Zaman-Kesirli Kadomtsev- Petviashvili Denkleminin Conformable Türev ile Yaklaşık Çözümleri

Abstract

Bu çalışmada, zaman-kesirli Kadomtsev-Petviashvili (K-P) diferansiyel denklemini çözmek için Rezidual Kuvvet Serisi Metodu (RPSM) kullanılmıştır. Çözüm prosedüründe, kesirli türevler, conformable kesirli türev tanımına göre hesaplanmıştır. Bu model yaklaşık olarak çözülmüş ve elde edilen sonuçlar, sub-equation metodu ile elde edilen tam çözümlerle karşılaştırılmıştır. Sonuçlar, mevcut yöntemin doğru, güvenilir, uygulanmasının basit olduğunu ve doğrusal olmayan kısmi diferansiyel denklemlerin çözümü için iyi bir alternatif olduğunu ortaya koymaktadır.

Approximate Solutions of the Time-Fractional Kadomtsev-Petviashvili Equation with Conformable Derivative

Abstract

In this study, residual power series method, namely RPSM, is applied to solve time-fractional Kadomtsev-Petviashvili (K-P) differential equation. In the solution procedure, the fractional derivatives are explained in the conformable sense. The model is solved approximately and the obtained results are compared with exact solutions obtained by the sub-equation method. The results reveal that the present method is accurate, dependable, simple to apply and a good alternative for seeking solutions of nonlinear fractional partial differential equations.

Keywords

• Kesirli kısmi diferansiyel denklemler,Kesirli Kadomtsev-Petviashvili denklemi,conformable kesirli türev

References

1. Abdeljawad, T. (2015). On conformable fractional calculus. Journal of computational and Applied Mathematics, 279, 57-66.
2. Ahmad, R. S. (2015). An analytical solution of the fractional Navier-Stokes equation by residual power series method, Doctoral dissertation, Zarqa University.
3. Alabsi T.Y. (2017). Solution of Conformable Fractional Navier-Stokes Equation, M.S. thesis, Zarqa University.
4. Alquran, M. (2014). Analytical solutions of fractional foam drainage equation by residual power series method. Mathematical sciences, 8(4), 153-160.
5. Alquran, M. (2015). Analytical solution of time-fractional two-component evolutionary system of order 2 by residual power series method. J. Appl. Anal. Comput, 5(4), 589-599.
6. Arikoglu, A., & Ozkol, I. (2009). Solution of fractional integro-differential equations by using fractional differential transform method. Chaos, Solitons & Fractals, 40(2), 521-529.
7. Arqub, O. A. (2013). Series solution of fuzzy differential equations under strongly generalized differentiability. Journal of Advanced Research in Applied Mathematics, 5(1), 31-52.
8. Abu Arqub, O., El-Ajou, A., Bataineh, A. S., & Hashim, I. (2013). A representation of the exact solution of generalized Lane-Emden equations using a new analytical method. In Abstract and Applied Analysis (Vol. 2013). Hindawi.

9. Atangana, A., Baleanu, D., & Alsaedi, A. (2015). New properties of conformable derivative. Open Mathematics, 13(1).
10. Caputo, M. (1967). Linear models of dissipation whose Q is almost frequency independent—II. Geophysical Journal International, 13(5), 529-539.
11. Das, S. (2011). Functional fractional calculus. Springer Science & Business Media.
12. Diethelm, K. (2010). The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. Springer Science & Business Media.
13. El-Ajou, A., Arqub, O. A., Zhour, Z. A., & Momani, S. (2013). New results on fractional power series: theories and applications. Entropy, 15(12), 5305-5323.
14. El-Sayed, A. M. A., Nour, H. M., Raslan, W. E., & El-Shazly, E. S. (2015). A study of projectile motion in a quadratic resistant medium via fractional differential transform method. Applied Mathematical Modelling, 39(10-11), 2829-2835.
15. Ghazanfari, B., & Veisi, F. (2011). Homotopy analysis method for the fractional nonlinear equations. Journal of King Saud University-Science, 23(4), 389-393.
16. Guo, S., & Mei, L. (2011). The fractional variational iteration method using He's polynomials. Physics Letters A, 375(3), 309-313.
17. Heaviside, O. (2008). Electromagnetic theory (Vol. 3). Cosimo, Inc.
18. Jafari, H., & Daftardar-Gejji, V. (2006). Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition. Applied Mathematics and Computation, 180(2), 488-497.
19. Jaradat, H. M., Al-Shara, S., Khan, Q. J., Alquran, M., & Al-Khaled, K. (2016). Analytical solution of time-fractional Drinfeld-Sokolov-Wilson system using residual power series method. IAENG Int. J. Appl. Math, 46(1), 64-70.
20. Khalil, R., Al Horani, M., Yousef, A., & Sababheh, M. (2014). A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264, 65-70.
21. Kumar, A., Kumar, S., & Singh, M. (2016). Residual power series method for fractional Sharma-Tasso-Olever equation. Communications in Numerical Analysis, 2016(1), 1-10.
22. Carpinteri, A., & Mainardi, F. (Eds.). (2014). Fractals and fractional calculus in continuum mechanics (Vol. 378). Springer.
23. Momani, S., & Shawagfeh, N. (2006). Decomposition method for solving fractional Riccati differential equations. Applied Mathematics and Computation, 182(2), 1083-1092.
24. Momani, S., Odibat, Z., & Erturk, V. S. (2007). Generalized differential transform method for solving a space-and time-fractional diffusion-wave equation. Physics Letters A, 370(5-6), 379-387.
25. Oldham, K. B. (2010). Fractional differential equations in electrochemistry. Advances in Engineering Software, 41(1), 9-12.
26. Podlubny, I. (1999). Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in science and engineering, 198, 261-300.