Geodesic
Spline in tension method for non-linear two point boundary value problems on a geometric mesh
Matematičeskoe modelirovanie, Tome 27 (2015) no. 3, pp. 33-48.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we discuss a new method based on spline in tension approximation for the numerical solution of two-point non-linear boundary value problems on uniform mesh. The method is of order four. We have discussed the derivation and the convergence of the proposed method in detail. The method is extended to non-uniform mesh. Numerical results are given to illustrate the efficiency of the proposed method.
Keywords: Spline in tension; Non polynomial spline; Convergence analysis; Root mean square errors; Variable mesh; Burgers’ equation. 
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J. Talwar; R. K. Mohanty. Spline in tension method for non-linear two point boundary value problems on a geometric mesh. Matematičeskoe modelirovanie, Tome 27 (2015) no. 3, pp. 33-48. http://geodesic.mathdoc.fr/item/MM_2015_27_3_a2/

[1] D. J. Fyfe, “The use of cubic splines in the solution of two point boundary value problems”, Comput. J., 12 (1969), 188–192 | DOI

[2] M. K. Jain, T. Aziz, “Spline function approximation for differential equations”, Comput. Methods Appl. Mech. Engrg., 26:2 (1981), 129–143 | DOI

[3] M. K. Jain, T. Aziz, “Cubic spline solution of two-point boundary value problems with significant first derivatives”, Comput. Methods Appl. Mech. Engrg., 39:1 (1983), 83–91 | DOI

[4] M. K. Jain, Numerical solution of differential equations, Second edition, A Halsted Press Book, John Wiley Sons, Inc., New York, 1984

[5] M. M. Chawla, R. Subramanian, “A New Spline method for Singular two point boundary value problems”, Int. J. Comput. Math., 24 (1988), 291–310 | DOI

[6] M. K. Kadalbajoo, R. K. Bawa, “Cubic spline method for a class of non-linear singularly perturbed boundary value problems”, J. Optim. Theory Appl., 76 (1993), 415–428 | DOI

[7] E. A. Al-Said, “Spline methods for solving a system of second order boundary value problems”, Int. J. Comput. Math., 70 (1999), 717–727 | DOI

[8] E. A. Al-Said, “The use of cubic splines in the numerical solution of a system of second order boundary value problem”, Comput. Math. Appl., 42 (2001), 861–869 | DOI

[9] D. J. Evans, “Group explicit methods for solving large linear systems”, Int. J. Comput. Math., 17 (1985), 81–108 | DOI

[10] R. K. Mohanty, J. Talwar, “A combined approach using coupled reduced alternating group explicit (CRAGE) algorithm and sixth order off-step discretization for the solution of two point nonlinear boundary value problem”, Journal of Applied Mathematics and Computation, 219 (2012), 248–259 | DOI

[11] A. Khan, “Parametric cubic spline solution of two point boundary value problems”, Appl. Math. Comput., 154 (2004), 175–182 | DOI

[12] M. Kumar, “Higher order method for singular boundary problems by using spline function”, Appl. Math. Comput., 192 (2007), 175–179 | DOI

[13] H. B. Keller, Numerical Methods for Two Point Boundary Value Problem, Blaisdell Pub. Co., Waltham, MA, 1992

[14] R. K. Mohanty, D. J. Evans, “Highly accurate two parameter CAGE parallel algorithms for non-linear singular two point boundary value problems”, Int. J. Comput. Math., 82 (2005), 433–444 | DOI

[15] D. J. Evans, “Iterative methods for solving non-linear two point boundary value problems”, Int. J. Comput. Math., 72 (1999), 395–401 | DOI

[16] J. H. Ahlberg, J. H. Nilson, E. N. Walsh, The Theory of Splines and Their Applications, Academic Press, San Diego, 1967

[17] E. L. Albasiny, W. D. Hoskins, “Cubic Spline Solutions to Two Point Boundary Value Problems”, Computer Journal, 12 (1969), 151–153 | DOI

[18] M. K. Jain, T. Aziz, “Numerical solution of stiff and convection diffusion equations using adaptive spline function approximation”, Appl. Math. Modelling, 7 (1983), 57–62 | DOI

[19] R. K. Mohanty, J. Talwar, “Compact alternating group explicit method for the cubic spline solution of two point boundary value problems with significant nonlinear first derivative terms”, Mathematical Sciences, 6:58 (2012)

[20] M. M. Chawla, R. Subramanian, “A New Fourth Order Cubic Spline Method for Nonlinear Two point Boundary Value Problems”, Inter. J. Computer Math., 22 (1987), 321–341 | DOI

[21] R. K. Mohanty, “A Family of Variable Mesh Methods for the Estimates of (du/dr) and the Solution of Non-linear Two Point Boundary Value Problems with Singularity”, J. Comput. Appl. Math., 182 (2005), 173–187 | DOI

[22] M. K. Kadalbajoo, K. C. Patidar, “Tension spline for the numerical solution of singularly perturbed non-linear boundary value problems”, Comput. Appl. Math., 21:3 (2002), 717–742

[23] M. K. Kadalbajoo, K. C. Patidar, “Tension spline for the solution of self-adjoint singular perturbation problems”, Int. J. Comput. Math., 79:7 (2002), 849–865 | DOI

[24] R. K. Mohanty, D. J. Evans, U. Arora, “Convergent spline in tension methods for singularly perturbed two-point singular boundary value problems”, Int. J. Comput. Math., 82:1 (2005), 55–66 | DOI

[25] R. K. Mohanty, U. Arora, “A family of non-uniform mesh tension spline methods for singularly perturbed two-point singular boundary value problems with significant first derivatives”, Appl. Math. Comput., 172:2 (2006), 531–554 | DOI

[26] M. Marusic, “A fourth/second order accurate collocation method for singularly perturbed two-point boundary value problems using tension splines”, Numer. Math., 88:1 (2001), 135–158 | DOI

[27] I. Khan, T. Aziz, “Tension spline method for second-order singularly perturbed boundary-value problems”, Int. J. Comput. Math., 82:12 (2005), 1547–1553 | DOI

[28] R. K. Mohanty, V. Gopal, “A fourth order finite difference method based on spline in tension approximation for the solution of one-space dimensional second order quasi-linear hyperbolic equations”, Advances in Difference Equations, 70 (2013)

[29] J. Rashidinia, R. Mohammadi, “Tension spline approach for the numerical solution of nonlinear Klein–Gordon equation”, Comput. Phys. Comm., 181:1 (2010), 78–91 | DOI

[30] J. Rashidinia, R. Mohammadi, “Tension spline solution of non-linear sine-Gordon equation”, Numer. Algorithms, 56:1 (2011), 129–142 | DOI