EFFECTIVE METHOD OF SOLVING ONE-THIRD THIRD-ORDER IVPS WITH TWO OFF-GRID POINTS ORDINARY DIFFERENTIAL EQUATIONS

Abstract

This article explores the effectiveness of a technique for solving one-third third-order initial value problems (IVPs) in ordinary differential equations by incorporating two off-grid points. A truncated power series was used as the basis function to construct the approximate solution. The resulting multistep methods were obtained through interpolation at selected grid points, while the third derivatives of the approximate solution were enforced at non-interpolating points through collocation. The system of linear equations arising from the collocation and interpolation of the approximate solution was solved to obtain the unknown coefficients. These computed values were then substituted back into the approximate expression to produce a continuous linear multistep method, from which the proposed scheme was derived through extrapolation. The method’s performance was tested on nonlinear problems involving general third-order ordinary differential equations, and the results showed good agreement with several existing methods documented in the literature.