Zaman kesirli KdV denklemlerinin residual kuvvet serisi yöntemi ile yaklaşık çözümü

Öz

Bu çalışmada zaman-kesirli Korteveg de Vries (KdV) denkleminin ve modifiye edilmiş halinin rezidual kuvvet serisi metodu (RPSM) ile yaklaşık çözümü elde edilmiştir.  Nümerik sonuçlar verilmiş ve bu sonuçlar tam çözümle nümerik ve grafiksel olarak karşılaştırılmıştır.  Bulunan sonuçlar kullanılan yöntemin gayet başarılı, etkili ve güvenilir olduğunu ortaya koymaktadır.

Anahtar Kelimeler

• Residual kuvvet serisi yöntemi,KdV denklemi,Caputo kesirli türevi,Kesirli kısmi diferansiyel denklem

Approximate solution of time-fractional KdV equations by residual power series method

Öz

In this article, approximate solutions of the time-fractional Korteveg de Vries (KdV) and modified version of it is obtained by using the residual power series method (RPSM). Numerical results are given and then they are compared with the exact solutions both numerically and graphically. The results show that the present method is very successful, effective and reliable.

Anahtar Kelimeler

• Residual power series method,KdV equation,Caputo fractional derivative,Fractional partial differential equation

Kaynakça

1. Ray, S.S., and Bera, R.K., An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Applied Mathematics and Computation, 167(1), 561-571, (2005).
2. Kexue, L., and Jigen, P., Laplace transform and fractional differential equations, Applied Mathematics Letters, 24(12), 2019-2023, (2011).
3. Zurigat, M., Momani, S., Odibat, Z., and Alawneh, A., The homotopy analysis method for handling systems of fractional differential equations, Applied Mathematical Modelling, 34(1), 24-35, (2010).
4. Yildirim, A., An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 10(4), 445-450, (2009).
5. Arikoglu, A., and Ozkol, I., Solution of fractional differential equations by using differential transform method, Chaos, Solitons & Fractals, 34(5), 1473-1481, (2007).
6. Şenol, M., and Dolapci, I.T., On the Perturbation–Iteration Algorithm for fractional differential equations, Journal of King Saud University-Science, 28(1), 69-74, (2016).
7. Kumar, A., Kumar, S., and Singh, M., Residual power series method for fractional Sharma-Tasso-Olever equation, Communications in Numerical Analysis, 1-10, (2016).
8. Jaradat, H.M., Al-Shara, S., Khan, Q.J., Alquran, M., and Al-Khaled, K., Analytical solution of time-fractional Drinfeld-Sokolov-Wilson system using residual power series method, IAENG International Journal of Applied Mathematics, 46(1), 64-70, (2016).

9. Alquran, M., Analytical solutions of fractional foam drainage equation by residual power series method, Mathematical sciences, 8(4), 153-160, (2015).
10. El-Ajou, A., Arqub, O.A., Zhour, Z.A., and Momani, S., New results on fractional power series: theories and applications, Entropy, 15(12), 5305-5323, (2013).
11. Ahmad, R.S., An analytical solution of the fractional Navier-Stokes equation by residual power series method, Zarqa University, Doctoral dissertation, 10-90, (2015).
12. Al Qurashi M.M., Korpinar, Z., Baleanu, D., and Inc, M., A new iterative algorithm on the time-fractional Fisher equation: Residual power series method, Advances in Mechanical Engineering, 9(9) 1–8, (2017).
13. Tchier, F., Inc, M., Korpinar, Z., and Baleanu, D., Solutions of the time fractional reaction–diffusion equations with residual power series method, Advances in Mechanical Engineering, 8(10) 1–10, (2016).
14. Inc, M., Korpinar, Z., Al Qurashi M.M., and Baleanu, D., A new method for approximate solutions of some nonlinear equations: Residual power series method, Advances in Mechanical Engineering, 8(4) 1–7, (2016).
15. Momani, S., Odibat, Z., and Alawneh, A., Variational iteration method for solving the space‐and time‐fractional KdV equation”, Numerical Methods for Partial Differential Equations, 24(1), 262-271, (2008).
16. Abdulaziz, O., Hashim, I., and Ismail, E.S., Approximate analytical solution to fractional modified KdV equations, Mathematical and Computer Modelling, 49(1-2), 136-145, (2009).