Adaptive Block Hybrid Method with Quasilinearization for the Accurate Solution of General Third-Order IVPs

Abstract

This paper introduces a novel variable-step block hybrid method for the direct solution of general third-order initial value problems. The proposed approach combines quasilinearization with an embedded procedure to enhance accuracy and computational efficiency. Rigorous analysis establishes the convergence and $A$-stability of the proposed method, providing a reliable numerical framework for a broad class of third-order initial value problems. Numerical experiments on a range of benchmark test problems demonstrate that the proposed method achieves high accuracy and competitive computational efficiency compared with several existing approaches. These results highlight the effectiveness and potential of the proposed method as a robust tool for the direct numerical solution of third-order differential equations arising in engineering and related applications.