Abstract
In this paper, we mainly study the numerical solution of linear fifth order boundary value problems by using cubic B-splines. Our algorithm develops not only the cubic spline approximation solution but also the approximation derivatives of first order to fourth order of the analytic solution at the same time. This new method has lower computational cost than many other methods and is second order convergent. Numerical examples are given to demonstrate the effectiveness of our method.
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Agarwal, R.P.: Boundary Value Problems for High Order Differential Equations. World Scientific, Singapore (1986)
Albasiny, E.L., Hoskins, W.D.: Cubic spline solutions to two point boundary value problems. Comput. J. 12, 151–153 (1969)
Al-Said, E.A.: Cubic spline method for solving two-point boundary-value problems, Korean J. Comput. Appl. Math. 5(3), 669–680 (1998)
Al-Said, E.A., Noor, M.A., Al-Shejari, A.A.: Numerical solutions for system of second order boundary value problems. Korean J. Comput. Appl. Math. 5(3), 659–667 (1998)
Al-Said, E.A., Noor, M.A.: Cubic splines method for a system of third order boundary value problems. Appl. Math. Comput. 142, 195–204 (2003)
Bickley, W.G.: Piecewise cubic interpolation and two point boundary value problems. Comput. J. 11, 206–208 (1968)
Caglar, H.N., Caglar, S.H., Twizell, E.H.: The numerical solution of fifth-order boundary value problems with sixth degree B-spline functions. Appl. Math. Lett. 12, 25–30 (1999)
Caglar, H., Caglar, N.: Solution of fifth order boundary value problems by using local polynomial regression. Appl. Math. Comput. 186, 952–956 (2007)
Chawla, M., Subramanian, R.: A new fourth order cubic spline method for non-linear two point boundary value problems. Int. J. Comput. Math. 22, 321–341 (1987)
Davies, A.R., Karageorghis, A., Phillips, T.N.: Spectral Galerkin methods for the primary two point boundary value problem in modelling viscoelastic flows. Int. J. Numer. Methods Eng. 26, 647–662 (1988)
De Boor, C.: Practical Guide to Splines. Springer-Verlag, Berlin (1978)
Fyfe, D.J.: The use of cubic splines in the solution of two point boundary value problems. Comput. J. 12, 188–192 (1969)
Islam, S., Khan, M.A.: A numerical method based on polynomial sextic spline functions for the solution of special fifth-order boundary-value problems. Appl. Math. Comput. 181, 356–361 (2006)
Jain, M.K., Aziz, T.: Cubic spline solution of two point boundary value problems with significant first derivatives. Comput. Methods Appl. Mech. Eng. 39, 83–91 (1983)
Karageorghis, A., Phillips, T.N., Davies, A.R.: Spectral collocation methods for the primary two point boundary value problem in modelling viscoelastic flows. Int. J. Numer. Methods Eng. 26, 805–813 (1988)
Khan, M.S.: Finite difference solutions of fifth order boundary value problems. Ph.D. thesis, Brunal University, England (1994)
Kumar, M., Gupta, Y.: Methods for solving singular boundary value problems using splines: a review. J. Appl. Math. Comput. 32, 265–278 (2010)
Lucas, T.R.: Error bounds for interpolating cubic splines under various end conditions. SIAM J. Numer. Anal. 11(3), 569–584 (1974)
Lamnii, A., Mraoui, H., Sbibih, D., Tijini, A.: Sextic spline solution of fifth order boundary value problems. Math. Comput. Simul. 77, 237–246 (2008)
Noor, M.A., Mohyud-Din, S.T.: An efficient algorithm for solving fifth-order boundary value problems. Math. Comput. Model. 45, 954–964 (2007)
Noor, M.A., Mohyud-Din, S.T.: Variational iteration technique for solving higher order boundary value problems. Appl. Math. Comput. 189, 1929–1942 (2007)
Noor, M.A., Mohyud-Din, S.T.: Variational iteration method for fifth-order boundary value problems using He’s polynomials. Math. Probl. Eng. (2008). doi:10.1155/2008/954794
Noor, M.A., Mohyud-Din, S.T.: A new approach to fifth-order boundary value problems. Int. J. Nonlinear Sci. 7(2), 143–148 (2009)
Noor, M.A., Mohyud-Din, S.T.: Modified decomposition method for solving linear and nonlinear fifth-order boundary value problems. Int. J. Appl. Math. Comput. Sci. In press
Papamichael, N., Worsey, A.J.: A cubic spline method for the solution of a linear fourth order two point boundary value problem. J. Comput. Appl. Math. 7, 187–189 (1981)
Rashidinia, J., Jalilian, R., Farajeyan, K.: Spline approximate solution of fifth order boundary value problem. Appl. Math. Comput. 192, 107–112 (2007)
Schoenberg, I.J.: Contribution to the problem of approximation of equidistant data by analytic functions. Quart. Appl. Math. 4, 45–99 (1946)
Schoenberg, I.J.: Contribution to the problem of approximation of equidistant data by analytic functions. Quart. Appl. Math. 4, 112–141 (1946)
Siddiqi, S.S., Akram, G.: Sextic spline solutions of fifth order boundary value problems. Appl. Math. Lett. 20, 591–597 (2007)
Siddiqi, S.S., Akram, G., Elahi, A.: Quartic spline solution of linear fifth order boundary value problems. Appl. Math. Comput. 196, 214–220 (2008)
Wazwaz, A.M.: The numerical solution of fifth-order boundary value problems by the decomposition method. J. Comput. Appl. Math. 136, 259–270 (2001)
Zhang, J.: The numerical solution of fifth-order boundary value problems by the variational iteration method. Comput. Math. Appl. 58, 2347–2350 (2009)
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Lang, FG., Xu, XP. A new cubic B-spline method for linear fifth order boundary value problems. J. Appl. Math. Comput. 36, 101–116 (2011). https://doi.org/10.1007/s12190-010-0390-y
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DOI: https://doi.org/10.1007/s12190-010-0390-y