Fractional spectral collocation methods for linear and nonlinear variable order FPDEs

Journal of Computational Physics

Volumn 293, Issue , 2014, Pages 312-338

  Zayernouri, Mohsen ^a   Karniadakis, George Em ^a  

^a BROWN UNIVERSITY   (United States)

Author keywords

Jacobi polyfractonomial; Penalty method; Riemann Liouville; Riesz and Caputo fractional derivatives; Space time fractional Lagrange interpolants; Variable order differentiation matrix

Indexed keywords

ADVECTION; CONSTRAINED OPTIMIZATION; CONVERGENCE OF NUMERICAL METHODS; DIFFERENTIATION (CALCULUS); INTERPOLATION; LAGRANGE MULTIPLIERS;

CAPUTO FRACTIONAL DERIVATIVES; DIFFERENTIATION MATRICES; JACOBI POLYFRACTONOMIAL; LAGRANGE INTERPOLANT; LIOUVILLE; PENALTY METHODS; RIEMANN-LIOUVILLE; RIESZ AND CAPUTO FRACTIONAL DERIVATIVE; SPACE-TIME FRACTIONAL LAGRANGE INTERPOLANT; SPACETIME; VARIABLE-ORDER DIFFERENTIATION MATRIX; VARIABLES ORDERING;

DIFFUSION;

EID: 84988268719     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2014.12.001     Document Type: Article

References (50)

1. Zayernouri M., Karniadakis G.E. Fractional Sturm-Liouville eigen-problems: theory and numerical approximations. J. Comput. Phys. 2013, 47(3):2108-2131.
2. Benson D.A., Wheatcraft S.W., Meerschaert M.M. Application of a fractional advection-dispersion equation. Water Resour. Res. 2000, 36(6):1403-1412.
3. Mainardi F. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models 2010, Imperial College Press.
4. Chester W. Resonant oscillations in closed tubes. J. Fluid Mech. 1964, 18(1):44-64.
5. Keller J.J. Propagation of simple non-linear waves in gas filled tubes with friction. Z. Angew. Math. Phys. 1981, 32(2):170-181.
6. Sugimoto N., Kakutani T. 'Generalized Burgers' equation' for nonlinear viscoelastic waves. Wave Motion 1985, 7(5):447-458.
7. Magin R.L. Fractional Calculus in Bioengineering 2006, Begell House Inc., Redding, CT.
8. Bouchaud J.P., Georges A. Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 1990, 195(4):127-293.
9. Metzler R., Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 2000, 339(1):1-77.
10. Klages R., Radons G., Sokolov I.M. Anomalous Transport: Foundations and Applications 2008, Wiley-VCH.
11. Sugimoto N. Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves. J. Fluid Mech. 1991, 225(631-653):4.
12. Barkai E., Metzler R., Klafter J. From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E 2000, 61(1):132.
13. Henry B., Wearne S. Fractional reaction-diffusion. Phys. A, Stat. Mech. Appl. 2000, 276(3):448-455.
14. Komatsu T. On stable-like processes. Probability Theory and Mathematical Statistics 1995, 210-219.
15. Kikuchi K., Negoro A. On Markov process generated by pseudo-differential operator of variable order. Osaka J. Math. 1997, 34(2):319-335.
16. Coimbra C. Mechanics with variable-order differential operators. Ann. Phys. 2003, 12(11-12):692-703.
17. Cooper G., Cowan D. Filtering using variable order vertical derivatives. Comput. Geosci. 2004, 30(5):455-459.
18. Tseng C. Design of variable and adaptive fractional order FIR differentiators. Signal Process. 2006, 86(10):2554-2566.
19. Ramirez L.E., Coimbra C.F. A variable order constitutive relation for viscoelasticity. Ann. Phys. 2007, 16(7-8):543-552.
20. Pedro H., Kobayashi M., Pereira J., Coimbra C. Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere. J. Vib. Control 2008, 14(9-10):1659-1672.
21. Sun H.G., Chen W., Chen Y.Q. Variable-order fractional differential operators in anomalous diffusion modeling. Phys. A, Stat. Mech. Appl. 2009, 388(21):4586-4592.
22. Chen C.-M., Liu F., Anh V., Turner I. Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation. SIAM J. Sci. Comput. 2010, 32(4):1740-1760.
23. Chen C.M., Liu F., Anh V., Turner I. Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term. Appl. Math. Comput. 2011, 217(12):5729-5742.
24. Zhang H., Liu F., Phanikumar M.S., Meerschaert M.M. A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model. Comput. Math. Appl. 2013, 66(5):693-701.
25. Chen W., Zhang J., Zhang J. A variable-order time-fractional derivative model for chloride ions sub-diffusion in concrete structures. Fract. Calc. Appl. Anal. 2013, 16(1):76-92.
26. Zhuang P., Liu F., Anh V., Turner I. Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 2009, 47(3):1760-1781.
27. Shen S., Liu F., Chen J., Turner I., Anh V. Numerical techniques for the variable order time fractional diffusion equation. Appl. Math. Comput. 2012, 218(22):10861-10870.
28. Chen C., Liu F., Burrage K., Chen Y. Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative. IMA J. Appl. Math. 2012, 1-21.
29. Shen S., Liu F., Anh V., Turner I., Chen J. A characteristic difference method for the variable-order fractional advection-diffusion equation. J. Appl. Math. Comput. 2013, 1-16.
30. X. Zhao, Z. Sun, G.E. Karniadakis, Second-order approximations for variable order fractional derivatives: algorithms and applications, Journal of Computational Physics, Accepted.
31. Lin Y., Xu C. Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 2007, 225(2):1533-1552.
32. Li X., Xu C. A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 2009, 47(3):2108-2131.
33. Li X., Xu C. Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 2010, 8(5):1016.
34. Khader M.M. On the numerical solutions for the fractional diffusion equation. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(6):2535-2542.
35. Piret C., Hanert E. A radial basis functions method for fractional diffusion equations. J. Comput. Phys. 2012, 71-81.
36. Doha E., Bhrawy A., Ezz-Eldien S. A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput. Math. Appl. 2011, 62(5):2364-2373.
37. Bhrawy A.H., Al-Shomrani M.M. A shifted Legendre spectral method for fractional-order multi-point boundary value problems. Adv. Differ. Equ. 2012, 2012(1):1-19.
38. Maleki M., Hashim I., Kajani M.T., Abbasbandy S. An adaptive pseudospectral method for fractional order boundary value problems. Abstr. Appl. Anal. 2012, 2012.
39. Baleanu D., Bhrawy A., Taha T. Two efficient generalized Laguerre spectral algorithms for fractional initial value problems. Abstr. Appl. Anal. 2013, 2013.
40. Bhrawy A., Alghamdia M. A new Legendre spectral Galerkin and pseudo-spectral approximations for fractional initial value problems. Abstr. Appl. Anal. 2013, 2013:306746. (10 pp.).
41. Deng W., Hesthaven J. Local discontinuous Galerkin methods for fractional diffusion equations. ESAIM: Math. Model. Numer. Anal. 2013, 1845-1864.
42. Xu Q., Hesthaven J. Stable multi-domain spectral penalty methods for fractional partial differential equations. J. Comput. Phys. 2014, 257:241-258.
43. Zayernouri M., Karniadakis G.E. Exponentially accurate spectral and spectral element methods for fractional ODEs. J. Comput. Phys. 2014, 257-Part A:460-480.
44. Zayernouri M., Karniadakis G.E. Discontinuous spectral element methods for time- and space-fractional advection equations. SIAM J. Sci. Comput. 2014, 36(4):B684-B707.
45. Zayernouri M., Cao W., Zhang Z., Karniadakis G.E. Spectral and discontinuous spectral element methods for fractional delay equations. SIAM J. Sci. Comput. 2014, 36(6):B904-B929.
46. Zayernouri M., Karniadakis G.E. Fractional spectral collocation method. SIAM J. Sci. Comput. 2014, 36(1):A40-A62.
47. Zayernouri M., Ainsworth M., Karniadakis G.E. A unified Petrov-Galerkin spectral method for fractional PDEs. Comput. Methods Appl. Mech. Eng. 2014, 10.1016/j.cma.2014.10.051.
48. Podlubny I. Fractional Differential Equations 1999, Academic Press, San Diego, CA, USA.
49. Askey R., Fitch J. Integral representations for Jacobi polynomials and some applications. J. Math. Anal. Appl. 1969, 26:411-437.
50. Szegö G. Orthogonal Polynomials, vol. 23 1992, AMS Bookstore.