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.
Similar content being viewed by others
References
Schechter, R. S.,The Variational Method in Engineering, McGraw-Hill Book Company, New York, New York, 1967.
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.
Villadsen, J., andMichelsen, M. L.,Solution of Differential Equation Models by Polynomial Approximation, Prentice-Hall, Englewood Cliffs, New Jersey, 1978.
Ross, B., andFarrell, O. J.,Solved Problems: Gamma and Beta Functions, Legendre Polynomials, Bessel Functions, The Macmillan Company, New York, New York, 1963.
Hwang, C.,Study of Operational Matrix Method in Dynamic Systems, National Cheng Kung University, Department of Chemical Engineering, PhD Thesis, 1981.
Chen, W. L.,Application of Walsh Functions to Time-Varying and Delay Systems, National Cheng Kung University, Department of Electrical Engineering, PhD Thesis, 1977.
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.
Kelley, H. J.,Gradient Theory of Optimal Flight Paths, AIAA Journal, Vol. 30, No. 10, pp. 947–954, 1960.
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.
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.
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.
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.
Gradshteyn, I. S., andRyzhik, I. M.,Tables of Integrals, Series, and Products, Academic Press, New York, New York, 1977.
Author information
Authors and Affiliations
Additional information
Communicated by A. Miele
Rights 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
Issue Date:
DOI: https://doi.org/10.1007/BF00934535