A finite element method (FEM) for multiterm fractional partial differential equations (MT-FPDEs) is studied for obtaining a numerical solution effectively. The weak formulation for MT-FPDEs and the existence and uniqueness of the weak solutions are obtained by the well-known Lax-Milgram theorem. The Diethelm fractional backward difference method (DFBDM), based on quadrature for the time discretization, and FEM for the spatial discretization have been applied to MT-FPDEs. The stability and convergence for numerical methods are discussed. The numerical examples are given to match well with the main conclusions.
1. Introduction
In recent years, the numerical treatment and supporting analysis of fractional order differential equations has become an important research topic that offers great potential. The FEMs for fractional partial differential equations have been studied by many authors (see [1–3]). All of these papers only considered single-term fractional equations, where they only had one fractional differential operator. In this paper, we consider the MT-FPDEs, which include more than one fractional derivative. Some authors also considered solving linear problems with multiterm fractional derivatives (see [4, 5]). This motivates us to consider their effective numerical solutions for such MT-FPDEs, which have been proposed in [6, 7].
Let Ω=(0,X)d, where d≥1 is the space dimension. We consider the MT-FPDEs with the Caputo time fractional derivatives as follows:
(1)P(DCt)u(t,x)-Δxu(t,x)=f(t,x),t∈[0,T],x∈Ω,(2)u(0,x)=u0(x),x∈Ω,(3)u(t,x)=0,t∈[0,T],x∈∂Ω,
where the operator P(CDt)u(t,x) is defined as
(4)P(DCt)u(t,x)=(D0Ctα+∑i=1sai0CDtαi)u(t,x),
with 0<αs<αs-1<⋯<α1<α<1 and {ai>0}i=1s. Here 0CDtαu(t,x) denotes the left Caputo fractional derivative with respect to the time variable t and Δx denotes the Laplace operator with respect to the space variable x.
Some numerical methods have been considered for solving the multiterm fractional differential equations. In [8], Liu et al. investigate some effective numerical methods for time fractional wave-diffusion and diffusion equations:
(5)0CDtαu(t,x)-kΔxu(t,x)=f(t,x),0<x<L,t>0,
where k and L are arbitrary positive constants and f(t,x) is a sufficiently smooth function. The authors consider the implicit finite difference methods (FDMs) and prove that it is unconditionally stable. The error estimate of the FDM is O(Δt+Δt2-α+Δx), where Δt and Δx are the time and space step size, respectively. They also investigate the fractional predictor-corrector methods (FPCMs) of the Adams-Moulton methods for multiterm time fractional differential equations (1) with order {αi}i=1,…,s by solving the equivalent Volterra integral equations. The error estimate of the FPCM is O(Δt+Δt1+min{αl}+Δx2). In recent years, there are some articles for the predictor-correction method for initial-value problems (see [9–14]). For the application of the FDMs, there have been many research articles as follows. In [15–20], Simos et al. investigate the numerical methods for solving the Schrödinger equation. In [21–24], the Runge-Kutta methods are considered and applied to get the numerical solution of orbital problems. For long-time integration, the Newton-Cotes formulae are considered in [25–27].
In [28], Badr investigate the FEM for linear multiterm fractional differential equations with one variable as follows:
(6)0CDt1+αu(t)+∑i=1sAi(x)Dαiu(x)=f(t),α≤n,αi<n-1,0<t<1,
where Ai(x) are known functions. The author gives the details of the modified Galerkin method for the above equations and makes the numerical example for checking the numerical method. In [29], Ford et al. consider the FEM for (5) with singular fractional order and obtain the error estimate O(Δt2-α+Δx2). In this paper, we follow the work in [29] and consider the FEM for solving MT-FPDEs (1)–(3). Then, we prove the stability and convergence of the FEM for MT-FPDEs and make the error estimate.
The paper is organized as follows. In Section 2, the weak formulation of the MT-FPDEs is given and the existence and uniqueness results for such problems are proved. In Section 3, we consider the convergence rate of time discretization of MT-FPDEs, based on the Diethelm fractional backward difference method (DFBDM). In Section 4, we propose an FEM based on the weak formulation and carry out the error analysis. In Section 5, the stability of this method is proven. Finally, the numerical examples are considered for matching well with the main conclusions.
2. Existence and Uniqueness
Let Γ(·) denote the gamma function. For any positive integer n and n-1<α<n, the Caputo derivative are the Riemann-Liouville derivative are, respectively, defined as follows [30].
The left Caputo derivatives:
(7)0CDtαv(t):=1Γ(n-α)∫0t1(t-τ)α-n+1(dndτnv(τ))dτ.
The left Riemann-Liouville derivatives:
(8)0RDtαv(t):=1Γ(n-α)dndtn∫0tv(τ)(t-τ)α-n+1dτ.
The right Riemann-Liouville derivatives:
(9)tRDTαv(t):=(-1)nΓ(n-α)dndtn∫tTv(τ)(τ-t)α-n+1dτ.
Let C∞(0,T) denote the space of infinitely differentiable functions on (0,T) and C0∞(0,T) denote the space of infinitely differentiable functions with compact support in (0,T). We use the expression A≲B to mean that A≤cB when c is a positive real number and use the expression A≅B to mean that A≲B≲A. Let L2(𝒬) be the space of measurable functions whose square is the Lebesgue integrable in 𝒬 which may denote a domain 𝒬=I×Ω, I or Ω. Here time domain I:=(0,T) and space domain Ω:=(0,X). The inner product and norm of L2(𝒬) are defined by
(10)(u,v)L2(𝒬):=∫𝒬uvd𝒬,∥u∥L2(𝒬):=(u,u)L2(𝒬)1/2,∀u,v∈L2(𝒬).
For any real σ>0, we define the spaces lH0σ(𝒬) and rH0σ(𝒬) to be the closure of C0∞(𝒬) with respect to the norms ∥v∥lH0σ(𝒬), and ∥v∥rH0σ(𝒬) respectively, where
(11)∥v∥lH0σ(𝒬):=(∥v∥L2(𝒬)2+|v|lH0σ(𝒬)2)1/2,|v|lH0σ(𝒬)2:=∥D0Rtσv∥L2(𝒬)2,∥v∥rH0σ(𝒬):=(∥v∥L2(𝒬)2+|v|rH0σ(𝒬)2)1/2,|v|rH0σ(𝒬)2:=∥DtRTσv∥L2(𝒬)2.
In the usual Sobolev space H0σ(𝒬), we also have the definition
(12)∥v∥H0σ(𝒬):=(∥v∥L2(𝒬)2+|v|H0σ(𝒬)2)1/2,|v|H0σ(𝒬)2:=(D0Rtσv,tRDTσv)L2(𝒬)cos(πσ).
From [3], for σ>0, σ≠n-1/2, the spaces lH0σ(𝒬), rH0σ(𝒬), and H0σ(𝒬) are equal, and their seminorms are all equivalent to |·|H0σ(𝒬). We first recall the following results.
Lemma 1 (see [<xref ref-type="bibr" rid="B3">3</xref>]).
Let 0<θ<2, θ≠1. Then for any w,v∈H0θ/2(0,T), then
(13)(D0Rtθw,v)L2(0,T)=(D0Rtθ/2w,tRDTθ/2v)L2(0,T).
From [3], we define the following space:
(14)Bα/2(I×Ω)=Hα/2(I,L2(Ω))∩Hα1/2(I,L2(Ω))∩⋯∩Hαs/2(I,L2(Ω))∩L2(I,H01(Ω))=Hα/2(I,L2(Ω))∩L2(I,H01(Ω)).
Here Bα/2(I×Ω) is a Banach space with respect to the following norm:
(15)∥v∥Bα/2(I×Ω)=(∥v∥Hα/2(I,L2(Ω))2+∥v∥L2(I,H01(Ω))2)1/2,
where Hα/2(I,L2(Ω)):={v;∥v(t,·)∥L2(Ω)∈Hα/2(I)}, endowed with the norm
(16)∥v∥Hα/2(I,L2(Ω)):=∥∥v(t,·)∥L2(Ω)∥Hα/2(I).
Based on the relation equation between the left Caputo and the Riemann-Liouville derivative in [31], we can translate the Caputo problem to the Riemann-Liouville problem. Then, we consider the weak formulation of (1) as follows. For f∈Bα/2(I×Ω)′, find u(t,x)∈Bα/2(I×Ω) such that
(17)𝒜(u,v)=ℱ(v),v∈Bα/2(I×Ω),
where the bilinear form is, by Lemma 1,
(18)𝒜(u,v)≔(D0Rtα/2u,tRDTα/2v)L2(I×Ω)+∑i=1sai(D0Rtαi/2u,tRDTαi/2v)L2(I×Ω)+(∇xu,∇xv)L2(I×Ω),
and the functional is ℱ(v):=(f-,v)L2(I×Ω), f-(t,x):=f(t,x)+(u0(x)t-α/Γ(1-α))+∑i=1sai(u0(x)t-αi/Γ(1-αi)).
Based on the main results in Subsection 3.2 in [32], we can prove the following existence and uniqueness theorem.
Theorem 2.
Assume that 0<α<1 and f-∈Bα/2(I×Ω)′. Then the system (17) has a unique solution in Bα/2(I×Ω). Furthermore,
(19)∥u∥Bα/2(I×Ω)≲∥f-∥Bα/2(I×Ω)′.
Proof.
The existence and uniqueness of the solution of (17) is guaranteed by the well-known Lax-Milgram theorem. The continuity of the bilinear form 𝒜 and the functional ℱ is obvious. Now we need to prove the coercivity of 𝒜 in the space Bα/2(I×Ω). From the equivalence of lH0α(I×Ω), rH0α(I×Ω) and H0α(I×Ω), for all u,v∈Bα/2(I×Ω), using the similar proof process in [32], we obtain
(20)𝒜(v,v)≳(D0Rtα/2v,0RDtα/2v)L2(I×Ω)+∑i=1sai(D0Rtαi/2v,0RDtαi/2v)L2(I×Ω)+(∇xu,∇xv)L2(I×Ω)≳∥v∥Bα/2(I×Ω)2.
Then we take v=u in (17) to get ∥u∥Bα/2(I×Ω)2≲(f-,u)L2(I×Ω) by the Schwarz inequality and the Poincaré inequality.
3. Time Discretization and Convergence
In this section, we consider DFBDM for the time discretization of (1)–(3), which is introduced in [33] for fractional ordinary differential equations. We can obtain the convergence order for the time discretization for the MT-FPDEs. Let A=-Δx, D(A)=H01(Ω)∩H2(Ω). Let u(t), f(t), and u(0) denote the one-variable functions as u(t,·), f(t,·), and u(0,·), respectively. Then (1) can be written in the abstract form, for 0<t<T, 0<αs<⋯<α1<α<1, with initial value u(0)=u0. Now we have
(21)0RDtα[u-u0](t)+∑i=1sai0RDtαi[u-u0](t)+Au(t)=f(t).
Let 0=t0<t1<⋯<tN=T be a partition of [0,T]. Then, for fixed tj, j=1,2,…,N, we have
(22)0RDtα[u-u0](tj)=tj-αΓ(-α)∫01g(τ)τ-1-αdτ,
where g(τ)=u(tj-tjτ)-u0. Here, the integral is a Hadamard finite-part integral in [33] and [34].
Now, for every j, we replace the integral by a first-degree compound quadrature formula with equispaced nodes 0,(1/j),(2/j),…,1 and obtain
(23)∫01g(τ)τ-1-αdτ=∑k=0jαkj(α)g(kj)+Rj(α)(g),
where the weights αkj(α) are
(24)α(1-α)j-ααkj(α)={-1,fork=0,2k1-α-(k-1)1-α-(k+1)1-α,fork=1,2,…,j-1,(α-1)k-α-(k-1)1-α+k1-α,fork=j,
and the remainder term Rj(α)(g) satisfies ∥Rj(α)(g)∥≤γαjα-2sup0≤t≤T∥g′′(t)∥, where γα>0 is a constant.
Thus, for ωkj(α)=j-ααkj(α)/Γ(-α), we have
(25)D0Rtα[u-u0](tj)=Δt-α∑k=0jωkj(α)(u(tj-tk)-u(0))+tj-αΓ(-α)Rj(α)(g).
Let t=tj, we can write (21) as
(26)Δt-α∑k=0jωkj(α)(u(tj-tk)-u(0))+∑i=1saiΔt-αi∑k=0jωkj(αi)(u(tj-tk)-u(0))+Au(tj)=f(tj)-tj-αΓ(-α)Rj(α)(g)-∑i=1saitj-αiΓ(-αi)Rj(αi)(g),j=1,2,3,….
Denote Uj as the approximation of u(tj) and fj=f(tj). We obtain the following equation:
(27)Δt-α∑k=0jωkj(α)(Uj-k-U0)+∑i=1saiΔt-αi∑k=0jωkj(αi)(Uj-k-U0)+AUj=fj.
Lemma 3 (see [<xref ref-type="bibr" rid="B34">34</xref>]).
For 0<α<1, let the sequence {dj}j=1,2,… be given by d1=1 and dj=1+α(1-α)j-α∑k=1j-1αkj(α)dj-k. Then, 1≤dj≤(sin(πα)/πα(1-α))jα, for j=1,2,3,….
Let ej=u(tj)-Uj denote the error in tj. Then we have the following error estimate.
Theorem 4.
Let Uj and u(tj) be the solutions of (27) and (21), respectively. Then one has ∥Uj-u(tj)∥≲Δt2-α.
Proof.
Subtracting (27) from (26), we obtain the error equation
(28)Δt-α∑k=0jωkj(α)(ej-k-e0)+∑i=1saiΔt-αi∑k=0jωkj(αi)(ej-k-e0)+Aej=-tj-αΓ(-α)Rj(α)(g)-∑i=1saitj-αiΓ(-αi)Rj(αi)(g).
Note that e0=u(0)-U0=0. Denote
(29)ej=(α0j(α)+∑i=1saiΔtα-αiΓ(-α)Γ(-αi)α0j(αi)+AtαΓ(-α))-1×(∑k=1jαkj(α)ej-k+∑i=1saiΔtα-αiΓ(-α)Γ(-αi)∑k=1jαkj(αi)ej-k-Rj(α)(g)-∑i=1saiΓ(-α)Γ(-αi)tjα-αiRj(αi)(g)).
Let ∥·∥ denote the L2-norm, then we have
(30)∥ej∥≤∥(α0j(α)+∑i=1saiΔtα-αiΓ(-α)Γ(-αi)α0j(αi)+AtαΓ(-α))-1∥×(∑k=1jαkj(α)∥ej-k∥+∑i=1saiΔtα-αiΓ(-α)Γ(-αi)∑k=1jαkj(αi)∥ej-k∥+∥Rj(α)(g)∥+∑i=1saiΓ(-α)Γ(-αi)tjα-αi∥Rj(αi)(g)∥∑k=1jαkj(α)).
Note that A is a positive definite elliptic operator with all of eigenvalues λ>0. Since α0j(α)<0 and Γ(-α)<0, we have
(31)∥(α0j(α)+∑i=1saiΔtα-αiΓ(-α)Γ(-αi)α0j(αi)+AtαΓ(-α))-1∥=supλ>0∥(α0j(α)+∑i=1saiΔtα-αiΓ(-α)Γ(-αi)α0j(αi)+λtαΓ(-α))-1∥≲(-α0j(α)-∑i=1saiΔtα-αiΓ(-α)Γ(-αi)α0j(αi))-1.
Hence,
(32)∥ej∥≤α(1-α)j-α∑k=1jαkj(α)∥ej-k∥+∑i=1sαi(1-αi)j-αi∑k=1jαkj(αi)∥ej-k∥+α(1-α)γαn-2sup0≤t≤T∥u′′∥+∑i=1sαi(1-αi)γαin-2sup0≤t≤T∥u′′∥.
Denote d1=1 and
(33)dj=1+α(1-α)j-α∑k=1j-1αkj(α)dj-k,j=2,3,…,n,dji=1+αi(1-αi)j-αi∑k=1j-1αkj(αi)dj-k,j=2,3,…,n,
where i=1,2,…,s. By induction and Lemma 3, then we have
(34)∥ej∥≲α(1-α)n-2sup0≤t≤T∥u′′(t)∥·dj+∑i=1sαi(1-αi)n-2sup0≤t≤T∥u′′(t)∥·dji≲n-2sup0≤t≤T∥u′′(t)∥sin(πα)πjα+∑i=1sn-2sup0≤t≤T∥u′′(t)∥sin(παi)πjαi≲Δt2-α+∑i=1sΔt2-αi.
4. Space Discretization and Convergence
In this section, we will consider the space discretization for MT-FPDEs (1) and show the complete process and details of numerical scheme. The variational form of (1) is to find u(t,·)∈H01(Ω), such that, for all v∈H01(Ω),
(35)(D0Rtαu(t,x),v)L2(Ω)+∑i=1sai(D0Rtαiu(t,x),v)L2(Ω)+(∇xu,∇xv)=(f-(t,x),v)L2(Ω).
Let h denote the maximal length of intervals in Ω and let r be any nonnegative integer. We denote the norm in Hr(Ω) by ∥·∥Hr(Ω). Let Sh⊂H0r be a family of finite element spaces with the accuracy of order r≥2, that is, Sh consists of continuous functions on the closure Ω- of Ω which are polynomials of degree at most r-1 in each interval and which vanish outside Ωh, such that for small h, v∈Hb(Ω)∩H01(Ω),
(36)infχ∈Sh(∥v-χ∥L2(Ω)+h∥∇x(v-χ)∥L2(Ω))≤Chb∥v∥Hb(Ω),1≤b≤r.
The semidiscrete problem of (1) is to find the approximate solution uh(t)=uh(t,·)∈Sh and f-(t)=f-(t,·) for each t such that
(37)(D0Rtαuh(t),χ)L2(Ω)+∑i=1sai(D0Rtαiuh(t),χ)L2(Ω)+(∇xuh(t),∇xχ)L2(Ω)=(f-(t),χ)L2(Ω),∀χ∈Sh.
Let UN=uh(tN,x). After the time discretization, we have
(38)Δt-α∑k=0NωkN(α)(UN,χ)L2(Ω)+∑i=1saiΔt-αi∑k=0NωkN(αi)(UN,χ)L2(Ω)+(∇xUN,∇xχ)L2(Ω)=(f-(t),χ)L2(Ω),∀χ∈Sh.
In terms of the basis {ψm}m=1M-1⊆Sh, choosing χ=ψm, writing
(39)uh(tN,x)=∑j=1M-1UjNψj(x),
and inserting it into (38), one obtains
(40)∑j=1M-1Δt-α∑k=1NωkN(α)UjN-k(ψj,ψm)L2(Ω)+∑i=1sai∑j=1M-1Δt-αi∑k=1NωkN(αi)UjN-k(ψj,ψm)L2(Ω)+∑j=1M-1UjN(∇xψj,∇xψm)L2(Ω)=(f-,ψm)L2(Ω),m=1,2,…,M-1.
Let UN=(U1N,U2N,…,UM-1N)T. From (40), we obtain a vector equation
(41)Ψ1(Δt-αω0N(α)UN+∑i=1saiΔt-αiω0N(αi)UN)+Ψ2UN=Ψ1(FN-Δt-α∑k=1NωkN(α)UN-k-∑i=1saiΔt-αi∑k=1NωkN(αi)UN-k),
where initial condition is U0=u(0,x), Ψ1:={(ψj,ψm)L2(Ω)}j,m=1M-1 is the mass matrix, Ψ2 is stiffness matrix as Ψ2:={(∇xψj,∇xψm)L2(Ω)}j,m=1M-1, and FN:=(f-1,…,f-M-1)T is a vector valued function. Then, we can obtain the solution UN at t=tN.
Let Rh:H1(Ω)→Sh be the elliptic projection, defined by (∇xRhu,∇xχ)L2(Ω)=(∇xu,∇xχ)L2(Ω), for all χ∈Sh.
Lemma 5 (see [<xref ref-type="bibr" rid="B35">35</xref>]).
Assume that (36) holds, then with Rh and v∈Hb(Ω)∩H01(Ω), we have ∥Rhv-v∥L2(Ω)+h∥∇x(Rhv-v)∥L2(Ω)≤Chb∥v∥Hb(Ω) for 1≤b≤r.
In virtue of the standard error estimate for the FEM of MT-FPDEs, one has the following theorem which can be proved easily by Lemma 5 and the similar proof in [35].
Theorem 6.
For 0<αs<⋯<α1<α<1, let uh∈Sh and u(t,·)∈H01(Ω) be, respectively, the solutions of (37) and (1), then ∥u-uh∥L2(Ω)≲h2∥u∥L2(Ω).
5. Stability of the Numerical Method
In this section, we analyze the stability of the FEM for MT-FPEDs (1)–(3). Now we do some preparations before proving the stability of the method. Based on the definition of coefficients ωkj(α) in Section 3, we can obtain the following lemma easily.
Lemma 7.
For 0<α<1, the coefficients ωkj(α), (k=1,…,j) satisfy the following properties:
ω0j(α)>0 and ωkj(α)<0 for k=1,2,…,j,
Γ(2-α)∑k=1jωkj(α)=(1-α)j-α+1.
Now we report the stability theorem of this FEM for MT-FPDEs in this section as follows.
Theorem 8.
The FEM defined as in (38) is unconditionally stable.
Proof.
In (38), let χ(·)=Uj(·) at t=tj and the right hand f-=0. We have(42)Δt-α{1Γ(2-α)(Uj,Uj)L2(Ω)+∑k=1j-1ωkj(α)(Uj-k,Uj)L2(Ω)+ωjj(α)(U0,Uj)L2(Ω)∑k=1j-1}+∑i=1saiΔt-αi{∑i=1s1Γ(2-αi)(Uj,Uj)L2(Ω)+∑k=1j-1ωkj(αi)(Uj-k,Uj)L2(Ω)+ωjj(αi)(U0,Uj)L2(Ω)∑k=1j-1}+(∇xUj,∇xUj)L2(Ω)=0.
Using, Cauchy-Schwarz inequality, ±(Uj-k,Uj)≤(1/2)(∥Uj-k∥L2(Ω)2+∥Uj∥L2(Ω)2) for k=0,1,2,…,N and Lemma 7, we get
(43)(∑i=1sΔt-α2Γ(2-α)(1+(1-α)j-α)+∑i=1saiΔt-αi2Γ(2-αi)(1+(1-αi)j-αi))∥Uj∥L2(Ω)2+∥∇xUj∥L2(Ω)2≤Δt-α2[-∑k=1j-1ωkj(α)∥Uj-k∥L2(Ω)2-ωjj(α)∥U0∥L2(Ω)2]+∑i=1saiΔt-αi2[-∑k=1j-1ωkj(αi)∥Uj-k∥L2(Ω)2-ωjj(αi)∥U0∥L2(Ω)2].
We prove the stability of (37) by induction. Since when j=1, we have
(44)(∑i=1s∑i=1sΔt-α2Γ(2-α)(1+(1-α))+∑i=1saiΔt-αi2Γ(2-αi)(1+(1-αi)))∥U1∥L2(Ω)2≤(∑i=1sΔt-α2Γ(2-α)(1-(1-α))+∑i=1saiΔt-αi2Γ(2-αi)(1-(1-αi)))∥U0∥L2(Ω).
The induction basis ∥U1∥L2(Ω)≤∥U0∥L2(Ω) is presupposed. For the induction step, we have ∥Uj∥L2(Ω)≤∥Uj-1∥L2(Ω)≤⋯≤∥U0∥L2(Ω). Then using this result, by Lemma 7, we obtain
(45)(∑i=1s∑i=1sΔt-α(1+(1-α)(j+1)-α)2Γ(2-α)+∑i=1saiΔt-αi2Γ(2-αi)(1+(1-αi)(j+1)-αi))∥Uj+1∥L2(Ω)2≤(∑i=1sΔt-α(1-(1-α)(j+1)-α)2Γ(2-α)+∑i=1saiΔt-αi2Γ(2-αi)(1-(1-αi)(j+1)-αi))∥U0∥L2(Ω)2.
Here 0<1-α<1. After squaring at both sides of the above inequality, we obtain ∥Uj+1∥L2(Ω)≤∥U0∥L2(Ω).
6. Numerical Experiments
In this section, we present the numerical examples of MT-FPDEs to demonstrate the effectiveness of our theoretical analysis. The main purpose is to check the convergence behavior of numerical solutions with respect to Δt and Δx, which have been shown in Theorem 4 and Theorem 6. It is noted that the method in [29] is a special case of the method in our paper for fractional partial differential equation with single fractional order. So, we just need to compare FEM in our paper with other existing methods in [8, 28].
Example 9.
For t∈[0,T], x∈(0,1), consider the MT-FPDEs with two variables as follows:
(46)0CDtαu(t,x)+D0Ctβu(t,x)-∂x2u(t,x)=f(t,x),u(0,x)=0,x∈(0,1),u(t,0)=u(t,1)=0,t∈[0,T],
where the right-side function f(t,x)=(2t2-α/Γ(3-α))sin(2πx)+(2t2-β/Γ(3-β))sin(2πx)+4π2sin(2πx)t2. The exact solution is u(t,x)=t2sin(2πx).
We use this example to check the convergence rate (c. rate) and CPU time (CPUT) of numerical solutions with respect to the fractional orders α and β.
In the first test, we fix T=1, α=0.9 and β=0.5 and choose Δx=0.001 which is small enough such that the space discretization errors are negligible as compared with the time errors. Choosing Δt=1/2i (i=2,4,…,7), we report that the convergence rate of FDM in time is nearly 1.15 in Table 1, which matches well with the result of Theorem 4. On the other hand, Table 2 shows that an approximate convergence rate is 2, by fixing Δt=0.001 and choosing Δx=1/2i (i=2,…,6), which matches well with the result of Theorem 6. In the second test, we give the convergence rate when α=0.5, β=0.25 for Δt in Table 3, and Δx in Table 4, respectively. We also report the L2-norm and H1-norm of errors in Figures 1 and 2, respectively.
Fixing Δx=0.001, α=0.9, and β=0.3 in (46), we compare the error and CPUT calculated by the FEM in this paper with the FDM in [8] and the FPCM in [8]. From Table 5, it can be seen that the FEM in this paper is computationally effective.
Convergence rate in time for (46) with α=0.9 and β=0.5.
Δx
Δt
H1-norm
L2-norm
c. rate
CPUT (seconds)
0.001
1/4
1.3815 × 10-2
1.8960 × 10-3
0.214
0.001
1/16
6.1890 × 10-3
8.4939 × 10-4
1.1585
0.357
0.001
1/32
2.7858 × 10-3
3.8234 × 10-4
1.1516
0.736
0.001
1/64
1.2571 × 10-3
1.7252 × 10-4
1.1481
1.438
0.001
1/128
5.6567 × 10-4
7.7635 × 10-5
1.1520
2.922
Convergence rate in space for (46) with α=0.9 and β=0.5.
Δt
Δx
H1-norm
L2-norm
c. rate
CPUT (seconds)
0.001
1/4
0.2294
3.0611 ×10-2
20.35
0.001
1/16
5.9763 ×10-2
8.0381 ×10-3
1.9291
21.85
0.001
1/32
1.5067 ×10-2
2.0442 ×10-3
1.9753
26.68
0.001
1/64
3.7356 ×10-3
5.0966 ×10-4
2.0039
32.72
0.001
1/128
8.9129 ×10-4
1.2198 ×10-4
2.0629
41.03
Convergence rate in time for (46) with α=0.5 and β=0.25.
Δx
Δt
H1-norm
L2-norm
c. rate
CPUT (seconds)
0.001
1/4
3.0985 ×10-3
4.2525 ×10-4
0.218
0.001
1/16
1.0789 ×10-3
1.4807 ×10-4
1.5221
0.413
0.001
1/32
3.6702 ×10-4
5.0372 ×10-5
1.5556
0.921
0.001
1/64
1.1772 ×10-4
1.6156 ×10-5
1.6406
1.855
0.001
1/128
3.0704 ×10-5
4.2139 ×10-6
1.6388
3.783
Convergence rate in space for (46) with α=0.5 and β=0.25.
Δt
Δx
H1-norm
L2-norm
c. rate
CPUT (seconds)
0.001
1/4
2.3325 ×10-1
3.1119 ×10-2
23.73
0.001
1/16
6.0836 ×10-2
8.1823 ×10-3
1.9272
26.29
0.001
1/32
1.5382 ×10-2
2.0870 ×10-3
1.9711
33.49
0.001
1/64
3.8569 ×10-3
5.2622 ×10-4
1.9877
41.68
0.001
1/128
9.6378 ×10-4
1.3189 ×10-4
1.9963
55.24
Comparison of error and CPUT for (46) with α=0.9 and β=0.3.
Δx
Δt
FEM
FDM [8]
FPCM [8]
Error
CPUT
Error
CPUT
Error
CPUT
0.001
1/4
3.7056 ×10-3
0.238
5.8723 ×10-3
0.897
2.2027 ×10-2
6.16
0.001
1/8
1.6794 ×10-3
0.481
2.6751 ×10-3
1.837
8.7467 ×10-3
16.63
0.001
1/16
7.6528 ×10-4
0.962
1.2159 ×10-3
3.512
3.4693 ×10-3
30.11
0.001
1/32
3.5009 ×10-4
1.335
5.5190 ×10-4
7.001
1.3765 ×10-3
52.71
0.001
1/64
1.6027 ×10-4
2.703
2.4997 ×10-4
14.45
5.4564 ×10-4
106.49
H1-norm and L2-norm of errors for (46) with α=0.9, β=0.5, Δx=0.001 (a), and Δt=0.001 (b).
H1-norm and L2-norm of errors for (46) with α=0.5, β=0.25, Δx=0.001 (a), and Δt=0.001 (b).
Example 10.
Consider the following multierm fractional differential problem:
(47)D0Ctαu(t)+t-0.2D0Ctβu(t)=f(t),u(0)=0,
where f(t)=(4t1.5/Γ(2.5))+(12t2/Γ(3.5)). For α=0.5 and β=0.3, the exact solution is u(t)=t2+t2.5.
For the problem (47), our method in this paper is just the DFBDM in Section 3. Therefore, we only need to compare M1 with the FEM in [28] (FEM2). In Table 6, although the convergence rate of FEM2 is higher than that of DFBDM, the error and CPUT of DFBDM are smaller than those of FEM2.
Comparison of error, convergence rate, and CPUT for (47) with α=0.5 and β=0.3.
Δt
DFBDM (Section 3)
FEM2 [28]
Error
c. rate
CPUT
Error
c. rate
CPUT
1/4
9.1975 ×10-4
0.000864
3.6606 ×10-3
0.001862
1/8
3.3037 ×10-4
1.4772
0.001986
7.8173 ×10-3
2.2274
0.004902
1/16
1.1375 ×10-4
1.5382
0.004649
1.6210 ×10-4
2.2697
0.051816
1/32
3.5112 ×10-5
1.6958
0.012112
3.2629 ×10-4
2.3127
0.518130
Acknowledgments
The authors are grateful to the referees for their valuable comments. This work is supported by the National Natural Science Foundation of China (11101109 and 11271102), the Natural Science Foundation of Hei-Long-Jiang Province of China (A201107), and SRF for ROCS, SEM.
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