{"title":"Non-Polynomial Spline Method for the Solution of Problems in Calculus of Variations","authors":"M. Zarebnia, M. Hoshyar, M. Sedaghati","volume":51,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":484,"pagesEnd":490,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/6590","abstract":"In this paper, a numerical solution based on nonpolynomial\r\ncubic spline functions is used for finding the solution of\r\nboundary value problems which arise from the problems of calculus\r\nof variations. This approximation reduce the problems to an explicit\r\nsystem of algebraic equations. Some numerical examples are also\r\ngiven to illustrate the accuracy and applicability of the presented\r\nmethod.","references":"[1] R. Weinstock, Calculus of Variations: With Applications to Physics and\r\nEngineering, Dover, 1974.\r\n[2] B. Horn, B. Schunck, Determining optical flow, Artificial Intelligence,\r\nvol. 17, no. (1-3), pp. 185-203, 1981.\r\n[3] K. Ikeuchi, B. Horn, Numerical shape from shading and occluding\r\nboundaries. Artificial Intelligence,vol. 17,no. (1-3), pp. 141-184, 1981.\r\n[4] L. Elsgolts, Differential Equations and Calculus of Variations, Mir,\r\nMoscow, 1977 (translated from the Russian by G. Yankovsky).\r\n[5] I.M. Gelfand, S.V. Fomin, Calculus of Variations, Prentice-Hall,\r\nEnglewood Cliffs, NJ, 1963.\r\n[6] C.F. Chen, C.H. Hsiao, A walsh series direct method for solving\r\nvariational problems, J. Franklin Inst.vol. 300, pp. 265-280, 1975.\r\n[7] R.Y. Chang, M.L.Wang, Shifted Legendre direct method for variational\r\nproblems, J. Optim. Theory Appl.vol. 39, pp. 299-306, 1983.\r\n[8] I.R. Horng, J.H. Chou, Shifted Chebyshev direct method for solving\r\nvariational problems, Internat. J. Systems Sci. vol. 16, pp. 855-861,1985.\r\n[9] C. Hwang, Y.P. Shih, Laguerre series direct method for variational\r\nproblems, J. Optim. Theory Appl. Vol. 39, no. 1, pp. 143-149, 1983.\r\n[10] S. Dixit, V.K. Singh, A.K. Singh, O.P. Singh, Bernstein Direct Method\r\nfor Solving Variational Problems, International Mathematical\r\nForum,vol. 5, 2351-2370, 2010.\r\n[11] M. Razzaghi, S. Yousefi, Legendre wavelets direct method for\r\nvariational problems, Mathematics and Computers in Simulation, vol.\r\n53, pp. 185-192, 2000.\r\n[12] M. Tatari, M. Dehghan, Solution of problems in calculus of variations\r\nvia He-s variational iteration method, Physics Letters A, vol. 362, pp.\r\n401-406, 2007.\r\n[13] J.H. Ahlberg, E.N. Nilson, J.L. Walsh, The Theory of Splines and Their\r\nApplications, Academic Press, New York, 1967.\r\n[14] T.N.E. Greville, Introduction to spline functions, in: Theory and\r\nApplication of Spline Functions, Academic Press, New York, 1969.\r\n[15] P.M. Prenter, Splines and Variational Methods, John Wiley & Sons\r\nINC., 1975\r\n[16] G. Micula, Sanda Micula, Hand Book of Splines, Kluwer Academic\r\nPublisher-s, 1999.\r\n[17] M.A. Ramadan, I.F. Lashien, W.K. Zahra, Polynomial and\r\nnonpolynomial spline approaches to the numerical solution of second\r\norder boundary value problems, Applied Mathematics and Computation\r\n,vol. 184, pp. 476-484, 2007.\r\n[18] A. Khan, Parametric cubic spline solution of two point boundary value\r\nproblems, Applied Mathematics and Computation,vol. 154, pp. 175-182,\r\n2004.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 51, 2011"}