Volume 6 • Issue 2 • PP: 17-28 • 2023
On Novel Spline Function Method for Solving Third-Order Boundary Value Problems
View PDF Abstract
In this paper, a numerical method is suggested for solving general a nonlinear third order boundary value problem (BVP). In this method, the given nonlinear third-order BVP will be transformed into two third-order initial value problems (IVPs), then spline function approximations are applied to both two IVP for finding the Spline solution and its derivatives up to third order of the given BVP. The study shows that the spline solution of the BVP is existent and unique, and the convergence order of the spline method is fourth with a local truncation error . The presented algorithm is designed for solving a general BVP, where it is applied to some types of nonlinear third-order differential equations. Comparisons of the results obtained by spline method with other methods show the efficiency and highly accurate of the proposed method.
Keywords
References
[1] CAGLAR, H.N., CAGLAR, S.H. and TWIZELL, E.H., The numerical solution of thirdorder boundary-value problems with fourth-degree b spline functions, Intern. J. Computer Math. 71, (1999) 373-381.
[2] KHAN, A.; AZIZ, T., The numerical solution of third-order boundary value problems using quintic splines, Intern. J. Computer Math., Vol. 79(9), (2002) 1025-1031.
[3] KHAN, A.; AZIZ, T., The numerical solution of third-order boundary value problems using quintic splines, Applied Mathematics and Computation 137 (2003) 253-260.
[4] TIRMIZI I. A., E. H. TWIZELL.; SIRAJ-UL-ISLAM, A numerical method for thirdorder non-linear boundary-value problems in engineering, Intern. J. Computer Math., Vol. 82, No. 1, (2005), 103-109.
[5] SIRAJ-UL-ISLAM and TIRMIZI I. A., A smooth approximation for the solution of special non-linear third-order boundary-value problems based on non-polynomial splines, Intern. J. Computer Math., Vol. 83, No. 4, (2006), 397- 407.
[6] GROSSINHO, M.R., F.M. MINH.; A.I. SANTOS, A third-order boundary value problem with one-sided Nagumo condition, Nonlinear Analysis 63, (2005) e247 - e256.
[7] MAHMOUD, S. M., Quintic spline methods with three collocation points for solving third-order boundary-value problems, Mu'tah Journal for Research and Studies Vol. (26) No (1) ,119-139, (2011).
[8] ARCINIEGA, A., Shooting methods for numerical solution of nonlinear stochastic boundary-value problems, Stochastic Analysis and Applications, Vol. 25, (2007) 187-200.
[9] VALARMATHI, S.; N. RAMANUJAM, A computational method for solving boundary value problems for third-order singularly perturbed ordinary differential equations, Applied Mathematics and Computation, 129 (2002) 345-373.
[10] LIANG, S.; JEFFREY, D.J. An analytical approach for solving nonlinear boundary value problems in finite domains, Department of Applied Mathematics, University of Western Ontario, London, ON, Canada, N6A 5B7, (2010).
[11] USMANI, R.A.; SAKAI, M., Quartic spline solution for two point boundary value problems involving third-order differential equation. Journal of Mathematical and Physical Sciences, 18(4), (1984) 365–380.
[12] BHRAWY A. H.; ABD-ELHAMEED W. M., New algorithm for the numerical solutions of nonlinear third-order differential equations using jacobi-gauss collocation method, Math. Problems in Engineering, Vol. 2011, (2011),1-14.
Cite This Article
Choose your preferred format