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

Öz

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.

Anahtar Kelimeler

• Cubic Hermite Spline
• Collocation method
• Caputo fractional derivative
• Rosenau Hyman equation.

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

Öz

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.

Anahtar Kelimeler

• Kübik Hermite Spline
• Kolokasyon metodu
• Caputo kesirli türev
• Rosenau Hyman denklemi

Kaynakça

1. Atangana A, Baleanu D (2016). New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Thermal Science, 20(2):763-769.
2. Attia N, Akgül A, Seba D, Nour A (2020). An efficient numerical technique for a biological population model of fractional order. Chaos, Solitons & Fractals, 141:110349.
3. Baleanu D, Jajarmi A, Mohammadi H, Rezapour S (2020). A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative. Chaos, Solitons & Fractals, 134:109705.
4. Caputo M (1967). Linear model of dissipation whose Q is almost frequency independent - II. Geophys. J. R. Astron. Soc., 13:529-539.
5. Clarkson PA, Mansfield EL, Priestley TJ (1997). Symmetries of a class of nonlinear third-order partial differential equations, Math. Comput. Modell. 25(8–9) 195–212.
6. Cinar M, Secer A, Bayram M (2021). An application of Genocchi wavelets for solving the fractional Rosenau-Hyman equation. Alexandria Engineering Journal, 60(6):5331-5340.
7. Dehghana M, Manafian J, Saadatmandi A (2012). Application of semi-analytical methods for solving the Rosenau-Hyman equation arising in the pattern formation in liquid drops. Int. J. Numer. Meth. Heat Fluid Flow, 22(6):777-790.
8. Douglas J, Dupont T (1973). A finite element collocation method for quasilinear parabolic equations. Math. Comput., 27(121):17-28.

9. El-Sayed AMA, Rida SZ, Arafa AAM (2009). Exact solutions of fractional-order biological population model. Communications in Theoretical Physics, 52(6):992.
10. Ganaie IA, Gupta B, Parumasur N, Singh P, Kukreja VK (2013). Asymptotic convergence of cubic Hermite collocation method for parabolic partial differential equation. Applied Mathematics and Computation, 220:560-567.
11. Ganaie IA, Kukreja VK (2014). Numerical solution of Burgers' equation by cubic Hermite collocation method. Applied Mathematics and Computation, 237:571-581.
12. Ganaie IA, Arora S, Kukreja VK (2016). Cubic Hermite collocation solution of Kuramoto-Sivashinsky equation. International Journal of Computer Mathematics, 93(1):223-235.
13. Garralón J, Villatoro FR (2012). Dissipative perturbations for the Rosenau-Hyman equation. Commun. Nonlinear Sci. Numer. Simul., 17(12):4642-4648.
14. Iyiola OS, Ojo GO, Mmaduabuchi O (2016). The fractional Rosenau-Hyman model and its approximate solution. Alexandria Eng. J., 55(2):1655-1659. Karay M, Sabir Z, Akkilic AN, Bulut H (2025). A stochastic neural network for the numerical solutions of the nonlinear fractional order Zika virus model using reservoirs and human motion. Computational Biology and Chemistry, 108629.
15. Kaur SP, Mittal AK, Kukreja VK, Kaundal A, Parumasur N, Singh P (2021). Analysis of a linear and nonlinear model for diffusion-dispersion phenomena of pulp washing by using quintic Hermite interpolation polynomials. Afrika Matematika, 32:997-1019.
16. Kumari A, Kukreja VK (2021a). Robust septic Hermite collocation technique for singularly perturbed generalized Hodgkin-Huxley equation. International Journal of Computer Mathematics, 2021.
17. Kumari A, Kukreja VK (2021b). Septic Hermite collocation method for the numerical solution of Benjamin-Bona-Mahony-Burgers equation. Journal of Difference Equations and Applications, 27:1193-1217.
18. Kutluay S, Yağmurlu NM, Karakaş AS (2022). An effective numerical approach based on cubic Hermite B-spline collocation method for solving the 1D heat conduction equation. New Trends in Mathematical Sciences, 10(4):20–31.
19. Kutluay S, Yağmurlu NM, Karakaş AS (2023). A powerful robust cubic Hermite collocation method for the numerical calculations and simulations of the equal width wave equation. arXiv preprint, arXiv:2309.02439.
20. Kutluay S, Yağmurlu NM, Karakaş AS (2024a). A novel perspective for simulations of the modified equal-width wave equation by cubic Hermite B-spline collocation method. Wave Motion, 129:103342.
21. Kutluay S, Yağmurlu M, Karakaş AS (2024b). A robust quintic Hermite collocation method for one-dimensional heat conduction equation. Journal of Mathematical Sciences and Modelling, 7(2):82-89.
22. Kutluay S, Yağmurlu NM, Karakaş AS (2024c). A robust septic Hermite collocation technique for Dirichlet boundary condition heat conduction equation. International Journal of Mathematics and Computer in Engineering, 3(2):253–266.
23. Mittal AK, Ganaie IA, Kukreja VK, Parumasur N, Singh P (2013). Solution of diffusion-dispersion models using a computationally efficient technique of orthogonal collocation on finite elements with cubic Hermite as basis. Computers and Chemical Engineering, 58:203-210.
24. Molliq RY, Noorani MS (2012). Solving the fractional Rosenau-Hyman equation via variational iteration method and homotopy perturbation method. Int. J. Diff. Eqs., 2012.
25. Murio DA (2008). Implicit finite difference approximation for time fractional diffusion equations. Comput. Math. Appl., 56(4):1138-1145.
26. Oldham KB, Spanier J (1974). The fractional calculus. Academic Press, New York - London.
27. Podlubny I (1999). Fractional differential equations. Academic Press, San Diego.
28. Prenter PM (1975). Splines and Variational Methods. John Wiley, New York.
29. Rosenau P, Hyman JM (1993). Compactons: Solitons with finite wavelength. Phys. Rev. Lett., 70(5):564-567.
30. Rubin SG, Graves RA (1975). A Cubic Spline Approximation for Problems in Fluid Mechanics. NASA TR R-436, Washington, DC.
31. Samko SG, Kilbas AA, Marichev OI (1993). Fractional Integrals and Derivatives (Vol. 1). Gordon and Breach Science Publishers, Yverdon, Switzerland.
32. Senol M, Tasbozan O, Kurt A (2019). Comparison of two reliable methods to solve fractional Rosenau-Hyman equation. Math. Methods Appl. Sci., 2019:13-20.
33. Singh J, Kumar D, Swroop R, Kumar S (2018). An efficient computational approach for time-fractional Rosenau-Hyman equation. Neural Comput. Appl., 30(10):3063-3070.