This paper is concerned with the construction of biorthogonal multiwavelet basis in the unit interval to form a biorthogonal flatlet multiwavelet system. Next a method to calculate integer and fractional derivatives of the dual flatlet multiwavelets by multiplying some matrices is suggested. The system is then used to solve a fractional convection–diffusion equation. The biorthogonality and high vanishing moments properties of this system result in efficient and accurate solutions. Finally, numerical results for some test problems with known solutions are presented and the absolute errors are compared with the errors resulting from the other bases.
AgrawalOP (2002) Solution for a fractional diffusion–wave equation defiend in a bounded domain. Journal of Nonlinear Dynamics29: 145–155.
2.
AlpertB (1993) A class of basis in L2 for the sparse representation of integral oerators. SIAM Journal of Mathematical Analysis24: 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.
AlpertBBeylkinGCoifmanRRRokhlinV (1993) Wavelet–like basis for the fast solution of second-kind integral equations. SIAM Journal of Scientific and Statistical Computing14: 159–184.
5.
AlpertBBeylkinGGinesDVozovoiL (2002) Adaptive solution of partial differential equations in multiwavelet bases. Journal of Computational Physics182: 149–190.
6.
BaleanuD (2006) Fractional Hamiltoian analysis of irregular systems. Signal Processing86: 2632–2636.
7.
BaleanuDAvkarT (2004) Lagrangians with linear velocities within Riemann–Liouville fractional derivatives. Nuovo Cimento B119: 73–79.
8.
BaleanuDDiethelmKScalasETrujilloJJ (2012) Fractional Calculus Models and Numerical Methods (Series on Complexity, Nonlinearity and Chaos), Singapore: World Scientific.
9.
BaleanuDGüvençZBTenreiro MachadoJA (2010) New Trends in Nanotechnology and Fractional Calculus Applications, New York: Springer.
10.
BaleanuDMuslihSI (2005a) Formulation of Hamiltonian equations for fractional variational problems. Czechoslovac Journal of Physics55: 633–642.
11.
BaleanuDMuslihS (2005b) Lagrangian formulation of classical fields within Riemann–Liouville fractional derivatives. Physica Scripta72: 119–121.
12.
BaleanuDTrujilloJJ (2008) On exact solutions of a class of fractional Euler–Lagrange equations. Nonlinear Dynamics52: 331–335.
13.
BenchohraMHamaniSNtouyasSK (2009) Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Analysis TMA71: 2391–2396.
14.
BeylkinGCoifmanRRokhlinV (1991) Fast wavelet transforms and numerical algorithms I. Communications on Pure and Applied Mathematics44: 141–183.
15.
CaputoM (1967) Linear models of dissipation whose Q is almost frequency independent, Part II. Journal of the Royal Australian Society13: 529–539.
16.
ChenYWuYCuiYWangZJinD (2010) Wavelet method for a class of fractional convection-diffusion equation with variable coefficients. Journal of Computational Science1: 146–149.
17.
CohenADahmenWDeVoreR (2001) Adaptive wavelet methods for elliptic operator equations - convergence rates. Mathematics of Computation70: 27–75.
18.
DahmenWHanBJiaRQKunothA (2000) Biorghogonal multiwavelets on the interval: cubic Hermite splines. Constructive Approximation16: 221–259.
19.
DaraniMRAAdibiHAgarwalRPSaadatiR (2008) Flatlet oblique multiwavelet for solving integro–differential equations. Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Matematical Analysis15: 755–768.
20.
DaraniMRAAdibiHLakestaniM (2010) Numerical solution of integro-differential equations usnig flatlet oblique multiwavelets. Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis17: 55–74.
21.
DasS (2011) Functional Fractional Calculus, New York: Springer.
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.
DongJWeiXZhangQZhaoY (2009) Pitch detection using circular average magnitude difference function based on wavelet transform. International Journal of Innovative Computing, Information and Control5: 2717–2724.
25.
GoodmanTNTLeeSL (1994) Wavelets of multiplicityr. Transactions of the American Mathematical Society342: 307–324.
26.
HanBJiangQT (2002) Multiwavelets on the interval. Applied and Computational Harmonic Analysis12: 100–127.
27.
KayediAEslahchiMRDehghanM (2012) A method for obtaining the operational matrix of fractional Jacobi functions and applications. Journal of Vibration and Control. in press, DOI: 10.1177/1077546312467049.
28.
KeinertF (2003) Wavelets and Multiwavelets, New York: Chapman and Hall/CRC.
29.
KhaderMM (2011) On the numerical solutions for the fractional diffusion equation. Communications in Nonlinear Science and Numerical Simulation16: 2535–2542.
30.
KilbasAASrivastavaHMTrujilloJJ (2006) Theory and Applications of Fractional Differential Equations, San Diego, CA: Elsevier.
31.
LakestaniMDehghanMIrandoust–PakchinS (2012) The construction of operational matrix of fractional derivatives using B-spline functions. Communications in Nonlinear Science and Numerical Simulation17: 1149–1162.
32.
LakestaniMRazzaghiMDehghanM (2006) Semiorthogonal spline wavelets approximation for Fredholm integro-differential equations. Mathematical Problems in Engineering2006: 96184.
33.
LakestaniMSarayBN (2010) Numerical solution of telegraph equation using interpolating scaling functions. Computers and Mathematics with Applications60: 1964–1972.
34.
LakestaniMSarayBNDehghanM (2011) Numerical solution for the weakly singular Fredholm integro-differential equations using Legendre multiwavelets. Journal of Computational and Applied Mathematics235: 3291–3303.
35.
LinYXuC (2007) Finite difference/spectral approximation for the time-fractional differential diffusion equation. Journal of Computational Physics255: 1533–1552.
36.
Li-hongCZheng-xingC (2005) A method of construction for biorthogonal multiwavelets system with 2r multiplicity. Applied Mathematics and Computation167: 901–918.
37.
LotfiADehghanMYousefiSA (2011) A numerical technique for solving fractional optimal control problems. Computers and Mathematics with Applications62: 1055–1067.
38.
MaginMFengXBaleanuD (2009) Solving the fractional order Bloch equation. Concepts in Magnetic Resonance: Part A34A(1): 16–23.
39.
MillerKSRossB (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations, New York: John Wiley and Sons.
40.
MuslihSBaleanuD (2005) Hamiltonian formulation of systems with linear velocities within Riemann–Liouville fractional derivatives. Journal of Mathematical Analysis and Applications304: 599–603.
41.
MustaphaOLefebvreDKhalilMHoblosGChafoukH (2009) Fault detection algorithm using DCS method combined with filters bank derived from the wavelet transform. International Journal of Innovative Computing, Information and Control5: 1313–1328.
42.
N'GuérékataGM (2009) A Cauchy problem for some fractional abstract differential equations with fractional order with nonlocal conditions. Nonlinear Analysis70: 1873–1876.
43.
OdibatZMomaniS (2008) A generalized differential transform method for linear partial differential equations of fractional order. Applied Mathematics Letters21: 194–199.
44.
OldhamKBSpanierJ (1974) The Fractional Calculus, New York: Academic Press.
45.
PodlubnyI (1999) Fractional Differential Equations, San Diego, CA: Academic Press.
46.
QinPZhangX (2008) A numerical method for the space–time fractional convection–diffusion equation. Mathematic Numerica Sinica30: 305–310. (in Chinese).
47.
SaadatmandiADehghanM (2010) A new operational matrix for solving fractional-order differential equations. Computers and Mathematics with Applications59: 1326–1336.
48.
SaadatmandiADehghanA (2011a) A Legendre collocation method for fractional integro–differential equations. Journal of Vibration and Control17: 2050–2058.
49.
SaadatmandiADehghanM (2011b) A tau approach for solution of the space fractional diffusion equation. Computers and Mathematics with Applications62: 1135–1142.
50.
SaadatmandiADehghanMAziziMR (2012) The Sinc–Legendre collocation method for a class of fractional convection–diffusion equations with variable coefficients. Communications in Nonlinear Science and Numerical Simulation17: 4125–4136.
51.
StrelaVHellerPNStrangGTopiwalaPHeilC (1999) The application of multiwavelet filter banks to image processing. IEEE Transactions on Image Processing8: 548–563.
WangQ (2006) Numerical solutions for fractional KdV–Burgers equation by Adomian decomposition method. Applied Mathematics and Computation182: 1048–1055.
54.
WuJL (2009) A wavelet operational method for solving fractional partial differential equations numerically. Applied Mathematics and Computation214: 31–40.
55.
XiaXGGeronimoJSHardinDPSuterBE (1996) Design of prefilters for discrete multiwavelet transform. IEEE Transactions on Signal Processing44: 25–35.
56.
XingYWuXXuZ (2008) Multiclass least squares auto-correlation wavelet support vector machines. ICIC Express Letters2: 345–350.
57.
YousefiSRazzaghiM (2005) Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations. Mathematics and Computers in Simulation70: 1–8.
58.
YousefiSALotfiADehghanM (2011) The use of Legendre multiwavelet collocation method for solving the fractional optimal control problems. Journal of Vibration and Control17: 2059–2065.
59.
ZhongWLinW (2010) Nonlocal and multiple-point boundary value problem for fractional differential equations. Computers and Mathematics with Applications59: 1345–1351.
60.
ZhouYJiaoFLiJ (2009) Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Analysis71: 3249–3256.
61.
ZhouYJiaoFLiJ (2009) Existence and uniqueness for p-type fractional neutral differential equations. Nonlinear Analysis71: 2724–2733.
62.
ZhuangPLiuF (2005) An explicit approximation for the spacetime fractional diffusion equation. Numerical Mathematics A: Journal of Chinese Universities27: 223–228.
