A Combining Method for solution of nonlinear boundary value problems Zekeriya Girgin a , Yasin Yilmaz a,⇑ , Ersin Demir b a Pamukkale University, Faculty of Engineering, Department of Mechanical Engineering, 20070 Kinikli, Denizli, Turkiye b Pamukkale University, Faculty of Technology, Department of Mechatronics Engineering, 20070 Kinikli, Denizli, Turkiye article info Keywords: Combining Method Differential Quadrature Integral Quadrature Boundary value problem Simulation technique Stiff equation abstract Modeling-based simulation techniques and numerical methods such as Differential Quadrature Method and Integral Quadrature Method are widely used for solution of ordinary differential equations. However simulation techniques do not allow to impose boundary conditions, and both Differential Quadrature Method and Integral Quadrature Method have some deficiencies in applying multiple boundary conditions at the same loca- tion. Moreover they are not convenient for the solution of non-linear Ordinary Differential Equations without using any linearization process such as Newton–Raphson technique and Frechet derivative which requires an iterative procedure increasing the time needed for solution. In this study, modeling-based simulation technique is combined with Differential Quadrature Method and/or Integral Quadrature Method to eliminate the aforementioned deficiencies. The proposed method is applied to four different nonlinear boundary value problems including a coupled nonlinear system, second and fourth order nonlinear bound- ary value problems and a stiff nonlinear ordinary differential equation. The numerical results obtained using Combining Method are compared with existing exact results and/or results of other methods. Comparison of the results show the potential of Combin- ing Method for solution of nonlinear boundary value problems with high efficiency and accuracy, and less computational work. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction In seeking a more efficient method using just a few grid points to obtain accurate numerical results, the Differential Quad- rature Method (DQM) was proposed by Bellman and Casti [1], Bellman et al. [2]. Bellman et al. [2] suggested two approaches to determine the weighting coefficients of the first order derivative. In the first approach, it is very difficult to obtain the weighting coefficients for a large number of grid points. The second one uses a simple algebraic formulation, but with the coordinates of grid points chosen as the roots of the shifted Legendre polynomial. In order to deal with these drawbacks, Quan and Chang [3] used Lagrange interpolation polynomials as test functions, and then obtained explicit formulations to determine the weighting coefficients of the first and second order derivatives. Later, Shu and Richards [4] derived an iden- tical formula for the weighting coefficients of the first order derivative and provided a recurrence relationship which can generate weighting coefficients for any second and high order derivatives from the first order derivative weighting coefficients. The method is so called as Generalized Differential Quadrature Method (GDQM). http://dx.doi.org/10.1016/j.amc.2014.01.133 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail address: yyilmaz@pau.edu.tr (Y. Yilmaz). Applied Mathematics and Computation 232 (2014) 1037–1045 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc