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Mathematics
Volume 13
Issue 19
10.3390/math13193151
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

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

by
Nadiyah Hussain Alharthi
1,
Anurag Kaur
2,*,
Theodore E. Simos
3,* and
Rubayyi T. Alqahtani
1
1
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
2
Department of Mathematics, Thapar Institute of Engineering and Technology (Deemed University), Patiala 147004, Punjab, India
3
Center for Applied Mathematics and Bioinformatics, Gulf University for Science and Technology, West Mishref, Hawally 32093, Kuwait
*
Authors to whom correspondence should be addressed.
Mathematics 2025, 13(19), 3151; https://doi.org/10.3390/math13193151
Submission received: 31 July 2025 / Revised: 14 September 2025 / Accepted: 29 September 2025 / Published: 2 October 2025
Versions Notes

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 numericalsolution; initial value problems (IVPs); block methods; phase fitting

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MDPI and ACS Style

Alharthi, N.H.; Kaur, A.; Simos, T.E.; Alqahtani, R.T. The Implicit Phase-Fitted and Amplification-Fitted Four-Point Block Methods for Oscillatory First-Order Problems. Mathematics 2025, 13, 3151. https://doi.org/10.3390/math13193151

AMA Style

Alharthi NH, Kaur A, Simos TE, Alqahtani RT. The Implicit Phase-Fitted and Amplification-Fitted Four-Point Block Methods for Oscillatory First-Order Problems. Mathematics. 2025; 13(19):3151. https://doi.org/10.3390/math13193151

Chicago/Turabian Style

Alharthi, Nadiyah Hussain, Anurag Kaur, Theodore E. Simos, and Rubayyi T. Alqahtani. 2025. "The Implicit Phase-Fitted and Amplification-Fitted Four-Point Block Methods for Oscillatory First-Order Problems" Mathematics 13, no. 19: 3151. https://doi.org/10.3390/math13193151

APA Style

Alharthi, N. H., Kaur, A., Simos, T. E., & Alqahtani, R. T. (2025). The Implicit Phase-Fitted and Amplification-Fitted Four-Point Block Methods for Oscillatory First-Order Problems. Mathematics, 13(19), 3151. https://doi.org/10.3390/math13193151

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