The aim of this paper is to solve some fractional differential problems hav-
ing time fractional derivative by means of a wavelet Galerkin method that
uses the fractional scaling functions introduced in a previpous paper as approximating
functions. These refinable functions, which are a generalization of the
fractional B-splines, have many interesting approximation properties.
In particular, their fractional derivatives have a closed form that involves
just the fractional difference operator. This allows us to construct accurate
and efficient numerical methods to solve fractional differential problems.
Some numerical tests on a fractional diffusion problem will be given

PEZZA, Laura

PITOLLI, Francesca

English

Archivio della ricerca- Università di Roma La Sapienza

MASCOT2015 - IMACS/ISGG Workshop IAC-CNR, Rome, ItalyA fractional wavelet Galerkin methodfor the fractional diffusion problemLaura Pezza, Francesca PitolliDip. SBAI, Universita` di Roma “La Sapienza”,Via A. Scarpa 16, 00161 Roma, Italylaura.pezza@sbai.uniroma1.it, francesca.pitolli@sbai.uniroma1.itAbstractThe aim of this paper is to solve some fractional differential problems hav-ing time fractional derivative by means of a wavelet Galerkin method thatuses the fractional scaling functions introduced in [13] as approximatingfunctions. These refinable functions, which are a generalization of thefractional B-splines [16], have many interesting approximation properties.In particular, their fractional derivatives have a closed form that involvesjust the fractional difference operator. This allows us to construct accurateand efficient numerical methods to solve fractional differential problems.Some numerical tests on a fractional diffusion problem will be given.Keywords: Fractional diffusion problem, Wavelet Galerkin method, Fractionalrefinable function1. IntroductionFractional calculus arises in several fields, from physics to continuum mechanics,from signal processing to electromagnetism. In particular, fractional differentialequations are recently used to model a variety of real world phenomena, such aswave propagation in porous materials, diffusive phenomena in biological tissue,viscoelasticity. For a survey on the subject see, for instance, [8, 9, 11, 15] andreferences therein.The goal of this paper is to numerically solve diffusion problems having frac-tional time derivative using the fractional refinable functions introduced in [13]as approximating functions. These functions, which are a generalization of thefractional B-splines [16], have many interesting approximation properties. Inparticular, their fractional derivatives have a closed form that involves just thefractional difference operator so that it can be easily evaluated.This allows us to construct an accurate and efficient numerical method, the1fractional wavelet Galerkin method, which uses a wavelet Galerkin method inspace and a collocation method in time.The paper is organized as follows. In Section 2, a space-time fractional diffusionproblem is presented and the definition of fractional derivative is given. Thenumerical method we propose to solve this fractional diffusion problem is intro-duced in Section 3. In Section 4, the class of fractional refinable functions we areinterested in is described and their main properties are recalled. We will exploitthe properties of the refinable functions in the class to efficiently evaluate thecoefficients of the linear, algebraic system arising from the fractional waveletGalerkin method. Finally, in Section 5 the numerical results obtained in sometests will be displayed.2. A fractional diffusion problemWe consider the space-fractional time differential diffusion problem [3, 10]Dγt u(t, x)−∂2∂x2u(t, x) = f(t, x) , t ∈ [0, T ] , x ∈ [0, 1] ,u(0, x) = 0 , x ∈ [0, 1] ,u(t, 0) = u(t, 1) = 0 , t ∈ [0, T ] ,(1)where Dγt u, 0 < γ < 1, denotes the partial fractional derivative with respect tothe time t. Here, we follow the Riemann-Liouville definition, i.e.Dγt u(t, x) =1Γ(1− γ)∂∂t∫ t0u(τ, x)(t− τ)γdτ , (2)where Γ is the Euler’s gamma functionΓ(γ + 1) =∫ ∞0sγ e−s ds . (3)We notice that due to the homogeneous initial condition, for the function u(t, x),solution of the differential problem (1), the Riemann-Liouville definition (2)coincides with the Caputo definition [14]. One of the advantage of the Riemann-Liouville definition is in that the usual differentiation operator in the Fourierdomain can be easily extended to the fractional case, i.e.F(Dγt v(t))= (iω)γF(v(t)) , (4)where F(v) denotes the Fourier transform of the function v. Moreover, theRiemann-Liouville definition coincides with the Grunwald-Letnikov definitionDγt u(t, x) = limh→01hnth∑k=0(−1)k(γk)u(t− kh, x) , (5)2where (γk):=Γ(γ + 1)k! Γ(γ − k + 1), k ∈ Z+0 , (6)are the generalized binomial coefficients (Z+0 denotes the set Z+ ∪ 0). Theyreduce to the usual binomial coefficients when γ is an integer. We note thatthe Grunwald-Letnikov definition (5) is easier to use when addressing numericalsolution of fractional differential problems.In the following section we will introduce the wavelet Galerkin method to nu-merically solve the fractional differential problem (1).3. The wavelet Galerkin methodLet us introduce the sequence of approximating spacesVj = span {ϕjk = ϕ(2j · −k), k ∈ Z} , j ∈ Z , (7)where ϕ is a refinable function, i.e. a function defined through a refinementmask a = {ak ∈ R, k ∈ Z} and a refinement equationϕ =∑k∈Zak ϕ(2 · −k) . (8)Suitable conditions on the mask coefficients {ak} ensure the existence of a uniquefunction ϕ solution of (8), belonging to L2(R) and such that the space sequence{Vj} forms a multiresolution analysis (see [12] for details).For any j held fix, the wavelet Galerkin method looks for an approximatingfunctionuj(t, x) =∑k∈Zck(t)ϕjk(x) ∈ Vj (9)that solves the variational problem(Dγt uj , ϕjk)−(∂∂x2uj , ϕjk)= (f, ϕjk) , k ∈ Z ,uj(0, x) = 0 , x ∈ [0, 1] ,uj(t, 0) = 0 , uj(t, 1) = 0 , t ∈ [0, T ] ,(10)where (f, g) =∫ 10f g. Now, writing (10) in a weak form and using (9) we getthe system of fractional ordinary differential equationsM Dγt C(t) = LC(t) + F (t) , t ∈ [0, T ] ,C(0) = 0 ,(11)3where C(t) = (ck(t))k∈Z is the unknown vector. The connecting coefficients,i.e. the entries of the mass matrix M = (mki)k,i∈Z, of the stiffness matrixL = (ℓki)k,i∈Z and of the load vector F (t) = (fk(t))k∈Z, are given bymki =∫ 10ϕjk ϕji , ℓki =∫ 10ϕ′jk ϕ′ji , fk(t) =∫ 10f(t, ·)ϕjk .Due to the refinability of ϕ, the entries of M and L can be evaluated by solvingan eigenvalue-eigenvector problem [2]. The entries of F can be evaluated byquadrature formulas especially designed for wavelet methods [1, 4].To solve the fractional differential system (11) we use a collocation methodon dyadic nodes. For an integer value of T , let tp = p/2s, p = 0, . . . , T 2s, wheres is a given integer greater than 1, be a set of dyadic nodes in the dyadic interval[0, T ]. Now, assumingck(t) =∑r∈Zλrk χsr(t) , k ∈ Z , (12)where χsr(t) = χ(2s t− r) is χ a refinable function, and collocating (11) on thenodes tp, we get the linear system{(M ⊗A− L⊗B) Λ = FXTΛ = 0(13)where Λ = (λrk)r,k,∈Z,A = (apr)0≤p≤2sT,r∈Z , apr = Dγt χsr(tp) , (14)B = (bpr)0≤p≤2sT,r∈Z , bpr = χsr(tp) , (15)F = (fk(tp))k,∈Z,0≤p≤2sT and X = (χsr(0))r∈Z. Since the refinable functionsare compactly supported or have fast dacay, in practical applications the linearsystem (13) has finite dimension and the unknown vector Λ can be recoveredby solving (13) in the least square sense.We notice that the entries of B and X, which involve just the values of χsr onthe dyadic nodes tp, can be evaluated by usual refinement techniques [12]. Onthe other hand, we must pay a special attention to the evaluation of the entriesof A since they involve the values of the fractional derivative Dαt χrs. In thefollowing section we will exploit the properties of a particular class of refinablefunctions to accurately evaluate the matrix A.4. A class of fractional refinable functionsIn [13] a class of fractional refinable functions was introduced. The refinablefunctions belonging to this class are identified through the family of refinement4Figure 1: Top panels: the fractional mask and the fractional refinable functionfor α = 3.5. Bottom panels: the fractional derivative when γ = 0.2, 0.8.masks a(α,h) = {a(α,h)k , k ∈ Z+} that depend on the real parameters α ≥ 2 andh ≥ α. The mask coefficients have an explicit espression given bya(α,h)k =12h[(α+ 1k)+ 4(2h−α − 1)(α− 1k − 1)], k ∈ Z+0 . (16)(We assume(αk)= 0 when k < 0.)For each admissible values of α and h, the corresponding refinable function turnsout to be the unique solution of the refinement equationϕ(α,h) =∑k∈Z+0a(α,h)k ϕ(α,h)(2 · −k) . (17)We note that setting h = α in (16) givesa(α)k := a(α,α)k =12α(α+ 1k), k ∈ Z+0 , (18)which are the mask coefficients of the fractional B-spline so that ϕ(α,α) coincideswith the fractional B-spline Bα introduced in [16]. Bα is a piecewise polynomialof real degree α and reduces to the classical integer degree polynomial B-splinefor integer values of α (see [16] for details). Thus, the class of functions ϕ(α,h)can be seen as a generalization of the fractional B-splines and this is the reasonwhy they are called fractional refinable functions.Even if ϕ(α,h) has compact support only when α has an integer value, forany admissible value of α and h, ϕ(α,h) belongs to L2(R) and decays to theinfinity rather rapidly. Moreover, ϕ(α,h) gives rise to a multiresolution analysis5of L2(R). As a consequence, the approximating spacesV(α,h)j = span {ϕ(α,h)jk = ϕ(α,h)(2j · −k), k ∈ Z} , j ∈ Z , (19)are suitable to be used in the wavelet Galerkin method.As for the order of exactness, we recall that ϕ(α,h) has order of polynomialexactness d = ⌈α⌉ − 1, i.e. there exist sequences {qlk} such thatxl =∑k∈Zqlk ϕ(α,h)(x− k) , l = 0, . . . , ⌈α⌉ − 2 , (20)(see [13] for details and further properties).A remarkable property of the fractional refinable functions, especially usefulin the numerical solution of fractional differential problems, is the followingdifferentiation rule [13]:Dγx ϕ(α,h) = ∆γ ϕ(α−γ,h) =∑k∈Z+(−1)k(αk)ϕ(α−γ,h)(· − k) , γ > 0 . (21)This makes fractional differentiation very easy since fractional derivatives can beevaluated just by a finite difference formula. As a consequence, the computationof the coefficients apr in (14) is very efficient, as we will show in the followingsection.In Figure 1 (upper panels) the mask coefficients a(3.5,h)k , k = 0, 1, . . . , 8, and thecorresponding refinable function ϕ(3.5,h) are displayed for different values of h.In Figure 1 (lower panels) the fractional derivatives Dγx ϕ(3.5,h) are displayed forγ = 0.2 and γ = 0.8. The pictures show that ϕ(3.5,h) decays rapidly to infinityand is almost positive.5. A case studyIn this section we will apply the fractional wavelet Galerkin method using thefractional refinable functions ϕ(α,h) to solve the fractional diffusion problem (1)when f(t, x) = 2Γ(3−γ) t2−α sin(2πx) + 4π2 t2 sin(2πx). In this case the exactsolution is u(t, x) = t2 sin(2πx) (cf. [3]).To numerically solve the problem we approximate u(t, x) by using the fractionalrefinable functions described in the previous section. More precisely, for somegiven set of admissible values H = (α1, h1, α2, h2), we assumeu(H)j (t, x) =∑k∈Zc(α1,h1)k (t)ϕ(α2,h2)jk (x) ∈ V(α2,h2)j (22)as approximating function withc(α1,h1)k (t) =∑r∈Zλrk ϕ(α1,h1)sr (t) , k ∈ Z . (23)6Figure 2: The refinable basis {ϕ(3,3)3k } in the interval [0,1] and its first derivative(upper panels) and the refinable basis {ϕ(3.5,3.5)4k } and its fractional derivativefor γ = 0.2 (lower panels).Since just the ordinary first derivative of ϕ(α2,h2)jk is involved in the numericalmethod, we can assume α2 integer. In this case ϕ(α2,h2) reduces to the compactlysupported refinable function belonging to the general class introduced in [5]so that the connecting coefficients can be evaluated as in [7]. Moreover, theexistence and uniqueness of the solution of the Galerkin problem (10) and theconvergence of the wavelet Galekin method follows from some results in [7].On the other hand, the fractional refinable function ϕ(α1,h1) has a fast decayto the infinity so that it can be assumed compactly supported and the series in(23) are well approximated by only few terms.In the following tests we set α1 = 3.5, h1 ≥ 3.5, α2 = 3, h2 ≥ 3. The refinablebasis for α2 = 3 and h2 = 3, i.e. the cubic B-spline basis, on the interval [0,1]at resolution level j = 3 is displayed in Figures 2-3 (upper panels) along itsfirst derivative (we refer to [6] for details on the construction of refinable baseson bounded intervals). The refinable basis for α2 = 3.5 and h2 = 3.5 on theinterval [0,1] at resolution level s = 4 is displayed in Figures 2-3 (lower panels)along its 0.2 and 0.8 fractional derivatives.In Figure 4, the numerical solution u(H)j and the error u − u(H)j obtainedwhen using the refinable basis in Figures 2-3 are displayed.6. ConclusionsWe exploit the properties of a class of fractional refinable functions to constructa wavelet method to solve a space-fractional time partial diffusion problem. Themain advantage in using fractional refinable functions is in that they satisfy an7Figure 3: The refinable basis {ϕ(3,3)3k } in the interval [0,1] and its first derivative(upper panels) and the refinable basis {ϕ(3.5,3.5)4k } and its fractional derivativefor γ = 0.8 (lower panels).exact and very simple formula for the evaluation of the fractional derivatives sothat the resulting fractional wavelet Galerkin method is accurate and efficient,as shown in the numerical tests in Section 4.In a forthcoming paper we will address the solution of fractional diffusion prob-lems in Rd, d ≤ 3, by the fractional wavelet Galerkin problem.References[1] F. Calabro`, C. Manni & F. Pitolli, Computation of quadrature rules forintegration with respect to refinable functions on assigned nodes, AppliedNumerical Mathematics, 90 (2015), 168–189.[2] W. Dahmen & C.A. Micchelli, Using the refinement equation for evaluatingintegrals of wavelets, SIAM Journal of Numerical Analysis, 30 (1993), 507–537.[3] N. J. Ford, J. Xiao & Y. Yan, A finite element method for time fractionalpartial differential equations, Fractional Calculus and Applied Analysis, 14(2011), 454–474.[4] W. Gautschi, L. Gori & F. Pitolli, Gauss quadrature for refinable weightfunctions, Applied and Computational Harmonic Analysis, 8 (2000), 249–257.[5] L. Gori & F. Pitolli, A class of totally positive refinable functions Rendicontidi Matematica, Serie VII, 20 (2000), 305–322.8Figure 4: The numerical solution and the relative error for γ = 0.2 (upperpanels) and γ = 0.8 (lower panels).9[6] L. Gori, L. Pezza & F. Pitolli, A class of totally positive blending B-bases, inCurve and Surface Design: Saint Malo 1999, (P.-J. Laurent, P. Sablonnire,L.L. Schumaker eds.), Vanderbilt University Press, Nashville, TN (2000),pp. 119–126.[7] L. Gori, F. Pitolli & L. Pezza, On the Galerkin method based on a particularclass of scaling functions, Numerical Algorithms, 28 (2001), 187–198.[8] A.A. Kilbas, H.M. Srivastava & J.J. Trujillo, Theory and applications offractional differential equations, North-Holland Mathematics Studies, Vol.204, Elsevier Science, Amsterdam (2006).[9] R. Hilfer (ed.), Applications of fractional calculus in physics, World ScientificPublishing Co., Inc., River Edge, NJ (2000).[10] X.J. Li & C.J. Xu, A space-time spectral method for the time fractionaldiffusion equation, SIAM J. Numer. Anal. 47 (2009), 2108–2131.[11] F. Mainardi, Fractional calculus and waves in linear viscoelasticity. An in-troduction to mathematical models., Imperial College Press, London (2010).[12] S. Mallat, A wavelet tour of signal processing, Academic Press (1999).[13] L. Pezza, Fractional GP refinable functions, Rendiconti di Matematica,Serie VII, 27 (2007), 73–87.[14] I. Podlubny, Fractional differential equations, Mathematics in Science andEngineering, Vol. 198, Academic Press (1999).[15] V.E. Tarasov, Fractional dynamics. Applications of fractional calculus todynamics of particles, fields and media, Nonlinear Physical Science. Springer,Heidelberg (2010).[16] M. Unser & T. Blu, Fractional splines and wavelets, in SIAM (1) 42 (2000),43–67.10

A fractional wavelet Galerkin method for the fractional diffusion problem

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A fractional wavelet Galerkin method for the fractional diffusion problem

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