An A-Stable Uniformly Order Seven Block Hybrid Method for Solving Nonlinear Initial Value Problems

Abstract

This study presents the development of a new A-stable uniformly order seven block hybrid method for solving Nonlinear Initial Value Problems (NIVPs) in Ordinary Differential Equations (ODEs). Traditional numerical methods, including Euler’s method and Runge-Kutta methods, often struggle with nonlinear problems due to stability and computational inefficiencies, especially when dealing with stiff equations. To address this limitation, the proposed method integrates the advantages of block hybrid techniques, ensuring A-stability and uniform order seven, which enhances both accuracy and computational efficiency. The formulation of the method involves applying a one-step linear multistep approach combined with interpolation and collocation techniques. Through extensive analysis, the method is shown to satisfy essential numerical properties such as consistency, zero-stability, and convergence. Numerical experiments demonstrate that the new method outperforms existing methods in terms of accuracy and computational cost, particularly for stiff nonlinear problems. The method’s performance is validated by applying it to various test cases, yielding results consistent with previous studies and showing significant improvements in error reduction.

Keywords

• A-stable
• Block Hybrid Method
• Nonlinear Initial Value Problems
• Numerical Methods
• Stiffness
• Accuracy
• Computational Efficiency

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