Non-Polynomial Spline Method for the Solution of Problems in Calculus of Variations
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.
[1] R. Weinstock, Calculus of Variations: With Applications to Physics and
Engineering, Dover, 1974.
[2] B. Horn, B. Schunck, Determining optical flow, Artificial Intelligence,
vol. 17, no. (1-3), pp. 185-203, 1981.
[3] K. Ikeuchi, B. Horn, Numerical shape from shading and occluding
boundaries. Artificial Intelligence,vol. 17,no. (1-3), pp. 141-184, 1981.
[4] L. Elsgolts, Differential Equations and Calculus of Variations, Mir,
Moscow, 1977 (translated from the Russian by G. Yankovsky).
[5] I.M. Gelfand, S.V. Fomin, Calculus of Variations, Prentice-Hall,
Englewood Cliffs, NJ, 1963.
[6] C.F. Chen, C.H. Hsiao, A walsh series direct method for solving
variational problems, J. Franklin Inst.vol. 300, pp. 265-280, 1975.
[7] R.Y. Chang, M.L.Wang, Shifted Legendre direct method for variational
problems, J. Optim. Theory Appl.vol. 39, pp. 299-306, 1983.
[8] I.R. Horng, J.H. Chou, Shifted Chebyshev direct method for solving
variational problems, Internat. J. Systems Sci. vol. 16, pp. 855-861,1985.
[9] C. Hwang, Y.P. Shih, Laguerre series direct method for variational
problems, J. Optim. Theory Appl. Vol. 39, no. 1, pp. 143-149, 1983.
[10] 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.
[11] M. Razzaghi, S. Yousefi, Legendre wavelets direct method for
variational problems, Mathematics and Computers in Simulation, vol.
53, pp. 185-192, 2000.
[12] 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.
[13] J.H. Ahlberg, E.N. Nilson, J.L. Walsh, The Theory of Splines and Their
Applications, Academic Press, New York, 1967.
[14] T.N.E. Greville, Introduction to spline functions, in: Theory and
Application of Spline Functions, Academic Press, New York, 1969.
[15] P.M. Prenter, Splines and Variational Methods, John Wiley & Sons
INC., 1975
[16] G. Micula, Sanda Micula, Hand Book of Splines, Kluwer Academic
Publisher-s, 1999.
[17] 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.
[18] A. Khan, Parametric cubic spline solution of two point boundary value
problems, Applied Mathematics and Computation,vol. 154, pp. 175-182,
2004.
[1] R. Weinstock, Calculus of Variations: With Applications to Physics and
Engineering, Dover, 1974.
[2] B. Horn, B. Schunck, Determining optical flow, Artificial Intelligence,
vol. 17, no. (1-3), pp. 185-203, 1981.
[3] K. Ikeuchi, B. Horn, Numerical shape from shading and occluding
boundaries. Artificial Intelligence,vol. 17,no. (1-3), pp. 141-184, 1981.
[4] L. Elsgolts, Differential Equations and Calculus of Variations, Mir,
Moscow, 1977 (translated from the Russian by G. Yankovsky).
[5] I.M. Gelfand, S.V. Fomin, Calculus of Variations, Prentice-Hall,
Englewood Cliffs, NJ, 1963.
[6] C.F. Chen, C.H. Hsiao, A walsh series direct method for solving
variational problems, J. Franklin Inst.vol. 300, pp. 265-280, 1975.
[7] R.Y. Chang, M.L.Wang, Shifted Legendre direct method for variational
problems, J. Optim. Theory Appl.vol. 39, pp. 299-306, 1983.
[8] I.R. Horng, J.H. Chou, Shifted Chebyshev direct method for solving
variational problems, Internat. J. Systems Sci. vol. 16, pp. 855-861,1985.
[9] C. Hwang, Y.P. Shih, Laguerre series direct method for variational
problems, J. Optim. Theory Appl. Vol. 39, no. 1, pp. 143-149, 1983.
[10] 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.
[11] M. Razzaghi, S. Yousefi, Legendre wavelets direct method for
variational problems, Mathematics and Computers in Simulation, vol.
53, pp. 185-192, 2000.
[12] 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.
[13] J.H. Ahlberg, E.N. Nilson, J.L. Walsh, The Theory of Splines and Their
Applications, Academic Press, New York, 1967.
[14] T.N.E. Greville, Introduction to spline functions, in: Theory and
Application of Spline Functions, Academic Press, New York, 1969.
[15] P.M. Prenter, Splines and Variational Methods, John Wiley & Sons
INC., 1975
[16] G. Micula, Sanda Micula, Hand Book of Splines, Kluwer Academic
Publisher-s, 1999.
[17] 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.
[18] A. Khan, Parametric cubic spline solution of two point boundary value
problems, Applied Mathematics and Computation,vol. 154, pp. 175-182,
2004.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:55674", author = "M. Zarebnia and M. Hoshyar and M. Sedaghati", title = "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.", keywords = "Calculus of variation; Non-polynomial spline functions; Numerical method", volume = "5", number = "3", pages = "354-6", }
