Truncation and convergence dynamics: KdV Burgers model in the sense of Caputo derivative
DOI:
https://doi.org/10.5269/bspm.47472Abstract
This study examines the time fractional KdV Burgers equation with the initial conditions by using the extended result on Caputo formula, finite difference method (FDM). For this reason, various fractional differential operators are defined and analyzed. In order to check the stability of the numerical scheme, the Fourier-von Neumann technique is used. By presenting an example of KdV Burgers equation above mentioned issues are discussed and numerical solutions of the error estimates have been found for the FDM. For the errors in $L_2$ and $L_\infty$ the method accuracy has been controlled. Moreover, the obtained results have been compared with the exact solution for different cases of non-integer order and the behavior of the potentials u is presented as a graph. The numerical results have been shown in tables.
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2. A. Yokus, M. Tuz, An application of a new version of (G′ /G)-expansion method. In AIP Conference Proceedings (Vol. 1798, No. 1, p. 020165) AIP Publishing (2017). https://doi.org/10.1063/1.4972757
3. A. Yokus, Comparison of Caputo and conformable derivatives for time-fractional Korteweg-de Vries equation via the finite difference method. International Journal of Modern Physics B, 32(29), 1850365 (2018). https://doi.org/10.1142/S0217979218503654
4. F. Dusunceli,, E. Celik, Numerical solution for high-order linear complex differential equations with variable coefficients. Numerical Methods for Partial Differential Equations, 34(5), 1645-1658 (2018). https://doi.org/10.1002/num.22222
5. H. Bulut, A. T. Sulaiman, H. M. Baskonus, A. A. Sandulyak, New solitary and optical wave structures to the (1+1)-dimensional combined KdV-mKdV equation. Optik, 135, 327-336 (2017). https://doi.org/10.1016/j.ijleo.2017.01.071
6. A. Yokus, H. Bulut, Numerical simulation of KdV equation by finite difference method. Indian Journal of Physics, 92(12), 1571-1575 (2018). https://doi.org/10.1007/s12648-018-1207-3
7. H. Bulut, F. B. M. Belgacem, H. M. Baskonus, Some New Analytical Solutions for the Nonlinear Time-Fractional KdV-Burgers-Kuramoto Equation. parameters, 2, 3 (2015).
8. A. Yokus, Numerical Solutions of Time Fractional Korteweg-de Vries Equation and Its Stability Analysis. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), 353-361 (2019). https://doi.org/10.31801/cfsuasmas.420771
9. R. Najafi, G. D. Kucuk, E. Celik, Modified iteration method for solving fractional gas dynamics equation. Mathematical Methods in the Applied Sciences, 40(4), 939-946 (2017). https://doi.org/10.1002/mma.4023
10. A. Yokus, H. M. Baskonus, T. A. Sulaiman, H. Bulut, Numerical simulation and solutions of the two-component second order KdV evolutionarysystem, Numerical Methods for Partial Differential Equations, In press, corrected proof (2017). https://doi.org/10.1002/num.22192
11. K. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993).
12. I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).
13. K. B. Oldham and J. Spanier, The Fractional Calculus. Academic Press, New York (2006).
14. A. Yokus and D. Kaya, Conservation laws and a new expansion method for sixth order Boussinesq equation, AIP Conf. Proc. 1676, 020062 (2015). https://doi.org/10.1063/1.4930488
15. E. Sousa, Finite Difference Approximates for a Fractional Advection Diffusion Problem. J. Comput. Phys. 228 4038-4054 (2009). https://doi.org/10.1016/j.jcp.2009.02.011
16. M. M. Meerschaert and C. Tadjeran, Finite Difference Approximations for Fractional Advection-Dispersion Flow Equations. J. Comput. Appl. Math. 172 65-77 (2004). https://doi.org/10.1016/j.cam.2004.01.033
17. I. Aziz, M. Asif, Haar wavelet collocation method for three-dimensional elliptic partial differential equations. Computers & Mathematics with Applications, 73(9), 2023-2034 (2017). https://doi.org/10.1016/j.camwa.2017.02.034
18. F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput. 191 2-20 (2007). https://doi.org/10.1016/j.amc.2006.08.162
19. N. Ozdemir, M. Yavuz, Numerical Solution of Fractional Black- Scholes Equation by Using the Multivariate Pade Approximation. Acta Physica Polonica A, 132(3) 1050-1053 (2017). https://doi.org/10.12693/APhysPolA.132.1050
20. A. Esen, T. A. Sulaiman, H. Bulut, H. M. Baskonus, Optical solitons to the space-time fractional (1+ 1)-dimensional coupled nonlinear Schr¨odinger equation. Optik, 167, 150-156 (2018). https://doi.org/10.1016/j.ijleo.2018.04.015
21. M. Suba¸sı, H. Durur, On the stability of the solution in an optimal control problem for a Schr¨odinger equation. Applied Mathematics and Computation, 249, 521-526 (2014). https://doi.org/10.1016/j.amc.2014.10.069
22. M. Yavuz, N. Ozdemir, A Quantitative Approach to Fractional Option Pricing Problems with Decomposition Series. Konuralp Journal of Mathematics, 6(1) 102-109 (2018).
23. L. Su, W. Wang, and Q. Xu, Finite difference methods for fractional dispersion equations, Appl. Math. Comput. 216 3329-3334 (2010). https://doi.org/10.1016/j.amc.2010.04.060
24. S.B. Yuste, Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216 264-274 (2006). https://doi.org/10.1016/j.jcp.2005.12.006
25. I. Aziz, M. Ahmad, Numerical solution of two-dimensional elliptic PDEs with nonlocal boundary conditions. Computers Mathematics with Applications, 69(3) 180-205 (2015). https://doi.org/10.1016/j.camwa.2014.12.003
26. A. M. Wazwaz, Handbook of Differential Equations: Evolutionary Equations, 4 (2008).
27. W. Chen, L. Ye and H. Sun, Fractional diffusion equations by the Kansa method, Comput. Math. Appl. 59 1614-1620 (2010). https://doi.org/10.1016/j.camwa.2009.08.004
28. A. Yokus, D. Kaya, Numerical and exact solutions for time fractional burgers equation, Journal of Nonlinear Sciences and Applications 10 (7), 3419-3428 (2017). https://doi.org/10.22436/jnsa.010.07.06
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2022-01-26
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