A Combining Method for solution of nonlinear boundary value problems

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 location. 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 boundary 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 Combining Method for solution of nonlinear boundary value problems with high efficiency and accuracy, and less computational work.