Quartic spline solution of a third order singularly perturbed boundary value problem

Singularly perturbed boundary value problems are solved using various techniques. The spline of degree four is used for the approximate solution of a third order self adjoint singularly perturbed boundary value problem. Convergence analysis is given and the method is proved to be second order convergent. Two examples numerically illustrate the method. References M. Cui and F. Geng, A computational method for solving third order singularly perturbed boundary value problems, Applied Mathematics and Computation, Vol. 198, (2008) pp. 896--903, doi:10.1016/j.amc.2007.09.023 Zengji Du, Singularly perturbed boundary value problem for nonlinear systems, Applied Mathematics and Computation, Vol. 189, (2007) pp. 869--877, doi:10.1016/j.amc.2006.11.167 F. A. Howers, Singular perturbation and differential inequalities, Memoris of the American Mathematical Socity, Providence, Rhode Island, Vol. 168, (1976). Mohan K. Kadalbajoo and Kailash C. Patidar, Numerical soution of singularly perturbed two-point boundary value problem by spline in tension, Applied Mathematics and Computation, Vol. 131, (2002) pp. 299--320, doi:10.1016/S0096-3003(01)00146-1 Petio Kelevedjiev, Existence of positive solutions to a singular second order boundary value problem, Nonlinear Analysis:Theory, Methods and Applications, Vol 50 (8), (2002) pp. 1107--1118, doi:10.1016/S0362-546X(01)00803-3 Arshad Khan, Islam Khan and Tariq Aziz, Sextic spline solution of singularly perturbed boundary value problem, Applied Mathematics and Computation, Vol. 181, (2006) pp. 432--439, doi:10.1016/j.amc.2005.12.059 Manoj Kumar, A fourth order finite differeence method for a class of singular two point boundary value problems, Applied Mathematics and Computation, Vol. 133, (2002) pp. 539--545, doi:10.1016/S0096-3003(01)00255-7 R. K. Mohanty and Navnit Jha, A class of variable mesh spline in compression methods for singularly perturberd two point singular boundary value problem, Applied Mathematics and Computation, Vol. 168, (2005) pp. 704--716, doi:10.1016/j.amc.2004.09.049 J. Rashidinia, R. Mohammadi and M. Ghasemi, Cubic spline solution of singularly perturbed boundary value problem with significant first derivatives, Applied Mathematics and Computation, Vol. 190, (2007) pp. 1762--1766, doi:10.1016/j.amc.2007.02.050, H. G. Roos, M. Stynes and L. Tobiska, Numerical methods for singularly perturbed difference equation, Springer verlag, (1996). Lin Su-rang, Tian Gen-bao and Lin Zong-chi, Singular perturbation of boundary value problem for Quasilinear third order ordinary differential equations involving two small parameters, Applied Mathematics and Mechanics, Vol. 22 (2), (2001) pp. 229--236, doi:10.1023/A:1015553219376 Muhammad I Syam and Basem S. Attili, Numerical solution of singularly perturbed fifth order two point boundary value problem, Applied Mathematics and Computation, Vol. 170 (2005), pp. 1085--1094, doi:10.1016/j.amc.2005.01.003 Ikram A. Tirmizi, Fazal-i-Haq and Siraj-ul-islam, Non-polynomial spline solution of singularly perturbed boundary-value problems, Applied Mathematics and Computation, Vol. 196, (2008) pp. 6--16, DOI: 10.1016/j.amc.2007.05.029,. Wenyan Wang, Minggen Cui and Bo Han, A new method for solving a class of singular two-point boundary value problem, Applied Mathematics and Computation, Vol. 206, (2008) pp. 721--727, doi:10.1016/j.amc.2008.09.01