Numerical Solutions of a Class of Second Order Boundary Value Problems on Using Bernoulli Polynomials

ABSTRACT

The aim of this paper is to find the numerical solutions of the second order linear and nonlinear differential equations with*Dirichlet*, *Neumann* and *Robin* boundary conditions. We use the Bernoulli polynomials as linear combination to the approximate solutions of 2nd order boundary value problems. Here the Bernoulli polynomials over the interval [0,1] are chosen as trial functions so that care has been taken to satisfy the corresponding homogeneous form of the *Dirichlet* boundary conditions in the Galerkin weighted residual method. In addition to that the given differential equation over arbitrary finite domain [*a,b*] and the boundary conditions are converted into its equivalent form over the interval [0,1]. All the formulas are verified by considering numerical examples. The approximate solutions are compared with the exact solutions, and also with the solutions of the existing methods. A reliable good accuracy is obtained in all cases.

The aim of this paper is to find the numerical solutions of the second order linear and nonlinear differential equations with

Cite this paper

nullM. Islam and A. Shirin, "Numerical Solutions of a Class of Second Order Boundary Value Problems on Using Bernoulli Polynomials,"*Applied Mathematics*, Vol. 2 No. 9, 2011, pp. 1059-1067. doi: 10.4236/am.2011.29147.

nullM. Islam and A. Shirin, "Numerical Solutions of a Class of Second Order Boundary Value Problems on Using Bernoulli Polynomials,"

References

[1] M. I. Bhatti and P. Bracken, “Solutions of Differential Equations in a Bernstein Polynomial Basis,” Journal of Computational and Applied Mathematics, Vol. 205, No. 1, 2007, pp. 272-280. doi:10.1016/j.cam.2006.05.002

[2] M. A. Ramadan, I. F. Lashien and W. K. Zahra, “Polynomial and Nonpolynomial Spline Approaches to the Numerical Solution of Second Order Boundary Value Problem,” Applied Ma-thematics and Computation, Vol. 184, No. 2, 2007, pp. 476-484. doi:10.1016/j.amc.2006.06.053.

[3] R.A. Usmani and M. Sakai, “A Connection between Quartic Spline and Numerov Solution of a Boundary Value Problem,” International Journal of Computer Mathematics, Vol. 26, No. 3, 1989, pp. 263-273. doi:10.1080/00207168908803700

[4] Arshad Khan, “Parametric Cubic Spline Solution of Two Point Boundary Value Problems,” Applied Mathematics and Com-putation, Vol. 154, No. 1, 2004, pp. 175-182. doi:10.1016/S0096-3003(03)00701-X.

[5] E. A. Al-Said, “Cubic Spline Method for Solving Two Point Boundary Value Problems,” Korean Journal of Computational and Applied Mathematics, Vol. 5, 1998, pp. 759-770.

[6] E. A. Al-Said, “Quadratic Spline Solution of Two Point Boun-dary Value Problems,” Journal of Natural Geometry, Vol. 12, 1997, pp. 125-134.

[7] D. J. Fyfe, “The Use of Cubic Splines in the Solution of Two Point Boundary Value Problems,” The Computer Journal, Vol. 12, No. 2, 1969, pp. 188-192. doi:10.1093/comjnl/12.2.188

[8] A. K. Khalifa and J. C. Eilbeck, “Collocation with Quadratic and Cubic Splines,” The IMA Journal of Numerical Analysis, Vol. 2, No. 1, 1982, pp. 111-121. doi:10.1093/imanum/2.1.111

[9] G. Mullenheim, “Solving Two-Point Boundary Value Problems with Spline Functions,” The IMA Journal of Numerical Analysis, Vol. 12, No. 4, 1992, pp. 503-518. doi:10.1093/imanum/12.4.503

[10] J. Reinkenhof, “Differentiation and Integration Using Bernstein’s Polynomials,” International Journal for Numerical Methods in Engineering, Vol. 11, No. 10, 1977, pp. 1627-1630. doi:10.1002/nme.1620111012

[11] E. Kreyszig, “Bernstein Polynomials and Numerical Integra-tion,” International Journal for Numerical Methods in Engi-neering, Vol. 14, No. 2, 1979, pp. 292-295. doi:10.1002/nme.1620140213

[12] R. A. Usmani, “Bounds for the Solution of a Second Order Differential Equation with Mixed Boundary Conditions,” Journal of Engineering Mathematics, Vol. 9, No. 2, 1975, pp. 159-164. doi:10.1007/BF01535397

[13] B. Bialecki, “Sinc-Collocation Methods for Two Point Boundary Value Problems,” The IMA Journal of Numerical Analysis, Vol. 11, No. 3, 1991, pp. 357-375. doi:10.1093/imanum/11.3.357

[14] K. E. Atkinson, “An Introduction to Numerical Analysis,” 2nd Edition, John Wiley and Sons, New York, 1989, pp- 284.

[15] P. E. Lewis and J. P. Ward, “The Finite Element Method, Prin-ciples and Applications,” Addison-Wesley, Boston 1991.

[16] R. L. Burden and J. D. Faires, “Numerical Analysis,” Brooks/Cole Publishing Co., Pacific Grove, 1992.

[17] M. K. Jain, “Numerical Solution of Differential Equations,” 2nd Edition, New Age International, New Delhi, 2000.

[1] M. I. Bhatti and P. Bracken, “Solutions of Differential Equations in a Bernstein Polynomial Basis,” Journal of Computational and Applied Mathematics, Vol. 205, No. 1, 2007, pp. 272-280. doi:10.1016/j.cam.2006.05.002

[2] M. A. Ramadan, I. F. Lashien and W. K. Zahra, “Polynomial and Nonpolynomial Spline Approaches to the Numerical Solution of Second Order Boundary Value Problem,” Applied Ma-thematics and Computation, Vol. 184, No. 2, 2007, pp. 476-484. doi:10.1016/j.amc.2006.06.053.

[3] R.A. Usmani and M. Sakai, “A Connection between Quartic Spline and Numerov Solution of a Boundary Value Problem,” International Journal of Computer Mathematics, Vol. 26, No. 3, 1989, pp. 263-273. doi:10.1080/00207168908803700

[4] Arshad Khan, “Parametric Cubic Spline Solution of Two Point Boundary Value Problems,” Applied Mathematics and Com-putation, Vol. 154, No. 1, 2004, pp. 175-182. doi:10.1016/S0096-3003(03)00701-X.

[5] E. A. Al-Said, “Cubic Spline Method for Solving Two Point Boundary Value Problems,” Korean Journal of Computational and Applied Mathematics, Vol. 5, 1998, pp. 759-770.

[6] E. A. Al-Said, “Quadratic Spline Solution of Two Point Boun-dary Value Problems,” Journal of Natural Geometry, Vol. 12, 1997, pp. 125-134.

[7] D. J. Fyfe, “The Use of Cubic Splines in the Solution of Two Point Boundary Value Problems,” The Computer Journal, Vol. 12, No. 2, 1969, pp. 188-192. doi:10.1093/comjnl/12.2.188

[8] A. K. Khalifa and J. C. Eilbeck, “Collocation with Quadratic and Cubic Splines,” The IMA Journal of Numerical Analysis, Vol. 2, No. 1, 1982, pp. 111-121. doi:10.1093/imanum/2.1.111

[9] G. Mullenheim, “Solving Two-Point Boundary Value Problems with Spline Functions,” The IMA Journal of Numerical Analysis, Vol. 12, No. 4, 1992, pp. 503-518. doi:10.1093/imanum/12.4.503

[10] J. Reinkenhof, “Differentiation and Integration Using Bernstein’s Polynomials,” International Journal for Numerical Methods in Engineering, Vol. 11, No. 10, 1977, pp. 1627-1630. doi:10.1002/nme.1620111012

[11] E. Kreyszig, “Bernstein Polynomials and Numerical Integra-tion,” International Journal for Numerical Methods in Engi-neering, Vol. 14, No. 2, 1979, pp. 292-295. doi:10.1002/nme.1620140213

[12] R. A. Usmani, “Bounds for the Solution of a Second Order Differential Equation with Mixed Boundary Conditions,” Journal of Engineering Mathematics, Vol. 9, No. 2, 1975, pp. 159-164. doi:10.1007/BF01535397

[13] B. Bialecki, “Sinc-Collocation Methods for Two Point Boundary Value Problems,” The IMA Journal of Numerical Analysis, Vol. 11, No. 3, 1991, pp. 357-375. doi:10.1093/imanum/11.3.357

[14] K. E. Atkinson, “An Introduction to Numerical Analysis,” 2nd Edition, John Wiley and Sons, New York, 1989, pp- 284.

[15] P. E. Lewis and J. P. Ward, “The Finite Element Method, Prin-ciples and Applications,” Addison-Wesley, Boston 1991.

[16] R. L. Burden and J. D. Faires, “Numerical Analysis,” Brooks/Cole Publishing Co., Pacific Grove, 1992.

[17] M. K. Jain, “Numerical Solution of Differential Equations,” 2nd Edition, New Age International, New Delhi, 2000.