Kesirli Diferansiyel Sistemler için Verilen Bazı Sınır Koşullarında Çözümlerin Varlığı ve Tekliği

Year 2021, Issue: 28, 1486 - 1491, 30.11.2021

Münevver Tuz

https://doi.org/10.31590/ejosat.1021579

Abstract

Bu makalede, periyodik problemlerin kesirli durumu tartışılmaktadır. Zaman kesirli ısı denklemi göz önünde bulundurularak, periyodik ve anti-periyodik sınır koşulları ile ters problemler oluşturulmuştur. Bu problemler için varlık ve teklik sonuçlarını elde etmek için Fourier yöntemi kullanılmıştır. Periyodik bir fonksiyonun kesirli türevi gerçek eksen boyunca analiz edilmiş ve lineer sistemlerin kesirler durumundaki periyodik davranışı araştırılmıştır.

Keywords

Periyodik, Anti-periyodik, Kesirli Türev, Fourier Metodu, Lineer Sistem

Existence and Uniqueness of Solutions in Some Boundary Conditions for Fractional Differential Systems

Abstract

In this article, fractional case of periodic problems is discussed. Considering the time fractional heat equation, inverse problems with periodic and anti-periodic boundary conditions were created. For these problems, the Fourier method was used to obtain existence and uniqueness results. The fractional derivative of a periodic function was analyzed along the real axis, and the periodic behavior of linear systems in case of fractions was investigated.

Keywords

Periodic, Anti-periodic, Fractional Derivative, Fourier Method, Linear System

References

• Al-Mdallal, Q. M., (2009). An Efficient Method for Solving Fractional Sturm–Liouville Problems, Chaos Solitons Fractals, Vol.40, 183–189. Boroomand, A., Menhaj, M.B., (2009). Fractional-Order Hopfield Neural Networks, Advances in Neuro-Information Processing, 5506, Part I, 883-890.
• Kilbas, A. A., Srivastava H. M., Trujillo, J. J., (2006). Theory and Application of Fractional Differential Equations, Elsevier Science, Amsterdam.
• Delavari, H. et al., (2012). Stability Analysis of Caputo Fractional-Order Non-Linear Systems Revisited, Non-Linear Dynamics, Vol. 67, No. 4, 2433-2439.
• Bouchaud, J. P., Georges, A., (1990). Anomalous Diffusion in Disordered Media: Statistical Mechanisms, Models and Physical Applications, Phys. Rep., Vol. 195(4-5), 127–293.
• Klimek, M., Agrawal, O. P., (2013). Fractional Sturm–Liouville Problem, Computers and Mathematics with Applications, Vol. 66, No. 5, 795–812.
• Podlubny, I., (1999). Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, Vol. 198, San Diego, California, USA.
• Zayernouri, M., Karniadakis, G. E., Fractional Sturm-Liouville Eigen-Problems, Theory and Numerical Approximation, Journal of Computational Physics, Vol. 252, 495–517.
• Metzler, R., Klafter, J., (2002). The Random Walk’s Guide to Anomalous Diffusion a Fractional Dynamics Approach, Phys. Rep.,Vol. 339, No. 1, 67–90.
• Bochner, S., Chandrasekharan, K., (1949). Fourier Transforms, Princeton University Press.
• Blaszczyk, T., Ciesielski, M., (2014). Numerical Solution of Fractional Sturm-Liouville Equation in Integral Form, Fractional Calculus and Applied Analysis, Vol.17, No. 2, 307–320.
• Mainardi, F., (2010). Fractional Calculus and Waves in Linear Viscoelasticity, An Introduction to Mathematical Models, World Scientific, Singapore, 368, 2010.
• Ibrahim, R., Momani, S., (2007). On the Existence and Uniqueness of Solutions of a Class of Fractional Differential Equations, J. Math. Anal. Appl., Vol. 334, No. 1, 1–10.
• Tapdıgoglu, R., (2019). Problemes inverses pour des equations differentielles aux derives fractionnaires, Docteur de thesis, Universite de la Rochelle, France.
• Samko, S. G., (2003). Fractional Weyl–Riesz Integro Differentiation of Periodic Functions of Two Variables Via the Periodization of the Riesz Kernel, Appl. Anal. J., Vol. 82, No. 3, 269–299.
• Pooseh, S., Rodrigues, H. S., Torres, D. F. M., (2011). Fractional Derivatives in Dengue Epidemics, AIP Conf. Proc., Vol. 1389, 739–742.