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Journals / Karamanoğlu Mehmetbey Üniversitesi Mühendislik ve Doğa Bilimleri Dergisi / 2026 / Cilt: 8 Sayı: 1

Solving the Time-Fractional Rosenau-Hyman Equation via a Cubic Hermite Spline Collocation Scheme

Zaman-Kesirli Rosenau-Hyman Denkleminin Kübik Hermite Spline Kolokasyon Yöntemi ile Çözümü

Murat Arı(Karmanoğlu Mehmetbey Üniversitesi)0000-0002-4039-5970
Pages
1–7
Publication Date
2026-06-27
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

The numerical results show that the method provides highly accurate approximations and exhibits good computational efficiency.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this study, a cubic Hermite spline collocation method is proposed for the numerical solution of the time-fractional Rosenau-Hyman equation involving the Caputo fractional derivative.
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Key Findings & Data:The proposed approach employs cubic Hermite spline basis functions to approximate the spatial derivatives, providing a smooth and accurate representation of the solution.
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Conclusion & Impact:The numerical results show that the method provides highly accurate approximations and exhibits good computational efficiency.
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:Kübik Hermite Spline · Kolokasyon metodu · Caputo kesirli türev

Abstract

In this study, a cubic Hermite spline collocation method is proposed for the numerical solution of the time-fractional Rosenau-Hyman equation involving the Caputo fractional derivative. The proposed approach employs cubic Hermite spline basis functions to approximate the spatial derivatives, providing a smooth and accurate representation of the solution. By applying the collocation technique, the governing nonlinear fractional partial differential equation is transformed into a system of algebraic equations that can be solved efficiently. The accuracy and performance of the method are evaluated using the error norms L2 and L∞. Several numerical experiments are presented to demonstrate the reliability and effectiveness of the proposed scheme, and the obtained results are compared with the available exact solutions. The numerical results show that the method provides highly accurate approximations and exhibits good computational efficiency. These findings indicate that the proposed cubic Hermite spline collocation method can be effectively applied to a wider class of nonlinear fractional differential equations.

Özet

Bu çalışmada, Caputo kesirli türevini içeren zaman-kesirli Rosenau-Hyman denkleminin sayısal çözümü için kübik Hermite spline kolokasyon yöntemi önerilmektedir. Önerilen yaklaşımda, uzaysal türevlerin yaklaşık hesaplanması için kübik Hermite spline baz fonksiyonları kullanılarak çözümün düzgün ve yüksek doğrulukta temsil edilmesi sağlanmıştır. Kolokasyon tekniğinin uygulanmasıyla, doğrusal olmayan kesirli kısmi diferansiyel denklem etkin bir şekilde çözülebilen bir cebirsel denklem sistemine dönüştürülmüştür. Yöntemin doğruluğu ve performansı L_2 ve L_∞ hata normları kullanılarak değerlendirilmiştir. Önerilen yöntemin güvenilirliğini ve etkinliğini göstermek amacıyla çeşitli sayısal deneyler sunulmuş ve elde edilen sonuçlar mevcut analitik çözümler ile karşılaştırılmıştır. Sayısal sonuçlar, yöntemin oldukça yüksek doğruluk sağladığını ve iyi bir hesaplama verimliliği sergilediğini göstermektedir. Bu bulgular, önerilen kübik Hermite spline kolokasyon yönteminin benzer doğrusal olmayan kesirli diferansiyel denklemlerin çözümünde de etkili bir şekilde uygulanabileceğini göstermektedir.

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Cite This Article

Arı, M. (2026). Solving the Time-Fractional Rosenau-Hyman Equation via a Cubic Hermite Spline Collocation Scheme. Karamanoğlu Mehmetbey Üniversitesi Mühendislik ve Doğa Bilimleri Dergisi, Cilt: 8 Sayı: 1, 1–7. https://doi.org/10.55213/kmujens.1882854

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