Shifted Chebyshev direct method for solving variational problems. (English) Zbl 0568.49019

Shifted Chebyshev polynomials for solving variational problems are given in this study. This technique reduces a variational problem to the solution of algebraic equations, and the computation is straightforward on a digital computer. Two illustrative examples are given. Only a small number of the shifted Chebyshev polynomials are needed to calculate the Chebyshev coefficients, and the result produced is very attractive and accurate.


49M05 Numerical methods based on necessary conditions
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
49K15 Optimality conditions for problems involving ordinary differential equations
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
65K10 Numerical optimization and variational techniques
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[1] ABRAMOWITZ M., Handbook of Mathematical Functions (1967)
[2] BRYSON A. E., J. appl. Mech. 84 pp 247– (1962) · Zbl 0112.20003 · doi:10.1115/1.3640537
[3] CHANG R. Y., J. optim. Theory Applic 39 pp 299– (1983) · Zbl 0481.49004 · doi:10.1007/BF00934535
[4] CHEN C. F., J. Franklin Inst. 300 pp 265– (1975) · Zbl 0339.49017 · doi:10.1016/0016-0032(75)90199-4
[5] HWANG C., J. optim. Theory Applic 39 pp 143– (1983) · Zbl 0481.49005 · doi:10.1007/BF00934611
[6] KELLEY H. J., AIA Aerospace J. 30 pp 947– (1960)
[7] MIELE A., J. optim. Theory Applic 17 pp 361– (1975) · Zbl 0296.49024 · doi:10.1007/BF00932781
[8] MIELE A., J. optim. Theory Applic 5 pp 235– (1970) · Zbl 0192.51802 · doi:10.1007/BF00927913
[9] MIELE A., J. optim. Theory Applic 10 pp 381– (1972) · Zbl 0233.49009 · doi:10.1007/BF00935401
[10] SHIH Y. M., J. Chinese Inst. Engng. 6 pp 135– (1983) · doi:10.1080/02533839.1983.9676735
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