Development and stability analysis of a block hybrid collocation method for numerical solution of the third-order quasi-linear ordinary differential equations

This paper presents the development and stability analysis of a block hybrid collocation method for the direct numerical solution of third-order quasi-linear ordinary differential equations subject to standard initial conditions. The proposed method is formulated without reducing the original problem to an equivalent system of first-order equations, thereby avoiding the associated increase in computational cost and implementation complexity. A continuous hybrid scheme is constructed using three interpolation points and four off-step collocation points, allowing the simultaneous approximation of the solution at multiple grid points within a block. Collocation conditions are imposed on the third derivative of the approximating polynomial to derive a unified block formulation. Theoretical properties of the method, including consistency, order of accuracy, zero-stability, and convergence, are rigorously analyzed. Numerical experiments involving several benchmark problems with known exact solutions are conducted to assess the method’s performance. The results demonstrate that higher accuracy is achieved with significantly fewer computational steps than those required by existing methods. These findings confirm that the block hybrid collocation method provides an efficient, stable, and reliable numerical framework for solving third-order quasi-linear ordinary differential equations.