# Numerical solution of fractional sub-diffusion and time-fractional diffusion-wave equations via fractional-order Legendre functions

• M. R. Hooshmandasl
• M. H. Heydari
• C. Cattani
Regular Article

## Abstract.

Fractional calculus has been used to model physical and engineering processes that are best described by fractional differential equations. Therefore designing efficient and reliable techniques for the solution of such equations is an important task. In this paper, we propose an efficient and accurate Galerkin method based on the fractional-order Legendre functions (FLFs) for solving the fractional sub-diffusion equation (FSDE) and the time-fractional diffusion-wave equation (FDWE). The time-fractional derivatives for FSDE are described in the Riemann-Liouville sense, while for FDWE are described in the Caputo sense. To this end, we first derive a new operational matrix of fractional integration (OMFI) in the Riemann-Liouville sense for FLFs. Next, we transform the original FSDE into an equivalent problem with fractional derivatives in the Caputo sense. Then the FLFs and their OMFI together with the Galerkin method are used to transform the problems under consideration into the corresponding linear systems of algebraic equations, which can be simply solved to achieve the numerical solutions of the problems. The proposed method is very convenient for solving such kind of problems, since the initial and boundary conditions are taken into account automatically. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

## References

1. 1.
2. 2.
K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order (Academic Press, New York, 1974)Google Scholar
3. 3.
S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science, 1993)Google Scholar
4. 4.
K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (John Wiley and Sons, 1993)Google Scholar
5. 5.
R. Du, W.R. Cao, Z.Z. Sun, Appl. Math. Model. 34, 2998 (2010)
6. 6.
F. Liu, V. Anh, I. Turner, J. Comput. Appl. Math. 166, 209 (2004)
7. 7.
F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Appl. Math. Comp. 191, 12 (2007)
8. 8.
C. Tadjeran, M.M. Meerschaert, H.-P. Scheffler, J. Comput. Phys. 213, 205 (2006)
9. 9.
P. Zhuang, F. Liu, V. Anh, I. Turner, SIAM J. Numer. Anal. 46, 1079 (2008)
10. 10.
F. Liu, C. Yang, K. Burrage, J. Comput. Appl. Math. 231, 160 (2009)
11. 11.
S. Yuste, L. Acedo, SIAM J. Numer. Anal. 42, 1862 (2005)
12. 12.
S. Yuste, J. Comput. Phys. 216, 264 (2006)
13. 13.
C.M. Chen, F. Liu, I. Turner, V. Anh, J. Comput. Phys. 227, 886 (2007)
14. 14.
R. Scherer, S. Kalla, L. Boyadjiev, B. Al-Saqabi, Appl. Numer. Math. 58, 1212 (2008)
15. 15.
T.A.M. Langlands, B. Henry, J. Comput. Phys. 205, 719 (2005)
16. 16.
J.K.R. Metzler, Phys. Rep. 339, 1 (2000)
17. 17.
M. Cui, J. Comput. Phys. 228, 7792 (2009)
18. 18.
G. Gao, Z. Sun, J. Comput. Phys. 230, 586 (2011)
19. 19.
J. Renand, Z. Sun, X. Zhao, J. Comput. Phys. 232, 456 (2013)
20. 20.
X. Zhao, Z. Sun, J. Comput. Phys. 230, 6061 (2011)
21. 21.
C. Chen, F. Liu, K. Burrage, J. Comput. Phys. 198, 754 (2008)
22. 22.
D. Murio, Comput. Math. Appl. 56, 1138 (2008)
23. 23.
T. Langlands, B. Henry, J. Comput. Phys. 205, 719 (2005)
24. 24.
P. Zhuang, F. Liu, V. Anh, I. Turner, IMA J. Appl. Math. 74, 645 (2009)
25. 25.
C. Chen, F. Liu, I. Turner, V. Anh, Numer. Algor. 54, 1 (2010)
26. 26.
C. Chen, F. Liu, I. Turner, V. Anh, SIAM J. Sci. Comput. 32, 1740 (2010)
27. 27.
Y. Zhang, Z. Sun, H. Wu, SIAM J. Numer. Anal. 49, 2302 (2011)
28. 28.
Y. Zhang, Z. Sun, J. Comput. Phys. 230, 8713 (2011)
29. 29.
M. Cui, J. Comput. Phys. 231, 2621 (2012)
30. 30.
X. Zhao, Q. Xu, Appl. Math. Model. 38, 3848 (2014)
31. 31.
A. Mohebbi, M. Abbaszadeh, M. Dehghan, Eng. Anal. Boundary Elem. 38, 72 (2014)
32. 32.
J. Chen, F. Liu, Q. Liu, X. Chen, V. Anh, I. Turner, K. Burrage, Appl. Math. Model. 38, 3695 (2014)
33. 33.
A.H. Bhrawy, E.H. Dohac, D. Baleanud, S.S. Ezz-Eldien, J. Comput. Phys. 293, 142 (2015)
34. 34.
J. Chen, F. Liu, V. Anh, S. Shen, Q. Liu, C. Liao, Appl. Math. Comput. 219, 1737 (2012)
35. 35.
F. Mainardi, Appl. Math. Lett. 9, 23 (1996)
36. 36.
W. Wess, J. Math. Phys. 27, 2782 (1996)
37. 37.
W.R. Schneider, W. Wess, J. Math. Phys. 30, 134 (1989)
38. 38.
Y.A. Rossikhin, M.V. Shitikova, Appl. Mech. Rev. 50, 15 (1997)
39. 39.
Z.Z. Sun, X. Wu, Appl. Numer. Math. 56, 193 (2006)
40. 40.
M. Cui, J. Comput. Phys. 6, 383 (2013)
41. 41.
X. Hu, L. Zhang, Comput. Phys. Commun. 182, 1645 (2011)
42. 42.
X. Hu, L. Zhang, Appl. Math. Comput. 218, 5019 (2012)
43. 43.
C.F.L. Godinho, J. Weberszpil, J.A. Helayel-Neto, Chaos Solitons Fractals 4, 765 (2012)
44. 44.
45. 45.
M.H. Heydari, M.R. Hooshmandasl, F.M.M. Ghaini, F. Fereidouni, Eng. Anal. Bound. Elem. 37, 1331 (2013)
46. 46.
C. Canuto, M. Hussaini, A. Quarteroni, T. Zang, Spectral methods in fluid dynamics (Springer, 1988)Google Scholar
47. 47.
A. Saadatmandi, M. Dehghan, Comput. Math. Appl. 59, 1326 (2010)
48. 48.
E.H. Doha, A.H. Bhrawy, S.S. Ezz-Eldien, Appl. Math. Model. 36, 4931 (2012)
49. 49.
E.H. Doha, A.H. Bhrawy, S.S. Ezz-Eldien, Comput. Math. Appl. 62, 2364 (2011)
50. 50.
D. Baleanu, A.H. Bhrawy, T.M. Taha, Abstr. Appl. Anal. 2013, 546502 (2013)
51. 51.
M.H. Heydari, M.R. Hooshmandasl, F.M.M. Ghaini, Comput. Math. Appl. 68, 269 (2014)
52. 52.
A.H. Bhrawy, M.A. Zaky, R.A.V. Gorder, Numer. Algor. 71, 151 (2016)
53. 53.
A.H. Bhrawy, E.H. Doha, S.S. Ezz-Eldien, M.A. Abdelkawy, Calcolo 53, 1 (2016)
54. 54.
M.H. Heydari, M.R. Hooshmandasl, F. Mohammadi, Appl. Math. Comput. 234, 267 (2014)
55. 55.
M.H. Heydari, M.R. Hooshmandasl, F. Mohammadi, Adv. Appl. Math. Mech. 6, 247 (2014)
56. 56.
M.H. Heydari, M.R. Hooshmandasl, F.M.M. Ghaini, C. Cattani, Phys. Lett. A 379, 71 (2015)
57. 57.
M.H. Heydari, M.R. Hooshmandasl, F. Mohammadi, C. Cattani, Commun. Nonlinear Sci. Numer. Simulat. 19, 37 (2014)
58. 58.
M.H. Heydari, M.R. Hooshmandasl, F.M.M. Ghaini, C. Cattani, Appl. Math. Comput. 286, 139 (2016)
59. 59.
C. Cattani, Int. J. Fluid Mech. Res. 5, 1 (2003)Google Scholar
60. 60.
C. Cattani, Comp. Math. Appl. 50, 1191 (2005)
61. 61.
C. Cattani, A. Kudreyko, Appl. Math. Comput. 215, 4164 (2010)
62. 62.
C. Cattani, Math. Modell. Anal. 11, 117 (2006)
63. 63.
C. Cattani, Math. Prob. Eng. 2008, 164808 (2008)
64. 64.
C. Cattani, Math. Prob. Eng. 2010, 408418 (2010)
65. 65.
C. Cattani, Math. Prob. Eng. 2012, 502812 (2012)
66. 66.
S. Kazem, S. Abbasbandy, S. Kumar, Appl. Math. Model. 37, 5498 (2013)
67. 67.
F. Liu, M.M. Meerschaert, R.J. McGough, P. Zhuang, Q. Liu, Fract. Calc. Appl. Anal. 16, 9 (2013)

© Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg 2016

## Authors and Affiliations

• M. R. Hooshmandasl
• 1
• 2
Email author
• M. H. Heydari
• 1
• 2
• C. Cattani
• 3
1. 1.Faculty of MathematicsYazd UniversityYazdIran
2. 2.The Laboratory of Quantum Information ProcessingYazd UniversityYazdIran
3. 3.Engineering School (DEIM)University of TusciaViterboItaly