A Continuous and Optimized Three-Step Third-Derivative Hybrid Block Method for Third-Order IVPs Based on Volterra Integral Equations of the Second Kind

Abstract

This paper presents an optimization of two hybrid points within a three-step method based on Volterra integral equations of the second kind for third-order initial value problems. The scheme is formulated by combining power series expansion with exponentially fitted functions as basic functions to construct a continuous hybrid formulation. This continuous scheme is then discretized into a block method using a three-step framework with appropriately selected hybrid points. The two off-grid points are optimized by equating the leading terms of the truncation error to zero, and the resulting error equations are used to determine the approximate values of the unknown parameters. The theoretical properties of the proposed method are investigated. The order of accuracy and associated error constant are derived, and the method is shown to satisfy the conditions of consistency and zero-stability, thereby establishing convergence. Stability analysis further confirms the robustness of the scheme. Numerical experiments conducted for solving stiff problems demonstrate that the proposed method provides highly accurate approximations with improved computational efficiency compared with existing block and hybrid methods. The results indicate that the method constitutes a reliable and efficient computational tool for solving stiff problems encountered in applied mathematics and scientific computing.