The fictitious time integration method to solve the space- and time-fractional burgers equations

Computers, Materials and Continua

Volumn 15, Issue 3, 2010, Pages 221-240

  Liu, Chein Shan ^a  

^a NATIONAL TAIWAN UNIVERSITY   (Taiwan)

Author keywords

Caputo derivative; Fictitious time integration method (FTIM); Fractional Burgers equation; Group preserving scheme; Riemann Liouville derivative

Indexed keywords

BURGERS EQUATIONS; CAPUTO DERIVATIVES; CAPUTO SENSE; DEPENDENT VARIABLES; FICTITIOUS TIME INTEGRATION METHODS; FRACTIONAL BURGERS EQUATION; FRACTIONAL DERIVATIVES; GRID POINTS; NUMERICAL SCHEME; SEMIDISCRETIZATION; SPATIAL TEMPORALS;

WAVELET TRANSFORMS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 77954550806     PISSN: 15462218     EISSN: 15462226     Source Type: Journal    
DOI: None     Document Type: Article

References (34)

1. Bateman, H. (1915): Some recent researches on the motion of fluids. Mon. Weather Rev., vol.43, pp. 163-170.
2. Biler, P.; Funaki, T.; Woyczynski, W. A. (1998): Fractal Burger equations. J.Differential Equations, vol.148, pp. 9-46.
3. Burgers, J. M. (1948): A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech., vol.1, pp. 171-199.
4. Burgers, J. M. (1974): The Nonlinear Diffusion Equation. Reidel, Boston, 1974.
5. Caputo, M. (1967): Linear models of dissipation whose Q is almost frequency independent, II. Geophys. J. R. Astron. Soc., vol.13, pp. 529-539.
6. Caputo, M; Mainardi, F. (1971): A new dissipation model based on memory mechanism. Pure Appl. Geophys., vol.91, pp. 134-147.
7. Chang, C. W.; Liu, C.-S. (2009): A fictitious time integration method for backward advection-dispersion equation. CMES: Computer Modeling in Engineering & Sciences, vol.51, pp. 261-276.
8. Chi, C. C.; Yeih, W.; Liu, C.-S. (2009): A novel method for solving the Cauchy problem of Laplace equation using the fictitious time integration method. CMES: Computer Modeling in Engineering & Sciences, vol.47, pp. 167-190.
9. Chen, W.; Holm, S. (2003): Modified Szabo's wave equation models for lossy media obeying frequency power law. J. Acoust. Soc. Am., vol.114, pp. 2570-2574.
10. Cole, J. D. (1951): On a quasi linear parabolic equation occurring in aerodynamics. Quart. Appl. Math., vol.9, pp. 225-236.
11. Gu, Y. T.; Zhuang, P.; Liu, F. (2010): An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation. CMES: Computer Modeling in Engineering & Sciences, vol.56, pp. 303-333.
12. He, J. H. (1999): Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Techno., vol.15, pp. 86-90.
13. x. Comm. Pure Appl. Math., vol.3, pp. 201-230.
14. Kou, C.; Yan, Y.; Liu, J. (2009): Stability analysis for fractional differential equations and their applications in the models of HIV-1 infection. CMES: Computer Modeling in Engineering & Sciences, vol.39, pp. 301-318.
15. Liu, C.-S. (2001): Cone of non-linear dynamical system and group preserving schemes. Int. J. Non-Linear Mech., vol.36, pp. 1047-1068.
16. Liu, C.-S. (2006a): A group preserving scheme for Burgers equation with very large Reynolds number. CMES: Computer Modeling in Engineering & Sciences, vol.12, pp. 197-211.
17. Liu, C.-S. (2006b): An efficient backward group preserving scheme for the backward in time Burgers equation. CMES: Computer Modeling in Engineering & Sciences, vol.12, pp. 55-65.
18. Liu, C.-S. (2008a): Solving an inverse Sturm-Liouville problem by a Lie-group method. Boundary Value Problems, vol.2008, Article ID 749865.
19. Liu, C.-S. (2008b): Identifying time-dependent damping and stiffness functions by a simple and yet accurate method. J. Sound Vib., vol.318, pp. 148-165.
20. Liu, C.-S. (2008c): A Lie-group shooting method for simultaneously estimating the time-dependent damping and stiffness coefficients. CMES: Computer Modeling in Engineering & Sciences, vol.27, pp. 137-149.
21. Liu, C.-S. (2008d): A Lie-group shooting method estimating nonlinear restoring forces in mechanical systems. CMES: Computer Modeling in Engineering & Sciences, vol.35, pp. 157-180.
22. Liu, C.-S. (2008e): A fictitious time integration method for two-dimensional quasilinear elliptic boundary value problems. CMES: Computer Modeling in Engineering & Sciences, vol.33, pp. 179-198.
23. Liu, C.-S. (2009): A fictitious time integration method for solving m-point boundary value problems. CMES: Computer Modeling in Engineering & Sciences, vol.39, pp. 125-154.
24. Liu, C.-S.; Atluri, S. N. (2008a): A novel time integration method for solving a large system of non-linear algebraic equations. CMES: Computer Modeling in Engineering & Sciences, vol.31, pp. 71-83.
25. Liu, C.-S.; Atluri, S. N. (2008b): A fictitious time integration method (FTIM) for solving mixed complementarity problems with applications to non-linear optimization. CMES: Computer Modeling in Engineering & Sciences, vol.34, pp. 155-178.
26. Liu, C.-S.; Atluri, S. N. (2008c): A novel fictitious time integration method for solving the discretized inverse Sturm-Liouville problems, for specified eigenvalues. CMES: Computer Modeling in Engineering & Sciences, vol.36, pp. 261-285.
27. Liu, C.-S.; Atluri, S. N. (2009): A Fictitious time integration method for the numerical solution of the Fredholm integral equation and for numerical differentiation of noisy data, and its relation to the filter theory. CMES: Computer Modeling in Engineering & Sciences, vol.41, pp. 243-261.
28. Miskinis, P. (2002): Some properties of fractional Burgers equation. Math. Model. Anal., vol.7, pp. 151-158.
29. Momani, S. (2006): Non-perturbative analytical solutions of the space- and timefractional Burgers equations. Chaos Solitons Fractals, vol.28, pp. 930-937.
30. Podlubny, I. (1999): Fractional Differential Equations. Academic Press, San Diego.
31. Samko, S. G.; Kilbas, A. A.; Marichev, O. I. (1987): Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science, New York.
32. Seredynska, M.; Hanyga, A. (2000): Nonlinear Hamiltonian equations with fractional damping. J. Math. Phys., vol.41, pp. 2135-2156.
33. Tsai, C. C.; Liu, C.-S.; Yeih, W. (2010): Fictitious time integration method of fundamental solutions with Chebyshev polynomials for solving Poisson-type nonlinear PDEs. CMES: Computer Modeling in Engineering & Sciences, vol.56, pp. 131-151.
34. Ye, H.; Ding, Y. (2009): Dynamical analysis of a fractional-order HIV model. CMES: Computer Modeling in Engineering & Sciences, vol.49, pp. 255-268.