Semi-analytical solutions of conformable fractional partial differential equations

Öz

In this study, semi-analytical solutions of conformable fractional partial differential equations, which are expressed with the definition of conformable fractional derivative, which has recently entered the literature, have been investigated. The method used to reach these solutions is the form of the differential transform method, which exists in the literature, adapted to conformable fractional partial differential equations. This derivative-based method aims to find the coefficients of the solution in the Taylor series form. The performance of the method has been tested on a conformable fractional partial equation and a conformable fractional partial equation system. The general operation of the method is to find the transformed form of each term in the differential equation according to the method. Therefore, in the first example, the conformable fractional partial differential equation was designed as an inverse problem with the desire to be an equation containing less common terms in the literature. The conformable fractional partial differential equation system in the other example is an equation system that models a known biochemical process in the literature and has a strong nonlinear term. The results obtained showed that the method stands out due to its ease of application and coding.

Anahtar Kelimeler

• The conformable fractional derivative
• fractional power series
• conformable fractional partial differential equations
• reaction-diffusion Brusselator system
• conformable fractional partial differential transform method

Uyumlu kesirli kısmi diferensiyel denklemlerin yarı-analitik çözümleri

Öz

Bu çalışmada, literatüre yakın zamanda giren uyumlu kesirli türev tanımı ile ifade edilen uyumlu kesirli kısmi diferensiyel denklemlerin, yarı-analitik çözümleri araştırılmıştır. Bu çözümlere ulaşmak için kullanılan yöntem, literatürde var olan diferensiyel dönüşüm yönteminin, uyumlu kesirli kısmi diferensiyel denklemlere uyarlanmış formudur. Türev tabanlı olan bu yöntem Taylor seri formundaki çözümün katsayılarını bulmayı hedeflemektedir. Yöntemin performansı, uyumlu kesirli kısmi bir denklem ve uyumlu kesirli kısmi bir denklem sistemi üzerinde test edilmiştir. Yöntemin genel işleyişi, diferensiyel denklemdeki her bir terimin yönteme göre dönüşmüş formunu bulmak şeklindedir. Bu nedenle ilk örnekte, uyumlu kesirli kısmi diferensiyel denklemin, literatürde daha az rastlanan terimleri içeren bir denklem olması isteğiyle bir ters problem şeklinde kurgulanmıştır. Diğer örnekteki uyumlu kesirli kısmi diferensiyel denklem sistemi ise literatürde bilinen bir biyokimyasal süreci modelleyen ve güçlü lineer olmayan terime sahip bir denklem sistemidir. Elde edilen sonuçlar, yöntemin uygulama ve kodlama kolaylığı ile ön plana çıktığını göstermiştir.

Anahtar Kelimeler

• Uyumlu kesirli türev
• kesirli kuvvet serisi
• uyumlu kesirli kısmi diferensiyel denklemler
• reaksiyon-difüzyon Brusselator sistem
• uyumlu kesirli kısmi diferensiyel dönüşüm yöntemi.

Kaynakça

1. Oldham, K.B., Spanier, J., The Fractional Calculus, Academic Press, San Diego, (1974).
2. Miller, K. S., An introduction to the fractional calculus and fractional differential equations, John Willey & Sons. (1993).
3. Podlubny, I., Fractional differential equations, mathematics in science and engineering, Academic Press, San Diego, USA, (1999).
4. Arditi, R., Ginzburg, L. R., Coupling in predator-prey dynamics: ratio-dependence, Journal of theoretical biology, 139(3), 311-326, (1989).
5. Güngör, H., A novel study on Caputo-Fabrizio fractional Cahn-Allen equation, Alexandria Engineering Journal, 119, 1-7, (2025).
6. Khalil, R., Al Horani, M., Yousef, A., Sababheh, M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264: 65-70, (2014).
7. Abdeljawad, T., On conformable fractional calculus, Journal of Computational and Applied Mathematics, 279, 57-66, (2015).
8. Chung, W. S., Fractional Newton mechanics with conformable fractional derivative, Journal of Computational and Applied Mathematics, 290, 150-158, (2015).

9. Arafa, A., A different approach for conformable fractional biochemical reaction-diffusion models, Applied Mathematics-A Journal of Chinese Universities, 35, 452-467, (2020).
10. Güngör, H., The Efficient Method to Solve the Conformable Time Fractional Benney Equation, Journal of Mathematics, 2024(1), Article ID: 676521, 12 pages, (2024).
11. Zhou, J. K., Differential Transformation and its Application for Electrical Circutits, Huarjung University Press, Wuuhahn China, (1986).
12. Ünal, E., Gökdoğan, A., Solution of conformable fractional ordinary differential equations via differential transform method, Optik, 128, 264-273, (2017).
13. Thabet, H., Kendre, S., Analytical solutions for conformable space-time fractional partial differential equations via fractional differential transform, Chaos, Solitons & Fractals, 109, 238-245, (2018).
14. Keskin, Y., Oturanc, G., Reduced differential transform method for partial differential equations, International Journal of Nonlinear Sciences and Numerical Simulation, 10(6), 741-750, (2009).
15. Jang, B., Solving linear and nonlinear initial value problems by the projected differential transform method, Computer Physics Communications, 181(5), 848-854, (2010).
16. Eslami, M., Taleghani, S.A., Differential transform method for conformable fractional partial differential equations, Iranian Journal of Numerical Analysis and Optimization, 9(2), 17-29, (2019).
17. Thabet, H., Kendre, S., Peters, J., Analytical solutions for nonlinear systems of conformable space-time fractional partial differential equations via generalized fractional differential transform, Vietnam Journal of Mathematics, 47, 487-507, (2019).
18. Fırat, Ö., Özkan, O., New Two Dimensional Differential Transform Method with Conformable Derivative for Fractional Partial Differential Equations, Journal of Inequalities & Special Functions, 13(4), 12-24, (2022).
19. Odibat, Z. M., Kumar, S., Shawagfeh, N., Alsaedi, A., Hayat, T., A study on the convergence conditions of generalized differential transform method, Mathematical Methods in the Applied Sciences, 40(1), 40-48, (2017).
20. Singh, J., Rashidi, M. M., Kumar, D., Swroop, R., A fractional model of a dynamical Brusselator reaction-diffusion system arising in triple collision and enzymatic reactions, Nonlinear Engineering, 5(4), 277-285, (2016).
21. Zellal, M., Belghaba, K., Analytical Study of Fractional Reaction-Diffusion Brusselator System, Çankaya University Journal of Science and Engineering, 19(2), 113-123, (2022).
22. Nisar, K. S., Jagatheeshwari, R., Ravichandran, C., Veeresha, P., An effective analytical method for fractional Brusselator reaction-diffusion system, Mathematical Methods in the Applied Sciences, 46(18), 18749-18758, (2023).