Numerical Solutions of Time Fractional Korteweg--de Vries Equation and Its Stability Analysis

Asıf Yokuş

doi:10.31801/cfsuasmas.420771

Research Article

Numerical Solutions of Time Fractional Korteweg--de Vries Equation and Its Stability Analysis

Year 2019, Volume: 68 Issue: 1, 353 - 361, 01.02.2019

Asıf Yokuş

https://doi.org/10.31801/cfsuasmas.420771

Cited By: 13

Abstract

In this study, the fractional derivative and finite difference operators are analyzed. The time fractional KdV equation with initial condition is considered. Discretized equation is obtained with the help of finite difference operators and used Caputo formula. The inherent truncation errors in the method are defined and analyzed. Stability analysis is explored to demonstrate the accuracy of the method. While doing this analysis, considering conservation law, with the help of using the definition discovered by Lax-Wendroff, von Neumann stability analysis is applied. The numerical solutions of time fractional KdV equation are obtained by using finite difference method. The comparison between obtained numerical solutions and exact solution from existing literature is made. This comparison is highlighted with the graphs as well. Results are presented in tables using the Mathematica software package wherever it is needed.

Keywords

Finite difference method, time fractional KdV equation, Caputo formula, numerical solutions, fractional partial differential equation

References

Podlubny, I., Fractional Differential Equations. Academic Press, San Diego (1999).
Oldham K. B. and Spanier, J., The Fractional Calculus. Academic Press, New York (2006).
Bertram, R., Fractional Calculus and Its Applications, Springer-Verlag, Berlin Heidelberg , (1975).
Kilbas, A.A., Srivastava, H.M. and Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier B. V., Amsterdam, Netherlands (2006).
Samko, S.G., Kilbas, A.A. and Marichev, O.I., Fractional Integrals and Derivatives-Theory and Applications, Gordon and Breach, Longhorne, PA (1993).
Feng B. and Mitsui, T., A finite difference method for the Korteweg-de Vries and the Kadomtsev-Petviashvili equations, J. Comput. Appl. Math. (1998) 95--116.
Miller K. and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993).
Zaslavsky, G.M., Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371 (2002) 461--580.
Sousa, E., Finite difference approximates for a fractional advection diffusion problem, J. Comput. Phys. 228 (2009) 4038--4054.
El-Wakil, S. A. Abulwafa, E. M., El-shewy E. K. and Mahmoud, A. A., Time-fractional KdV equation for electron-acoustic waves in plasma of cold electron and two different temperature isothermal ions, Astrophys Space Sci. (2011) 333: 269--276
Liu, F., Zhuang, Anh, P. V., Turner, I. and Burrage, K., Stability and convergence of the difference methods for the space-time fractional advection--diffusion equation, Appl. Math. Comput. 191 (2007) 2--20.
Su, L., Wang, W. and Yang, Z., Finite difference approximations for the fractional advection--diffusion equation, Phys. Lett. A 373 (2009) 4405--4408.
Benson, D., Wheatcraft, S. and Meerschaert, M., Application of a fractional advection--dispersion equation, Water Resour. Res. 36 (2000) 1403--1412.
Mainardi, F., Raberto, Gorenflo, M. R., and Scalas, E., Fractional calculus and continuous-time finance II: the waiting-time distribution, Physica A 287 (2000) 468--481.
Kaya, D., An application of the decomposition method for the KdVb equation. Applied Mathematics and Computation, 152(1) (2004) 279--288.
Scalas, E., Gorenflo, R. and Mainardi, F., Fractional calculus and continuous-time finance, Phys. A 284 (2000) 376--384.
Su, L., Wang, W. and Xu, Q., Finite difference methods for fractional dispersion equations, Appl. Math. and Comput. 216 (2010) 3329--3334.
Yokus, A., Baskonus, H. M., Sulaiman, T. A. and Bulut, H., Numerical simulation and solutions of the two-component second order KdV evolutionarysystem. Numerical Methods for Partial Differential Equations, 34(1) (2018), 211--227.
Yuste, S.B., Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys. 216 (2006) 264--274.
Aziz, I. and Asif, M., Haar wavelet collocation method for three-dimensional elliptic partial differential equations. Computers & Mathematics with Applications, 73(18) (2017), 2023--2034.
Meerschaert M.M. and Tadjeran, C., Finite difference approximations for fractional advection--dispersion flow equations, J. Comput. Appl. Math. 172 (2004) 65--77.
Sajjadian, M., Numerical solutions of Korteweg de Vries and Korteweg de Vries-Burger's equations using computer programming, Int. J. Nonlinear Sci, 15 (2013) 69--79.
Chen, W., Ye, L. and Sun, H., Fractional diffusion equations by the Kansa method. Comput. Math. Appl. 59 (2010) 1614--1620.
Odibat Z.M. and Shawagfeh, N.T., Generalized Taylor's formula, Appl. Math. Comput. 186 (2007) 286--293.
Debnath, L., Linear Partial Differential Equations for Scientists and Engineers, Birkhäuser Boston (2007)
Schielen, R.M.J., Nonlinear Stability Analysis and Pattern Formation in Morphological Models, Ph.D. Thesis, Universiteit Utrecht (1995).

Year 2019, Volume: 68 Issue: 1, 353 - 361, 01.02.2019

Asıf Yokuş

https://doi.org/10.31801/cfsuasmas.420771

Cited By: 13

Abstract

References

Podlubny, I., Fractional Differential Equations. Academic Press, San Diego (1999).
Oldham K. B. and Spanier, J., The Fractional Calculus. Academic Press, New York (2006).
Bertram, R., Fractional Calculus and Its Applications, Springer-Verlag, Berlin Heidelberg , (1975).
Kilbas, A.A., Srivastava, H.M. and Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier B. V., Amsterdam, Netherlands (2006).
Samko, S.G., Kilbas, A.A. and Marichev, O.I., Fractional Integrals and Derivatives-Theory and Applications, Gordon and Breach, Longhorne, PA (1993).
Feng B. and Mitsui, T., A finite difference method for the Korteweg-de Vries and the Kadomtsev-Petviashvili equations, J. Comput. Appl. Math. (1998) 95--116.
Miller K. and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993).
Zaslavsky, G.M., Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371 (2002) 461--580.
Sousa, E., Finite difference approximates for a fractional advection diffusion problem, J. Comput. Phys. 228 (2009) 4038--4054.
El-Wakil, S. A. Abulwafa, E. M., El-shewy E. K. and Mahmoud, A. A., Time-fractional KdV equation for electron-acoustic waves in plasma of cold electron and two different temperature isothermal ions, Astrophys Space Sci. (2011) 333: 269--276
Liu, F., Zhuang, Anh, P. V., Turner, I. and Burrage, K., Stability and convergence of the difference methods for the space-time fractional advection--diffusion equation, Appl. Math. Comput. 191 (2007) 2--20.
Su, L., Wang, W. and Yang, Z., Finite difference approximations for the fractional advection--diffusion equation, Phys. Lett. A 373 (2009) 4405--4408.
Benson, D., Wheatcraft, S. and Meerschaert, M., Application of a fractional advection--dispersion equation, Water Resour. Res. 36 (2000) 1403--1412.
Mainardi, F., Raberto, Gorenflo, M. R., and Scalas, E., Fractional calculus and continuous-time finance II: the waiting-time distribution, Physica A 287 (2000) 468--481.
Kaya, D., An application of the decomposition method for the KdVb equation. Applied Mathematics and Computation, 152(1) (2004) 279--288.
Scalas, E., Gorenflo, R. and Mainardi, F., Fractional calculus and continuous-time finance, Phys. A 284 (2000) 376--384.
Su, L., Wang, W. and Xu, Q., Finite difference methods for fractional dispersion equations, Appl. Math. and Comput. 216 (2010) 3329--3334.
Yokus, A., Baskonus, H. M., Sulaiman, T. A. and Bulut, H., Numerical simulation and solutions of the two-component second order KdV evolutionarysystem. Numerical Methods for Partial Differential Equations, 34(1) (2018), 211--227.
Yuste, S.B., Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys. 216 (2006) 264--274.
Aziz, I. and Asif, M., Haar wavelet collocation method for three-dimensional elliptic partial differential equations. Computers & Mathematics with Applications, 73(18) (2017), 2023--2034.
Meerschaert M.M. and Tadjeran, C., Finite difference approximations for fractional advection--dispersion flow equations, J. Comput. Appl. Math. 172 (2004) 65--77.
Sajjadian, M., Numerical solutions of Korteweg de Vries and Korteweg de Vries-Burger's equations using computer programming, Int. J. Nonlinear Sci, 15 (2013) 69--79.
Chen, W., Ye, L. and Sun, H., Fractional diffusion equations by the Kansa method. Comput. Math. Appl. 59 (2010) 1614--1620.
Odibat Z.M. and Shawagfeh, N.T., Generalized Taylor's formula, Appl. Math. Comput. 186 (2007) 286--293.
Debnath, L., Linear Partial Differential Equations for Scientists and Engineers, Birkhäuser Boston (2007)
Schielen, R.M.J., Nonlinear Stability Analysis and Pattern Formation in Morphological Models, Ph.D. Thesis, Universiteit Utrecht (1995).

There are 26 citations in total.

Details

Primary Language	English
Journal Section	Review Articles
Authors	Asıf Yokuş 0000-0002-1460-8573
Publication Date	February 1, 2019
Submission Date	November 16, 2017
Acceptance Date	January 30, 2018
Published in Issue	Year 2019 Volume: 68 Issue: 1

Cite

APA	Yokuş, A. (2019). 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. https://doi.org/10.31801/cfsuasmas.420771
AMA	Yokuş A. Numerical Solutions of Time Fractional Korteweg--de Vries Equation and Its Stability Analysis. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. February 2019;68(1):353-361. doi:10.31801/cfsuasmas.420771
Chicago	Yokuş, Asıf. “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, no. 1 (February 2019): 353-61. https://doi.org/10.31801/cfsuasmas.420771.
EndNote	Yokuş A (February 1, 2019) 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.
IEEE	A. Yokuş, “Numerical Solutions of Time Fractional Korteweg--de Vries Equation and Its Stability Analysis”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 1, pp. 353–361, 2019, doi: 10.31801/cfsuasmas.420771.
ISNAD	Yokuş, Asıf. “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 (February 2019), 353-361. https://doi.org/10.31801/cfsuasmas.420771.
JAMA	Yokuş A. Numerical Solutions of Time Fractional Korteweg--de Vries Equation and Its Stability Analysis. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:353–361.
MLA	Yokuş, Asıf. “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, vol. 68, no. 1, 2019, pp. 353-61, doi:10.31801/cfsuasmas.420771.
Vancouver	Yokuş A. Numerical Solutions of Time Fractional Korteweg--de Vries Equation and Its Stability Analysis. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(1):353-61.