Ağırlıklı iç çarpım ile zaman kesirli problem

Öz

Bu çalışmada, kesirli mertebeden kısmi diferansiyel denklemler içeren homojen başlangıç sınır değer probleminin analitik çözümünü araştırıyoruz. Homojen başlangıç sınır değeri problemi Caputo kesirli mertebe türevini içerdiğinden klasik başlangıç ve sınır koşullarına sahiptir. Değişkenlerine ayırma yöntemi ve L^2 [0,l] de tanımlanan ağırlıklı iç çarpım ile çözüm, bu çalışmada kullanılan Caputo anlamında kesirli türevi içeren bir Sturm-Liouville özdeğer probleminin özfonksiyonlarına göre bir Fourier serisi şeklinde oluşturulmuştur. Fourier serisindeki katsayıları elde etmek için ağırlıklı fonksiyona sahip yeni bir iç çarpım tanımlanmıştır. Çözülen örnek, değişkenlerine ayırma yönteminin kesirli matematik problemleri üzerindeki uygulanabilirliğini ve etkisini göstermektedir.

Anahtar Kelimeler

• Caputo kesirli türev
• dirichlet sınır koşulları
• değişkenlerine ayırma
• spektral method
• ağırlıklı iç çarpım

Time fractional problem via inner product including weighted function

Öz

In this research, we discuss the construction of analytic solution of homogenous initial boundary value problem including PDEs of fractional order. Since homogenous initial boundary value problem involves Caputo fractional order derivative, it has classical initial and boundary conditions. By means of separation of variables method and the inner product defined on L^2 [0,l], the solution is constructed in the form of a Fourier series with respect to the eigenfunctions of a corresponding Sturm-Liouville eigenvalue problem including fractional derivative in Caputo sense used in this study. We defined a new inner product with a weighted function to get coefficients in the Fourier series. Illustrative example presents the applicability and influence of separation of variables method on fractional mathematical problems.

Anahtar Kelimeler

• Caputo fractional derivative
• dirichlet boundary conditions
• separation of variables
• spectral method
• weighted ınner product

Kaynakça

1. Cetinkaya, S., Demir, A. and Kodal Sevindir, H., The analytic solution of initial boundary value problem including time-fractional diffusion equation, Facta Universitatis Ser. Math. Inform, 35, 1, 243-252, (2020).
2. Cetinkaya, S., Demir, A. and Kodal Sevindir, H., The analytic solution of sequential space-time fractional diffusion equation including periodic boundary conditions, Journal of Mathematical Analysis, 11, 1, 17-26, (2020).
3. Cetinkaya, S. and Demir, A., The Analytic Solution of Time-Space Fractional Diffusion Equation via New Inner Product with Weighted Function, Communications in Mathematics and Applications, 10, 4, 865-873, (2019).
4. Cetinkaya, S., Demir, A. and Kodal Sevindir, H., The Analytic Solution of Initial Periodic Boundary Value Problem Including Sequential Time Fractional Diffusion Equation, Communications in Mathematics and Applications, 11, 1, 173-179, (2020).
5. Cetinkaya, S. and Demir, A., Sequential Space Fractional Diffusion Equation's solutions via New Inner Product, Asian-European Journal of Mathematics, (2020), doi:10.1142/S1793557121501217
6. Cetinkaya, S. and Demir, A., Time Fractional Diffusion Equation with Periodic Boundary Conditions, Konuralp Journal of Mathematics, 8, 2, 337-342, (2020).
7. Cetinkaya, S. and Demir, A., Time Fractional Equation with Non-homogenous Dirichlet Boundary Conditions, Sakarya University Journal of Science SAUJS, 24, 6, 1185-1190, (2020).
8. Cetinkaya, S. and Demir, A., Diffusion Equation Including Local Fractional Derivatıve and Non-Homogenous Dirichlet Boundary Conditions, Journal of Scientific Reports-A, 45, 101-110, (2020).

9. Cetinkaya, S. and Demir, A., Equation Including Local Fractional Derivative and Neumann Boundary Conditions, Kocaeli Journal of Science and Engineering, 3, 2, 59-63, (2020).
10. Cetinkaya, S. and Demir, A., Solution of hybrid time fractional diffusion problem via weighted inner product, Journal of Applied Mathematics and Computational Mechanics, 20, 2, 17-27, (2021).
11. Cetinkaya, S. and Demir, A., On Solutions of Hybrid Time Fractional Heat Problem, Bulletin of the Institute of Mathematics Academia Sinica New Series, 16, 1, 49-62, (2021).
12. Cetinkaya, S., Demir, A. and Kodal Sevindir, H., Solution of Space-Time-Fractional Problem by Shehu Variational Iteration Method, Advances in Mathematical Physics, Article ID 5528928, (2021).
13. Cetinkaya, S. and Demir, A., Sequential time space fractional diffusion equation including nonhomogenous initial boundary conditions, Tbilisi Mathematical Journal, 14, 2, 83-91, (2021).
14. Bisquert, J., Interpretation of a fractional diffusion equation with nonconserved probability density in terms of experimental systems with trapping or recombination, Physical Review E, 72, 173-179, (2005).
15. Aguilar, J.F.G. and Hernández, M.M., Space-Time Fractional Diffusion-Advection Equation with Caputo Derivative, Abstract and Applied Analysis, 2014, Article ID 283019, (2014).
16. Nadal, E., Abisset-Chavanne, E., Cueto, E. and Chinesta, F., On the physical interpretation of fractional diffusion, Comptes Rendus Mecanique, 346, 581-589, (2018).
17. Yavuz, M. and Abdeljawad, T., 2020. Nonlinear regularized long-wave models with a new integral transformation applied to the fractional derivative with power and Mittag-Leffler kernel, Adv. Differ. Equ., 367, (2020).
18. Jena, R.M., Chakraverty, S. and Yavuz, M., Two-Hybrid Techniques Coupled with an Integral Transformation for Caputo Time-Fractional Navier Stokes Equations, Progress in Fractional Differentiation and Applications, 6, 3, 201-213, (2020).
19. Yavuz, M., European option pricing models described by fractional operators with classical and generalized Mittag-Leffler kernels, Numerical Methods for Partial Differential Equations, (2020).
20. Evirgen, F., Conformable Fractional Gradient Based Dynamic System for Constrained Optimization Problem, Acta Physica Polonica A, 132, 1066-1069, (2017).
21. Yavuz, M., Characterizations of two different fractional operators without singular kernel, Mathematical Modelling of Natural Phenomena, 14, 3, Article number 302, (2019).
22. Evirgen, F., Yavuz, M., Cattani, C., Atangana, A., Bulut, H., Hammouch, Z., Baskonus, H.M., Mekkaoui, T. and Agoujil, S., An alternative approach for nonlinear optimization problem with Caputo-Fabrizio derivative, In ITM Web of Conferences (Vol. 22, p. 01009), EDP Sciences, (2018).
23. Sarp, Ü., Evirgen, F. and İkikardeş S., Applications of differential transformation method to solve systems of ordinary and partial differential equations, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 20, 2, 135-156, (2018).
24. Yavuz, M. and Özdemir N., Analysis of an epidemic spreading model with exponential decay law, Mathematical Sciences and Applications E-Notes, 8, 1, 142-154, (2020).
25. Yavuz, M. and Özdemir N., Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel, Discrete & Continuous Dynamical Systems-S, 13, 3, 995-1006, (2020).
26. Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J., Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, Holland, (2006).
27. Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, USA, (1999).