# A Numerical Method for the Solution of General Third Order Boundary Value Problem in Ordinary Differential Equations

## Abstract

In this article we have considered general third order boundaryvalue problems and proposed an efficient difference method for numerical solution of the problems. We have shown under appropriate conditions that proposed method is convergent and second order accurate. The numerical results in experiment on test problems show the simplicity and efficiency of the method.

## References

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[2] R. P. Agarwal. Boundary Value Problems for Higher Order Differential Equations. World Scientific, Singapore, 1986.

[3] C. P. Gupta and V. Lakshmikantham. Existence and uniqueness theorems for a third-order three point boundary value problem. Nonlinear Anal.: Theory, Meth. Appl., 16(11)(1991), 949-957.

[4] K. N. Murty and Y. S. Rao. A theory for existence and uniqueness of solutions to three-point boundary value problems. J. Math. Anal. Appl., 167(1)(1992), 43-48.

[5] M. Feckan. Singularly perturbed higher order boundary value problems. J. Diff. Equat., 111(1)(1994), 79-102.

[6] J. Henderson and K. R. Prasad. Existence and uniqueness of solutions of three-point boundary value problems on time scales. Nonlin. Studies, 8(2001), 1-12.

[7] E. A. Al-Said. Numerical solutions for system of third-order boundary value problems. Int. J. Comp. Math., 78(1)(2001), 111-121.

[8] A. Khan and T. Aziz. The Numerical Solution of Third- Order Boundary-Value Problems Using Quintic Splines. Appl. Math. Comp., 137(2-3)(2003), 253-260.

[9] P. K. Pandey. Solving third-order Boundary Value Problems with Quartic Splines. Panday SpringerPlus, 5(1)(2016), 1-10 .

[10] S. Islam, M. A. Khan, I. A. Tirmizi and E. H. Twizell. Non-polynomial splines approach to the solution of a system of third-order boundary-value problems. Appl. Math. Comp.,

168(1)(2005), 152-163.

[11] F. Gao and C. M. Chi. Solving third-order obstacle problems with quartic B-splines. Appl. Math. Comp., 180(1)(2006), 270-274.

[12] Fazal-i-Haq, I. Hussain and A. Ali. A Haar Wavelets Based Numerical Method for Third-order Boundary and Initial Value Problems. World Appl. Sci. J., 13(10)(2011), 2244-2251.

[13] M. A. Noor and A. K. Khalifa. A numerical approach for odd-order obstacle problems. Int. J. Comp. Math., 54(1)(1994), 109-116.

[14] X. Li and B. Wu. Reproducing kernel method for singular multi-point boundary value problems. Math. Sc. 2012, 6:16, Article ID doi:10.1186/2251-7456-6-16

[15] P. K. Pandey. The Numerical Solution of Third Order Differential Equation Containing the First Derivative. Neural Parallel & Scientific Comp., 13(2005), 297-304.

[16] M. K. Jain, S. R. K. Iyenger and R. K. Jain. Numerical Methods for Scientific and Engineering Computation (2/e). Willey Eastern Limited, New Delhi, 1987.

[17] R. S. Varga. Matrix Iterative Analysis, Second Revised and Expanded Edition. Springer-Verlag, Heidelberg, 2000.

[18] R. A. Horn and C. R.Johnson. Matrix Analysis. Cambridge University Press, New York, NY 10011, USA, 1990.

[19] A. Ghazala, T. Muhammad, S. S. Shahid and U. R. Hamood. Solution of a Linear Third order Multi-Point Boundary Value Problem using RKM. British J. Math. Comp. Sci., 3(2)(2013),

180-194.

[2] R. P. Agarwal. Boundary Value Problems for Higher Order Differential Equations. World Scientific, Singapore, 1986.

[3] C. P. Gupta and V. Lakshmikantham. Existence and uniqueness theorems for a third-order three point boundary value problem. Nonlinear Anal.: Theory, Meth. Appl., 16(11)(1991), 949-957.

[4] K. N. Murty and Y. S. Rao. A theory for existence and uniqueness of solutions to three-point boundary value problems. J. Math. Anal. Appl., 167(1)(1992), 43-48.

[5] M. Feckan. Singularly perturbed higher order boundary value problems. J. Diff. Equat., 111(1)(1994), 79-102.

[6] J. Henderson and K. R. Prasad. Existence and uniqueness of solutions of three-point boundary value problems on time scales. Nonlin. Studies, 8(2001), 1-12.

[7] E. A. Al-Said. Numerical solutions for system of third-order boundary value problems. Int. J. Comp. Math., 78(1)(2001), 111-121.

[8] A. Khan and T. Aziz. The Numerical Solution of Third- Order Boundary-Value Problems Using Quintic Splines. Appl. Math. Comp., 137(2-3)(2003), 253-260.

[9] P. K. Pandey. Solving third-order Boundary Value Problems with Quartic Splines. Panday SpringerPlus, 5(1)(2016), 1-10 .

[10] S. Islam, M. A. Khan, I. A. Tirmizi and E. H. Twizell. Non-polynomial splines approach to the solution of a system of third-order boundary-value problems. Appl. Math. Comp.,

168(1)(2005), 152-163.

[11] F. Gao and C. M. Chi. Solving third-order obstacle problems with quartic B-splines. Appl. Math. Comp., 180(1)(2006), 270-274.

[12] Fazal-i-Haq, I. Hussain and A. Ali. A Haar Wavelets Based Numerical Method for Third-order Boundary and Initial Value Problems. World Appl. Sci. J., 13(10)(2011), 2244-2251.

[13] M. A. Noor and A. K. Khalifa. A numerical approach for odd-order obstacle problems. Int. J. Comp. Math., 54(1)(1994), 109-116.

[14] X. Li and B. Wu. Reproducing kernel method for singular multi-point boundary value problems. Math. Sc. 2012, 6:16, Article ID doi:10.1186/2251-7456-6-16

[15] P. K. Pandey. The Numerical Solution of Third Order Differential Equation Containing the First Derivative. Neural Parallel & Scientific Comp., 13(2005), 297-304.

[16] M. K. Jain, S. R. K. Iyenger and R. K. Jain. Numerical Methods for Scientific and Engineering Computation (2/e). Willey Eastern Limited, New Delhi, 1987.

[17] R. S. Varga. Matrix Iterative Analysis, Second Revised and Expanded Edition. Springer-Verlag, Heidelberg, 2000.

[18] R. A. Horn and C. R.Johnson. Matrix Analysis. Cambridge University Press, New York, NY 10011, USA, 1990.

[19] A. Ghazala, T. Muhammad, S. S. Shahid and U. R. Hamood. Solution of a Linear Third order Multi-Point Boundary Value Problem using RKM. British J. Math. Comp. Sci., 3(2)(2013),

180-194.

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## Published

2017-01-01

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