Numerical solution for a class of fractional convection-diffusion equations using the flatlet oblique multiwavelets

JVC/Journal of Vibration and Control

Volumn 20, Issue 6, 2014, Pages 913-924

  Irandoust Pakchin, Safar ^a   Dehghan, Mehdi ^b   Abdi Mazraeh, Somayeh ^a   Lakestani, Mehrdad ^a  

^a UNIVERSITY OF TABRIZ   (Iran)

^b AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

Author keywords

biorthogonal system; Caputo derivative; flatlet oblique multiwavelets; Fractional order differential equations; operational matrix

Indexed keywords

BIORTHOGONAL; CAPUTO DERIVATIVES; FRACTIONAL-ORDER DIFFERENTIAL EQUATIONS; MULTI-WAVELETS; OPERATIONAL MATRICES;

DIFFUSION IN LIQUIDS; ERRORS; FOURIER ANALYSIS; PARTIAL DIFFERENTIAL EQUATIONS;

HEAT CONVECTION;

EID: 84896885047     PISSN: 10775463     EISSN: 17412986     Source Type: Journal    
DOI: 10.1177/1077546312470473     Document Type: Article

References (62)

1. Agrawal OP. Solution for a fractional diffusion-wave equation defiend in a bounded domain. Journal of Nonlinear Dynamics. 2002; 29: 145-155
2. Alpert B. A class of basis in L2 for the sparse representation of integral oerators. SIAM Journal of Mathematical Analysis. 1993; 24: 246-262
3. Alpert B, Beylkin G, Coifman R and Rokhlin R (1990) Wavelets for the Fast Solution of Second-kind Integral Equations. Technical Report, Department of Computer Science, Yale University, New Haven, CT.
4. Alpert B, Beylkin G, Coifman RR, Rokhlin V. Wavelet-like basis for the fast solution of second-kind integral equations. SIAM Journal of Scientific and Statistical Computing. 1993; 14: 159-184
5. Alpert B, Beylkin G, Gines D, Vozovoi L. Adaptive solution of partial differential equations in multiwavelet bases. Journal of Computational Physics. 2002; 182: 149-190
6. Baleanu D. Fractional Hamiltoian analysis of irregular systems. Signal Processing. 2006; 86: 2632-2636
7. Baleanu D, Avkar T. Lagrangians with linear velocities within Riemann-Liouville fractional derivatives. Nuovo Cimento B. 2004; 119: 73-79
8. Baleanu D, Diethelm K, Scalas E, Trujillo JJ Fractional Calculus Models and Numerical Methods (Series on Complexity, Nonlinearity and Chaos). Singapore: World Scientific; 2012
9. Baleanu D, Güvenç ZB, Tenreiro Machado JA New Trends in Nanotechnology and Fractional Calculus Applications. New York: Springer; 2010
10. Baleanu D, Muslih SI. Formulation of Hamiltonian equations for fractional variational problems. Czechoslovac Journal of Physics. 2005a; 55: 633-642
11. Baleanu D, Muslih S. Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives. Physica Scripta. 2005b; 72: 119-121
12. Baleanu D, Trujillo JJ. On exact solutions of a class of fractional Euler-Lagrange equations. Nonlinear Dynamics. 2008; 52: 331-335
13. Benchohra M, Hamani S, Ntouyas SK. Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Analysis TMA. 2009; 71: 2391-2396
14. Beylkin G, Coifman R, Rokhlin V. Fast wavelet transforms and numerical algorithms I. Communications on Pure and Applied Mathematics. 1991; 44: 141-183
15. Caputo M. Linear models of dissipation whose Q is almost frequency independent, Part II. Journal of the Royal Australian Society. 1967; 13: 529-539
16. Chen Y, Wu Y, Cui Y, Wang Z, Jin D. Wavelet method for a class of fractional convection-diffusion equation with variable coefficients. Journal of Computational Science. 2010; 1: 146-149
17. Cohen A, Dahmen W, DeVore R. Adaptive wavelet methods for elliptic operator equations-convergence rates. Mathematics of Computation. 2001; 70: 27-75
18. Dahmen W, Han B, Jia RQ, Kunoth A. Biorghogonal multiwavelets on the interval: cubic Hermite splines. Constructive Approximation. 2000; 16: 221-259
19. Darani MRA, Adibi H, Agarwal RP, Saadati R. Flatlet oblique multiwavelet for solving integro-differential equations. Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Matematical Analysis. 2008; 15: 755-768
20. Darani MRA, Adibi H, Lakestani M. Numerical solution of integro-differential equations usnig flatlet oblique multiwavelets. Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis. 2010; 17: 55-74
21. Das S Functional Fractional Calculus. New York: Springer; 2011
22. Dehghan M, Manafian J and Saadatmandi A (2010a) Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numerical Methods for Partial Differential Equations 26: 448-479.
23. Dehghan M, Manafian J and Saadatmandi A (2010b) The solution of the linear fractional partial differential equations using the homotopy analysis method. Zeitschrift fur Naturforschung 65a: 935-945.
24. Dong J, Wei X, Zhang Q, Zhao Y. Pitch detection using circular average magnitude difference function based on wavelet transform. International Journal of Innovative Computing, Information and Control. 2009; 5: 2717-2724
25. Goodman TNT, Lee SL. Wavelets of multiplicityr. Transactions of the American Mathematical Society. 1994; 342: 307-324
26. Han B, Jiang QT. Multiwavelets on the interval. Applied and Computational Harmonic Analysis. 2002; 12: 100-127
27. Kayedi A, Eslahchi MR, Dehghan M. A method for obtaining the operational matrix of fractional Jacobi functions and applications. Journal of Vibration and Control. 2012;
28. Keinert F Wavelets and Multiwavelets. New York: Chapman and Hall/CRC; 2003
29. Khader MM. On the numerical solutions for the fractional diffusion equation. Communications in Nonlinear Science and Numerical Simulation. 2011; 16: 2535-2542
30. Kilbas AA, Srivastava HM, Trujillo JJ Theory and Applications of Fractional Differential Equations. San Diego, CA: Elsevier; 2006
31. Lakestani M, Dehghan M, Irandoust-Pakchin S. The construction of operational matrix of fractional derivatives using B-spline functions. Communications in Nonlinear Science and Numerical Simulation. 2012; 17: 1149-1162
32. Lakestani M, Razzaghi M, Dehghan M. Semiorthogonal spline wavelets approximation for Fredholm integro-differential equations. Mathematical Problems in Engineering. 2006; 2006: 96184
33. Lakestani M, Saray BN. Numerical solution of telegraph equation using interpolating scaling functions. Computers and Mathematics with Applications. 2010; 60: 1964-1972
34. Lakestani M, Saray BN, Dehghan M. Numerical solution for the weakly singular Fredholm integro-differential equations using Legendre multiwavelets. Journal of Computational and Applied Mathematics. 2011; 235: 3291-3303
35. Lin Y, Xu C. Finite difference/spectral approximation for the time-fractional differential diffusion equation. Journal of Computational Physics. 2007; 255: 1533-1552
36. Li-hong C, Zheng-xing C. A method of construction for biorthogonal multiwavelets system with 2r multiplicity. Applied Mathematics and Computation. 2005; 167: 901-918
37. Lotfi A, Dehghan M, Yousefi SA. A numerical technique for solving fractional optimal control problems. Computers and Mathematics with Applications. 2011; 62: 1055-1067
38. Magin M, Feng X, Baleanu D. Solving the fractional order Bloch equation. Concepts in Magnetic Resonance: Part A. 2009; 34A (1). 16-23
39. Miller KS, Ross B An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: John Wiley and Sons; 1993
40. Muslih S, Baleanu D. Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives. Journal of Mathematical Analysis and Applications. 2005; 304: 599-603
41. Mustapha O, Lefebvre D, Khalil M, Hoblos G, Chafouk H. Fault detection algorithm using DCS method combined with filters bank derived from the wavelet transform. International Journal of Innovative Computing, Information and Control. 2009; 5: 1313-1328
42. N'Guérékata GM. A Cauchy problem for some fractional abstract differential equations with fractional order with nonlocal conditions. Nonlinear Analysis. 2009; 70: 1873-1876
43. Odibat Z, Momani S. A generalized differential transform method for linear partial differential equations of fractional order. Applied Mathematics Letters. 2008; 21: 194-199
44. Oldham KB, Spanier J The Fractional Calculus. New York: Academic Press; 1974
45. Podlubny I Fractional Differential Equations. San Diego, CA: Academic Press; 1999
46. Qin P, Zhang X. A numerical method for the space-time fractional convection-diffusion equation. Mathematic Numerica Sinica. 2008; 30: 305-310
47. Saadatmandi A, Dehghan M. A new operational matrix for solving fractional-order differential equations. Computers and Mathematics with Applications. 2010; 59: 1326-1336
48. Saadatmandi A, Dehghan A. A Legendre collocation method for fractional integro-differential equations. Journal of Vibration and Control. 2011a; 17: 2050-2058
49. Saadatmandi A, Dehghan M. A tau approach for solution of the space fractional diffusion equation. Computers and Mathematics with Applications. 2011b; 62: 1135-1142
50. Saadatmandi A, Dehghan M, Azizi MR. The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients. Communications in Nonlinear Science and Numerical Simulation. 2012; 17: 4125-4136
51. Strela V, Heller PN, Strang G, Topiwala P, Heil C. The application of multiwavelet filter banks to image processing. IEEE Transactions on Image Processing. 1999; 8: 548-563
52. Vasundhara Devi J, Lakshmikantham V. Nonsmooth analysis and fractional differential equations. Nonlinear Analysis. 2009; 70: 4151-4157
53. Wang Q. Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method. Applied Mathematics and Computation. 2006; 182: 1048-1055
54. Wu JL. A wavelet operational method for solving fractional partial differential equations numerically. Applied Mathematics and Computation. 2009; 214: 31-40
55. Xia XG, Geronimo JS, Hardin DP, Suter BE. Design of prefilters for discrete multiwavelet transform. IEEE Transactions on Signal Processing. 1996; 44: 25-35
56. Xing Y, Wu X, Xu Z. Multiclass least squares auto-correlation wavelet support vector machines. ICIC Express Letters. 2008; 2: 345-350
57. Yousefi S, Razzaghi M. Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations. Mathematics and Computers in Simulation. 2005; 70: 1-8
58. Yousefi SA, Lotfi A, Dehghan M. The use of Legendre multiwavelet collocation method for solving the fractional optimal control problems. Journal of Vibration and Control. 2011; 17: 2059-2065
59. Zhong W, Lin W. Nonlocal and multiple-point boundary value problem for fractional differential equations. Computers and Mathematics with Applications. 2010; 59: 1345-1351
60. Zhou Y, Jiao F, Li J. Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Analysis. 2009; 71: 3249-3256
61. Zhou Y, Jiao F, Li J. Existence and uniqueness for p-type fractional neutral differential equations. Nonlinear Analysis. 2009; 71: 2724-2733
62. Zhuang P, Liu F. An explicit approximation for the spacetime fractional diffusion equation. Numerical Mathematics A: Journal of Chinese Universities. 2005; 27: 223-228