Atangana-Baleanu Caputo Anlamında Üçüncü Mertebeden Kesirli Türevli Diferansiyel Denklemler için Implicit Rather Fark Metodu

Abstract

Atangana-Baleanu Caputo (ABC) türevi ile tanımlı üçüncü mertebeden kesirli kısmi diferansiyel denklemin tam çözümü başlangıç ve sınır değerlerine bağlı olarak hesaplandı. Bu denklem için kararlılık kestirimleri verildi. Bu denklem Implicit Rather fark metodu ile çözüldü. Problem için fark şemalarının kararlılığı gösterildi. Bu teknik ABC üçüncü mertebeden kısmi diferansiyel denklemin α=0.001,0.1,0.5,0.99,0.999 için kesirli türev değerlerine karşılık uygulanmıştır. Yaklaşık çözüm, tekniğin doğruluğunu ve etkinliğini onaylar.

Keywords

• Kesirli Diferansiyel Denklemi
• Implicit Rather Fark Şeması
• Kararlılık Kestirimi
• Yaklaşık Çözüm
• Tam Çözüm

Implicit Rather Difference Method for Third Order Differential Equations in the Sense of Atangana-Baleanu Caputo Fractional Derivative

Abstract

The exact solution of the third order partial differential equation defined by Atangana-Baleanu Caputo (ABC) fractional derivative is calculated for depending on the initial and boundary values. Stability estimates are obtained for this equation. Implicit Rather difference schemes are constructed for this problem. The stability of difference schemes for this problem is presented. This technique has been applied by ABC fractional orders α=0.001,0.1,0.5,0.99,0.999. Approximation solution confirms the accuracy and effectiveness of the technique.

Keywords

• Fractional Differential Equation
• Implicit Rather Difference Schemes
• Stability Estimates
• Approximation Solution
• Exact Solution

References

1. Celik, C. (2012). Cranck-Nicholson Method for fractional Equation with the Riezs Fractional Derivative, Journal of Computational Physics,231: 1743- 1750.
2. Gorial, I. I. (2011). Numerical methods for fractional reaction- dispersion equation with Riesz space fractional derivative, Engineering and Technology. Journal, 29, 709-715.
3. Jafari, H. W.,Gejii, V. D. (2006). Solving Linear and Nonlinear Fractional Diffusion and Wave Equations by Adomian Decomposition, Applied Mathematics and Computation, 180, 488-497.
4. Bagley, R.(1983 ).A The oretical Basis for the Application of Fractional Derivative Viscoelasticitiy, Journal of Rheology, 27, 201.
5. Caputo, M.,Fabrizio, M. (2015). A New Definition of Fractional Derivative with out Singular Kernel, Prog. Fract Differ,1(2), 1- 13.
6. Atangana, A. On the New Fractional Derivative and Application to Nonlinear Fisher’s Reaction- Diffusion Equation, Applied Mathematics Computation, 273,948-956.
7. Atangana, A.,Nietto, J. J.(2015). Numerical Solution for of the RLC Circuit Via the Fractional Derivative without Singular Kernel, Advances Mechanical Engineering, 7(10),DOI: 10.1177/1687814015613758.
8. Alqahtani, R. T. (2016). Atangana-Baleanu derivative with fractional order applied to the model of groundwater within an unconfined aquifer, J. Nonlinear Sci. Appl, 9(6), 3647-3654.

9. Atangana, A.,Baleanu, D.(2016). New Fractional Derivatives with Nonlocal and Nonsigular Kernel Theory and Application to Heat Transfer Model, Arxih preprint, Arxiv: 1602.03408.
10. Atangana, A., Baleanu, D. (2016). Caputo- Fabrizio Derivative Applied to Groundwater Flow within Confined Aquifer, Journal of Nonlinear Science and Applications, 9, 3647- 3654.
11. Atangana, A., Gomez- Aguilar, J. F. (2018). A Mathematical Analysis of a Circular Pipe in Rate Type Fluid Via Hankel Transform, Numerical Methods for Partial Differential Equations, 34 (5), 1502- 1523.
12. Modanli, M. (2018).Two Numerical Methods for Fractional Partial Differential Equation with Nonlocal Boundary, Advances in Difference Equations, 19 (1), 333.
13. Atangana, A., Koca, I. (2016). Chaos in a Simple Nonlinear System with Atangana- Baleanu Derivatives with Fractional Order, Chaos, Solitons and Fractals, 89, 447- 454.
14. Akgül, A.,Modanli, M. (2019). Cranck- Nicholson Difference Scheme Method and Reproducing Kernel Function for Third Order Differential Equations in the Sense of Atangana-Baleanu Caputo Derivative, Chaos, Solitons and Fractals, 127, 10-16.
15. Qureshi, S.,Atangana, A. (2019). Mathematical Analysis of Dengue Fever Outbreak by Novel Fractional Operators with Field Data, Physica A: Statistical Mechanics and its Applications, DOI: 10.1016/j.physa.2019.121127.
16. Abro, K. A., Khan, I., Gomez- Aguilar, J. F. (2018). A Mathematical Analysis of a Circular Pipe in Rate Fluid via Hankel Transform, The European Physical Journal Plus, 133 (10), 397.
17. Gomez- Aguilar, J. F., Abro, K. A., Kolebaje, O., Yıldırım, A. (2019). Chaos in a Calcium Oscillation Model via Atangana- Baleanu Operator with Strong Memory, The European Physical Journal Plus, 134 (4), 140.
18. Yusuf, A., Qureshi, S., Inc, M., Aliyu A. I., Baleanu, D., Shaikh A. A. (2018).Two Strain Epidemic Model Involving Fractional Derivative with Mittag- Leffler Kernel, Chaos, 28 (12), 123121.
19. Qureshi, S., Yusuf, A. (2019). Fractional Derivatives Applied to MSEIR problems: Comparative Study with Real Word Data, The European Physical Journal Plus, 134 (4), 171.
20. Qureshi, S., Yusuf, A. (2019). Modeling Chickenpox Disease with Fractional Derivatives: From Caputo Atangana- Baleanu, Chaos, Solitons and Fractals, 122, 111- 118.
21. Durur, H., Yokuş, A. (2020). Analytical solutions of Kolmogorov–Petrovskii–Piskunov equation, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 22(2), 628-636.
22. Modanli, M. (2019). On the numerical solution for third order fractional partial differential equation by difference scheme method. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 9(3), 1-5.