Exact Implementation of Multiple Initial Conditions in the DQ Solution of Higher-Order ODEs

Abstract

The differential quadrature method (DQM) is one of the most elegant and useful approximate methods for solving initial and/or boundary value problems. It is easy to use and also straightforward to implement. However, the conventional DQM is well-known to have some difficulty in implementing multiple initial and/or boundary conditions at a given discrete point. To overcome this difficulty, this paper presents a simple and accurate differential quadrature methodology in which the higher-order initial conditions are exactly implemented. The proposed methodology is very elegant and uses a set of simple polynomials with a simple transformation to incorporate the higher-order initial conditions at the initial discrete time point. The order of accuracy of the proposed method for solving an <em>r</em>th order ordinary differential equation is “<em>m</em> + <em>r</em> – 1," where <em>m</em> being the number of discrete time points. This is better than the accuracy of the CBCGE (direct Coupling the Boundary/initial Conditions with the discrete Governing Equations) and MWCM (Modifying Weighting Coefficient Matrices) approaches whose order is in general “<em>m</em> – 1." Some test problems are also provided to highlight the superiority of the proposed method over the CBCGE and MWCM approaches.