamoulakis, J. N.,Sequential Gradient Restoration Algorithm for Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 5, No. 4, pp. 235–282, 1970.

  11. 11.

    Miele, A., Tietze, J. L., andLevy, A. V.,Summary and Comparison of Gradient-Restoration Algorithms for Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 10, No. 6, pp. 381–403, 1972.

  12. 12.

    Miele, A.,Recent Advances in Gradient Algorithms for Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 17, Nos. 5/6, pp. 361–430, 1975.

  13. 13.

    Gradshteyn, I. S., andRyzhik, I. M.,Tables of Integrals, Series, and Products, Academic Press, New York, New York, 1977.

Download references

Author information

Additional information

Communicated by A. Miele

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chang, R.Y., Wang, M.L. Shifted Legendre direct method for variational problems. J Optim Theory Appl 39, 299–307 (1983). https://doi.org/10.1007/BF00934535

Download citation

Key Words

  • Shifted Legendre polynomials
  • operational matrix
  • variational problems
  • optimization