Numerical Treatment of Nonlinear Third Order Boundary Value Problem

ABSTRACT

In this paper, the boundary value problems for nonlinear third order differential equations are treated. A generic approach based on nonpolynomial quintic spline is developed to solve such boundary value problem. We show that the approximate solutions of such problems obtained by the numerical algorithm developed using nonpolynomial quintic spline functions are better than those produced by other numerical methods. The algorithm is tested on a problem to demonstrate the practical usefulness of the approach.

In this paper, the boundary value problems for nonlinear third order differential equations are treated. A generic approach based on nonpolynomial quintic spline is developed to solve such boundary value problem. We show that the approximate solutions of such problems obtained by the numerical algorithm developed using nonpolynomial quintic spline functions are better than those produced by other numerical methods. The algorithm is tested on a problem to demonstrate the practical usefulness of the approach.

KEYWORDS

Nonlinear Third Order Boundary Value Problem, Nonpolynomial Quintic Spline, Draining and Coating Flows

Nonlinear Third Order Boundary Value Problem, Nonpolynomial Quintic Spline, Draining and Coating Flows

Cite this paper

nullP. Srivastava and M. Kumar, "Numerical Treatment of Nonlinear Third Order Boundary Value Problem,"*Applied Mathematics*, Vol. 2 No. 8, 2011, pp. 959-964. doi: 10.4236/am.2011.28132.

nullP. Srivastava and M. Kumar, "Numerical Treatment of Nonlinear Third Order Boundary Value Problem,"

References

[1] A. Khan and T. Aziz, “The Numerical Solution of Third- Order Boundary-Value Problems Using Quintic Splines,” Applied Mathematics and Computation, Vol. 137, No. 2-3, 2003, pp. 253-260. doi:10.1016/S0096-3003(02)00051-6

[2] S. Valarmathi and N. Ramanujam, “A Computational Method for Solving Boundary Value Problems for Third —Order Singularly Perturbed Ordinary Differential Equations,” Applied Mathematics and Computation, Vol. 129, No. 2-3, 2002, pp. 345-373. doi:10.1016/S0096-3003(01)00044-3

[3] T. Y. Na, “Computational Method in Engineering Boundary Value Problems,” Academic Press, New York, 1979.

[4] N. S. Asaithambi, “A Numerical Method for the Solution of the Falkner Equation,” Applied Mathematics and Computation, Vol. 81, No. 2-3, 1997, pp. 259-264. doi:10.1016/S0096-3003(95)00325-8

[5] X. Q. Li and M. G. Cui, “Existence and Numerical Method for Nonlinear Third-Order Boundary Value Problem in the Reproducing Kernel Space,” Boundary Value Problems, Article ID 459754, 2010, pp. 1-19.

[6] N. H. Shuaib, H. Power and S. Hibberd, “BEM Solution of Thin Film Flows on an Inclined Plane with a Bottom Outlet,” Engineering Analysis with Boundary Elements, Vol. 33, No. 3, 2009, pp. 388-398. doi:10.1016/j.enganabound.2008.06.007

[7] A. Cabada, M. R. Grossinho and F. Minhos, “External Solutions for Third-Order Nonlinear Problems with Upper and Lower Solutions in Reversed Order,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 62, 2005, pp. 1109-1121.

[8] M. Pei and S. K. Chang, “Existence and Uniqueness of Solutions for Third—Order Nonlinear Boundary Value Problems,” Journal of Mathematical Analysis and Application, Vol. 327, 2007, pp. 23-35. doi:10.1016/j.jmaa.2006.03.057

[9] F. M. Minhos, “On Some Third Order Nonlinear Boundary Value Problems: Existence, Location and Multiplicity Results,” Journal of Mathematical Analysis and Application, Vol. 339, 2008, pp. 1342-1353. doi:10.1016/j.jmaa.2007.08.005

[10] M. Cui and Z. Deng, “Solutions to the Definite Solutions Problem of Differential Equations in Space ,” Advances in Mathematics, Vol. 17, 1988, pp. 327-329.

[11] C. I. Li and M.-G. Cui, “The Exact Solution for Solving a Class Nonlinear Operator Equations in the Reproducing Kernel Space,” Applied Mathematics and Computation, Vol. 143, No. 2-3, 2003, pp. 393-399. doi:10.1016/S0096-3003(02)00370-3

[12] M. Kumar and P. K. Srivastava, “Computational Techniques for Solving Differential Equations by Quadratic, Quartic and Octic Spline,” Advances in Engineering Software, Vol. 39, No. 8, 2008, pp. 646-653. doi:10.1016/j.advengsoft.2007.09.001

[13] M. Kumar and P. K. Srivastava, “Computational Techniques for Solving Differential Equations by Cubic, Quintic and Sextic Spline,” International Journal for Computational Methods in Engineering Science & Mechanics, Vol. 10, No. 1, 2009, pp. 108-115. doi:10.1080/15502280802623297

[14] J. Rashidinia, R. Jalilian and R. Mohammadi, “Non- Polynomial Spline Methods for the Solution of a System of Obstacle Problems,” Applied Mathematics and Computation, Vol. 188, No. 2, 2007, pp. 1984-1990. doi:10.1016/j.amc.2006.11.074

[1] A. Khan and T. Aziz, “The Numerical Solution of Third- Order Boundary-Value Problems Using Quintic Splines,” Applied Mathematics and Computation, Vol. 137, No. 2-3, 2003, pp. 253-260. doi:10.1016/S0096-3003(02)00051-6

[2] S. Valarmathi and N. Ramanujam, “A Computational Method for Solving Boundary Value Problems for Third —Order Singularly Perturbed Ordinary Differential Equations,” Applied Mathematics and Computation, Vol. 129, No. 2-3, 2002, pp. 345-373. doi:10.1016/S0096-3003(01)00044-3

[3] T. Y. Na, “Computational Method in Engineering Boundary Value Problems,” Academic Press, New York, 1979.

[4] N. S. Asaithambi, “A Numerical Method for the Solution of the Falkner Equation,” Applied Mathematics and Computation, Vol. 81, No. 2-3, 1997, pp. 259-264. doi:10.1016/S0096-3003(95)00325-8

[5] X. Q. Li and M. G. Cui, “Existence and Numerical Method for Nonlinear Third-Order Boundary Value Problem in the Reproducing Kernel Space,” Boundary Value Problems, Article ID 459754, 2010, pp. 1-19.

[6] N. H. Shuaib, H. Power and S. Hibberd, “BEM Solution of Thin Film Flows on an Inclined Plane with a Bottom Outlet,” Engineering Analysis with Boundary Elements, Vol. 33, No. 3, 2009, pp. 388-398. doi:10.1016/j.enganabound.2008.06.007

[7] A. Cabada, M. R. Grossinho and F. Minhos, “External Solutions for Third-Order Nonlinear Problems with Upper and Lower Solutions in Reversed Order,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 62, 2005, pp. 1109-1121.

[8] M. Pei and S. K. Chang, “Existence and Uniqueness of Solutions for Third—Order Nonlinear Boundary Value Problems,” Journal of Mathematical Analysis and Application, Vol. 327, 2007, pp. 23-35. doi:10.1016/j.jmaa.2006.03.057

[9] F. M. Minhos, “On Some Third Order Nonlinear Boundary Value Problems: Existence, Location and Multiplicity Results,” Journal of Mathematical Analysis and Application, Vol. 339, 2008, pp. 1342-1353. doi:10.1016/j.jmaa.2007.08.005

[10] M. Cui and Z. Deng, “Solutions to the Definite Solutions Problem of Differential Equations in Space ,” Advances in Mathematics, Vol. 17, 1988, pp. 327-329.

[11] C. I. Li and M.-G. Cui, “The Exact Solution for Solving a Class Nonlinear Operator Equations in the Reproducing Kernel Space,” Applied Mathematics and Computation, Vol. 143, No. 2-3, 2003, pp. 393-399. doi:10.1016/S0096-3003(02)00370-3

[12] M. Kumar and P. K. Srivastava, “Computational Techniques for Solving Differential Equations by Quadratic, Quartic and Octic Spline,” Advances in Engineering Software, Vol. 39, No. 8, 2008, pp. 646-653. doi:10.1016/j.advengsoft.2007.09.001

[13] M. Kumar and P. K. Srivastava, “Computational Techniques for Solving Differential Equations by Cubic, Quintic and Sextic Spline,” International Journal for Computational Methods in Engineering Science & Mechanics, Vol. 10, No. 1, 2009, pp. 108-115. doi:10.1080/15502280802623297

[14] J. Rashidinia, R. Jalilian and R. Mohammadi, “Non- Polynomial Spline Methods for the Solution of a System of Obstacle Problems,” Applied Mathematics and Computation, Vol. 188, No. 2, 2007, pp. 1984-1990. doi:10.1016/j.amc.2006.11.074