The Implicit Phase-Fitted and Amplification-Fitted Four-Point Block Methods for Oscillatory First-Order Problems

Nadiyah Hussain Alharthi, Anurag Kaur (), Theodore E. Simos () and Rubayyi T. Alqahtani
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Nadiyah Hussain Alharthi: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
Anurag Kaur: Department of Mathematics, Thapar Institute of Engineering and Technology (Deemed University), Patiala 147004, Punjab, India
Theodore E. Simos: Center for Applied Mathematics and Bioinformatics, Gulf University for Science and Technology, West Mishref, Hawally 32093, Kuwait
Rubayyi T. Alqahtani: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia

Mathematics, 2025, vol. 13, issue 19, 1-44

Abstract: This study introduces a family of implicit four-point block methods for solving first-order initial value problems (IVPs) with oscillatory solutions. In addition to an eighth-order block method, amplification-fitted and phase-fitted implicit block methods are also derived. The methods are implemented in a predictor–corrector framework, where the predictor is a four-point explicit block method constructed with the corresponding properties. A comprehensive stability analysis is carried out to assess the robustness of the proposed approaches. Comparative evaluations with existing methods demonstrate the superior efficiency of the new algorithms. Numerical experiments further confirm that the proposed techniques provide significant improvements over traditional methods, particularly for oscillatory IVPs.

Keywords: numericalsolution; initial value problems (IVPs); block methods; phase fitting (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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