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Non-Polynomial Spline Method for the Solution of Problems in Calculus of Variations
Abstract:In this paper, a numerical solution based on nonpolynomial cubic spline functions is used for finding the solution of boundary value problems which arise from the problems of calculus of variations. This approximation reduce the problems to an explicit system of algebraic equations. Some numerical examples are also given to illustrate the accuracy and applicability of the presented method.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1063202Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1574
 R. Weinstock, Calculus of Variations: With Applications to Physics and Engineering, Dover, 1974.
 B. Horn, B. Schunck, Determining optical flow, Artificial Intelligence, vol. 17, no. (1-3), pp. 185-203, 1981.
 K. Ikeuchi, B. Horn, Numerical shape from shading and occluding boundaries. Artificial Intelligence,vol. 17,no. (1-3), pp. 141-184, 1981.
 L. Elsgolts, Differential Equations and Calculus of Variations, Mir, Moscow, 1977 (translated from the Russian by G. Yankovsky).
 I.M. Gelfand, S.V. Fomin, Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, 1963.
 C.F. Chen, C.H. Hsiao, A walsh series direct method for solving variational problems, J. Franklin Inst.vol. 300, pp. 265-280, 1975.
 R.Y. Chang, M.L.Wang, Shifted Legendre direct method for variational problems, J. Optim. Theory Appl.vol. 39, pp. 299-306, 1983.
 I.R. Horng, J.H. Chou, Shifted Chebyshev direct method for solving variational problems, Internat. J. Systems Sci. vol. 16, pp. 855-861,1985.
 C. Hwang, Y.P. Shih, Laguerre series direct method for variational problems, J. Optim. Theory Appl. Vol. 39, no. 1, pp. 143-149, 1983.
 S. Dixit, V.K. Singh, A.K. Singh, O.P. Singh, Bernstein Direct Method for Solving Variational Problems, International Mathematical Forum,vol. 5, 2351-2370, 2010.
 M. Razzaghi, S. Yousefi, Legendre wavelets direct method for variational problems, Mathematics and Computers in Simulation, vol. 53, pp. 185-192, 2000.
 M. Tatari, M. Dehghan, Solution of problems in calculus of variations via He-s variational iteration method, Physics Letters A, vol. 362, pp. 401-406, 2007.
 J.H. Ahlberg, E.N. Nilson, J.L. Walsh, The Theory of Splines and Their Applications, Academic Press, New York, 1967.
 T.N.E. Greville, Introduction to spline functions, in: Theory and Application of Spline Functions, Academic Press, New York, 1969.
 P.M. Prenter, Splines and Variational Methods, John Wiley & Sons INC., 1975
 G. Micula, Sanda Micula, Hand Book of Splines, Kluwer Academic Publisher-s, 1999.
 M.A. Ramadan, I.F. Lashien, W.K. Zahra, Polynomial and nonpolynomial spline approaches to the numerical solution of second order boundary value problems, Applied Mathematics and Computation ,vol. 184, pp. 476-484, 2007.
 A. Khan, Parametric cubic spline solution of two point boundary value problems, Applied Mathematics and Computation,vol. 154, pp. 175-182, 2004.