
Citation: Serguei Chiriaev, Nis Dam Madsen, Horst-Günter Rubahn, Shuang Ma Andersen. Helium Ion Microscopy of proton exchange membrane fuel cell electrode structures[J]. AIMS Materials Science, 2017, 4(6): 1289-1304. doi: 10.3934/matersci.2017.6.1289
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In the past, researchers have conducted extensive studies in the fields of fractional calculus and fractional differential equations (FDEs)[1,2,3,4]. To date, numerous scholars have extensively explored the realm of fractional integrals and derivatives in numerous forms, including Riemann-Liouville, Caputo, and Riesz integrals and derivatives, among others. However, there exists another type of fractional derivative that incorporates a logarithmic function in its definition, known as the Hadamard fractional derivative, which is defined as [2]
CHDαa,tv(x,t)=∫taω1−α(logt−logs)δv(x,s)dss, 0<a<t, |
where 0<α<1,ωβ(t)=tβ−1Γ(β), δv(x,s)=(s∂∂s)v(x,s), and Γ(⋅) representing the Gamma function.
Compared to the Riemann-Liouville and Caputo derivatives, the Caputo-Hadamard derivative, first introduced in 1892, more accurately captures some complex processes in practical applications. This includes Lomnitz logarithmic creep law [5,6] and ultra-slow mechanics [7,8], among others. In particular, the logarithmic increase in the mean square displacement of particles during ultra-slow diffusion has been demonstrated in [9,10,11]. As a result, the Hadamard fractional operator, whose kernel is a logarithmic function, has emerged as a natural model for ultra-slow diffusion processes and has garnered significant attention.
Let Ω=(xl,xr),Λ=(a,T) with a>0, and we will pay attention to the numerical approximation for the following Caputo-Hadamard fractional reaction sub-diffusion equation:
CHDαa,tu(x,t)−∂2xu(x,t)+κu(x,t)=f(x,t), (x,t)∈Ω×Λ, | (1.1) |
u(x,a)=φ(x), x∈Ω, | (1.2) |
u(xl,t)=u(xr,t)=0, t∈Λ. | (1.3) |
Here, the real constant κ∈R is the reaction coefficient, and the source term f(x,t) and the initial data φ(x) are given functions.
Recent studies have explored various numerical methods for tackling Caputo-Hadamard FDEs, encompassing the L1 scheme [12], L1-2 scheme, and L2-1σ scheme [13]. Moreover, Zhao et al. [14] introduced a spectral collocation method utilizing mapped Jacobi log orthogonal functions as basis functions, resulting in an efficient algorithm for solving Hadamard-type FDEs. Based on block-by-block approach, Ye et al. [15] proposed and analyzed a high order time stepping scheme having the convergence order more than three for the Caputo-Hadamard fractional differential equations.
Actually, the aforementioned studies mainly concentrate on the Caputo-Hadamard FDEs that possess smooth solutions. Recently, numerous efficient numerical methods have been developed for Caputo FDEs with weakly singular solutions, including the nonuniform L1 scheme [16,17], the nonuniform Alikhanov scheme [18,19,20], convolution quadrature method [21], and the spectral method [22]. Interested readers can also consult some recent references [23,24,25,26,27] for more numerical methods about FDE, such as the Alternating Direction Implicit method, extrapolation method, meshless method, and so on.
Nevertheless, numerical simulations for Caputo-Hadamard fractional reaction sub-diffusion equation (1.1)–(1.3) with weakly singular solutions remain relatively limited. For Eqs (1.1)–(1.3) without reaction term, Li et al. [28] proposed an L1 scheme on nonuniform meshes to approximate the time Caputo-Hadamard fractional derivative and employed the local discontinuous Galerkin method to approximate the spatial derivative. Later, the Alikhanov scheme with nonuniform time meshes for Caputo-Hadamard fractional sub-diffusion equations with an initial singularity was investigated in [29]. The stability and convergence of the resulting discrete scheme were analyzed, but the error bounds generally contain a constant factor Γ(1−α) or 1/(1−α) which will blow up as α approaches 1−.
Zhang et al. [30] derived a novel α-robust error analysis for convolution-type schemes with general nonuniform time step for Caputo fractional reaction sub-diffusion equations. By virtue of the ideas derived in [30], this paper will extend the nonuniform L1 and Alikhanov scheme presented in [28] and [29] to Caputo-Hadamard fractional sub-diffusion equations with reaction term, and then we will consider the α-robust error analysis of the proposed schemes. This means that the derived error bounds will not contain any blowup factor and will remain valid as α→1−.
Throughout this paper, we employ C to represent a generic constant that is independent of the mesh, and it may take different values at different places. Additionally, C exhibits α-robustness, meaning it is influenced by α, yet as α approaches 1, the value of C remains finite, avoiding any potential explosion. The proposed method in this paper is analyzed under the following regularity assumptions: for all (x,t)∈Ω×Λ, it holds
|∂lu(x,t)∂xl|≤C, l=0,1,2,3,4. | (1.4) |
|δlu(x,t)|≤C(1+(logta)σ−l), l=0,1,2, | (1.5) |
with the regularity parameter σ∈(0,1)∪(1,2), and δlu(x,t)=(t∂∂t)lu(x,t).
The remainder of this paper is organized as follows. In Section 2, we describe the detailed construction of the general convolution-type scheme. After that, the abstract result for graded mesh is applied to two typical numerical schemes, i.e., the widely used L1 scheme and Alikhanov's scheme. In Sections 3 and 4, we give a rigorous analysis of the stability and convergence of the L1 scheme and Alikhanov's scheme, and derive α-robust error estimates under specific regularity conditions imposed on the exact solution. In Section 5, some numerical examples are provided to support the theoretical statement. Some concluding remarks are given in the final section.
To develop a finite difference scheme for solving (1.1)–(1.3), we first divide the spatial interval [xl,xr] into M subintervals with grid size h=xr−xlM. Set discrete grid Ωh={xi|0≤i≤M} with xi=xl+ih. For any grid function w={wi|wi=w(xi), xi∈Ωh}, let δ2xwi be the standard second-order approximation of ∂2xw(xi), i.e.,
∂2xw(xi)≈δ2xwi:=wi−1−2wi+wi+1h2. |
Denote the space of grid functions W={w|w0=wM=0}. For any w,v∈W, the discrete L2 inner product and the associate L2 norm are given as
⟨w,v⟩:=hM∑i=0wivi, ‖w‖:=√⟨w,w⟩. |
Denote
∇xwi=wi−wi−1h, 1≤i≤M, |
for any grid functions w,v∈W, and it holds
−⟨δ2xw,v⟩=⟨∇xw,∇xv⟩. | (2.1) |
We now proceed to the discretization of time. First, we partition the interval [a,T] arbitrarily with a=t0<t1<⋯<tk−1<tk<⋯<tN=T, and set
τk=logtk−logtk−1, 1≤k≤N,ρk=τkτk+1, 1≤k≤N−1. |
Based on this partition, we derive L1 and Alikhanov's scheme, which are given in (3.1) and (4.1), respectively. Second, we further study and discuss these two formulas in the following divisions on the interval [a,T]:
tk=a(Ta)(k/N)r, k=0,1,⋯,N,r≥1, |
and, correspondingly,
logtk=loga+(logTa)(kN)r, k=0,1,⋯,N. | (2.2) |
Define vn:=v(x,tn),tn−θ:=θtn−1+(1−θ)tn, and vn−θ:=θvn−1+(1−θ)vn with an offset parameter θ∈[0,1). Then, the Caputo-Hadamard derivative operator in the problem (1.1) can be approximated as a convolution as detailed in the succeeding article [28]:
CHDαa,tv(x,tn−θ)≈CHDαa,τvn−θ=n∑k=1A(n)n−k∇τvk, | (2.3) |
where the difference operator is ∇τvk=vk−vk−1 for k≥1. To conduct our error analysis, we require the following three assumptions.
A1. The discrete kernels are monotone, meaning that
A(n)0≥A(n)1≥A(n)2≥...≥A(n)n−1for1≤n≤N. |
A2. There exists a constant πA>0 such that the discrete kernels satisfy a lower bound
A(n)n−k≥1πAτk∫tktk−1ω1−α(logtn−logs)dssfor1≤k≤n≤N. |
A3. There exists a constant ρ>0 for which the step size ratios ρk satisfy
ρk≤ρfor1≤k≤N−1. |
For the graded mesh, it can be checked that the step size ratio is ρk<1. In fact, by using the mean value theorem, we have
ρk=logtk−logtk−1logtk+1−logtk=kr−(k−1)r(k+1)r−kr=ηr−11ηr−12<1, |
with η1∈(k−1,k),η2∈(k,k+1).
Next, to derive the global consistency error, we introduce the discrete complementary convolution (DCC) kernels [16]
P(n)n−k=1A(k)0{1,k=n,∑nj=k+1(A(j)j−k−1−A(i)j−k)P(n)n−j,1≤k≤n−1, | (2.4) |
which are specifically chosen to enforce the identity
n∑j=mP(n)n−jA(j)j−m≡1,for1≤m≤n≤N. | (2.5) |
Futher, a discrete fractional Gronwall inequality is given as follows.
Lemma 2.1. Assume that A1-A3 hold, θ∈[0,1), and the nonnegative sequences (gn)Nn=1, (λl)N−1l=0,(vk)Nk=0 satisfy
n∑k=1A(n)n−k∇τ(vk)2≤n∑k=1λn−k(vk−θ)2+vn−θgn,1≤n≤N. | (2.6) |
If a constant Π satisfies Π≥∑N−1l=0λl and the maximum step size satisfies
max1≤n≤Nτn≤1α√2max{1,ρ}πAΓ(2−α)Π, | (2.7) |
then it holds that
vn≤Eα(2max{1,ρ}πAΠ(logtn)α)(v0+max1≤k≤nk∑j=1P(k)k−jgn), | (2.8) |
where Eα(⋅)=∞∑j=0(⋅)jΓ(jα+1) represents the special Mittag-Leffler function.
The proof of this lemma is similar to that in [19].
Lemma 2.2. [31] For ν>0,β>0,b>a>0, there holds
∫ba(logbs)ν−1(logsa)β−1dss=Γ(ν)Γ(β)Γ(ν+β)(logba)ν+β−1. |
Lemma 2.3. [19] If g is monotone increasing and h is monotone decreasing on the interval [a,b], and if both functions are integrable, then
(b−a)∫bag(s)h(s)ds≤∫bag(t)dt∫bah(s)ds. |
In the following, we derive two important lemmas which are useful in the convergence analysis later on.
Lemma 2.4. Assume that A1-A2 hold, then we have
n∑j=1P(n)n−j≤πAω1+α(logtn−loga). |
Proof. Denote h(t)=ω1+α(logt−loga), then it holds δh(t)=ωα(logt−loga). In one side, it follows from the definition of the Caputo-Hadamard derivative that
CHDαa,th(tj)=j∑k=1∫tktk−1ω1−α(logtj−logs)ωα(logs−loga)dss=j∑k=1∫logtklogtk−1ω1−α(logtj−τ)ωα(τ−loga)dτ. |
It is easy to check that ω1−α(logtj−τ) is monotone increasing and ωα(τ−loga) is monotone decreasing on the interval [logtk−1,logtk]. Thus, by Lemma 2.3 and A2, we obtain
CHDαa,th(tj)≤j∑k=11τk∫logtklogtk−1ω1−α(logtj−τ)dτ∫logtklogtk−1ωα(τ−loga)dτ=j∑k=11τk∫tktk−1ω1−α(logtj−logt)dtt∫tktk−1ωα(logs−loga)dss≤πAj∑k=1A(j)j−k∫tktk−1ωα(logs−loga)dss=πAj∑k=1A(j)j−k∫tktk−1δh(s)dss. | (2.9) |
In another side, by the definition of the Caputo-Hadamard derivative and Lemma 2.2, we get again that
CHDαa,th(tj)=∫tjaω1−α(logtj−logs)ωα(logs−loga)dss=1Γ(1−α)Γ(α)∫tja(logtjs)−α(logsa)α−1dss=1. | (2.10) |
Using (2.9), (2.10), and (2.5), we have
n∑j=1P(n)n−j=n∑j=1P(n)n−j⋅CHDαa,th(tj)≤n∑j=1P(n)n−jπAj∑k=1A(j)j−k∫tktk−1δh(s)dss=πAn∑k=1∫tktk−1δh(s)dssn∑j=kP(n)n−jA(j)j−k=πA∫tnt0δh(s)dss=πAω1+α(logtn−loga). |
Lemma 2.5. Assume that A1–A2 hold, and for any positive sequence (υk)nk=1, one has
n∑k=1P(n)n−kυk≤Γ(2−α)πAn∑j=1τjmaxj≤k≤n(υk(logtka)α−1). | (2.11) |
Proof. It follows from A2 that
k∑j=1A(k)k−jτj≥1πA∫tkt0ω1−α(logtk−logs)dss=(logtk−loga)1−απAΓ(2−α). | (2.12) |
Thus, using (2.12) and (2.5), we have
n∑k=1P(n)n−kυk⋅1≤n∑k=1P(n)n−kυkπAΓ(2−α)k∑j=1(logtk−loga)α−1A(k)k−jτj≤πAΓ(2−α)n∑j=1τjn∑k=jP(n)n−kA(k)k−j(logtk−loga)α−1υk≤Γ(2−α)πAn∑j=1τjmaxj≤k≤n(υk(logtka)α−1)⋅n∑k=jP(n)n−kA(k)k−j=Γ(2−α)πAn∑j=1τjmaxj≤k≤n(υk(logtka)α−1). |
We are now in a position to consider the α-robust error estimate of the L1 scheme. Following [28], the L1 approximation to the Caputo-Hadamard derivative CHDαa,tvn is given by
CHDαa,τvn=n∑k=1A(n)n−k∇τvk, | (3.1) |
where the discrete coefficients A(n)n−k are defined by
A(n)n−k=1τk∫tktk−1ω1−α(logtn−logs)dss. | (3.2) |
It is easy to see that Assumption A2 holds with πA=1. Using the integral mean-value theorem, one has
A(n)n−k−1−A(n)n−k=1Γ(1−α)(1τk+1∫tk+1tk(logtns)−αdss−1τk∫tktk−1(logtns)−αdss)=1Γ(1−α)[(logtnξk+1)−α−(logtnξk)−α]>0, | (3.3) |
with ξk+1∈(tk,tk+1),ξk∈(tk−1,tk). Thus Assumption A1 holds.
Let uni be the discrete approximation of solution u(xi,tn) for xi∈Ωh,0≤n≤N. The fully discrete scheme for problem (1.1)–(1.3) is given as
CHDαa,τuni−δ2xuni+κuni=fni,1≤i≤M−1,1≤n≤N, | (3.4) |
u0i=φ(xi),0≤i≤M, | (3.5) |
un0=unM=0,1≤n≤N, | (3.6) |
where fni=f(xi,tn).
We aim to demonstrate the stability and convergence of the scheme (3.4)–(3.6). The stability is established in the following theorem.
Theorem 3.1. Assume that the assumptions (1.4), (1.5), and A3 hold. Let κ+:=max{−κ,0}, and vni be the solutions of the following difference equation:
(CHDαa,τ−δ2x+κ)vni=gni,1≤i≤M−1,1≤n≤N, | (3.7) |
v0i=φ(xi),0≤i≤M, | (3.8) |
vn0=vnM=0,1≤n≤N. | (3.9) |
If the maximum step size is τ≤B1 with
B1=1α√4κ+max{1,ρ}Γ(2−α), | (3.10) |
we have
‖vn‖≤2Eα(4κ+max{1,ρ}(logtn)α)(‖v0‖+2max1≤k≤nk∑j=1P(k)k−j‖gj‖),1≤n≤N. | (3.11) |
Proof. After taking the inner product with 2vn on both sides of (3.7), we obtain
2(CHDαa,τvn,vn)−2(δ2xvn,vn)+2κ(vn,vn)=2(gn,vn), |
By employing the definition of the discrete Caputo-Hadamard derivative given in (3.1) and utilizing the monotone property given in (3.3), it can be inferred that
(CHDαa,τvn,vn)=2A(n)0‖vn‖2−2n−1∑k=1(A(n)n−k−1−A(n)n−k)(vk,vn)−2A(n)n−1(v0,vn)≥2A(n)0‖vn‖2−n−1∑k=1(A(n)n−k−1−A(n)n−k)‖vn‖2−A(n)n−1‖vn‖2−n−1∑k=1(A(n)n−k−1−A(n)n−k)‖vk‖2−A(n)n−1‖vk‖2=n∑k=1A(n)n−k∇τ‖vk‖2. | (3.12) |
It follows that
n∑k=1A(n)n−k∇τ‖vk‖2−2(δ2xvn,vn)+2κ(vn,vn)≤2(gn,vn), |
Noting (2.1), and using the Schwarz inequality, we obtain
n∑k=1A(n)n−k∇τ‖vk‖2≤2κ+‖vn‖2+2‖gn‖‖vn‖. |
By applying Lemma 2.1 to the above inequality, we obtain the desired estimate (3.11), thereby completing the proof.
We now consider the convergence of the scheme. To this end, let u(x,t) be the exact solution of (1.1)–(1.3), and let {uni|0≤i≤M,0≤n≤N} be the solution of problem (3.4)–(3.6). Set
eni=u(xi,tn)−uni, 0≤i≤M, 0≤n≤N. |
It is straightforward to obtain the following error equation:
CHDαa,τeni−δ2xeni+κeni=(Rt)ni+(Rs)ni,1≤i≤M−1,1≤n≤N, | (3.13) |
e0i=0,0≤i≤M, | (3.14) |
en0=enM=0,1≤n≤N, | (3.15) |
where
(Rt)ni=CHDαa,tu(xi,tn)−CHDαa,τuni, | (3.16) |
(Rs)ni=∂2xu(xi,tn)−δ2xuni. | (3.17) |
After conducting stability analysis in Theorem 3.1, if the maximum step size satisfies (3.10), one has, with ˉC=2Eα(4κ+max{1,ρ}(logtn)α),
‖en‖≤ˉC(‖e0‖+2max1≤k≤nk∑j=1P(k)k−j(‖(Rt)j‖+‖(Rs)j‖)), 1≤n≤N. | (3.18) |
Under the assumption of spatial regularity given by (1.4), the Taylor expansion provides a direct demonstration that ‖(Rs)n‖≤Ch2. When combined with Lemma 2.4, we can further deduce that
n∑j=1P(n)n−j‖(Rs)n‖≤C(logtna)αh2. | (3.19) |
Now, we only need to estimate the term ∑nj=1P(n)n−j‖(Rt)n‖, which will be achieved through the following lemmas.
Lemma 3.1. [29, Lemma 3.1] Suppose that f(x) has a continuous δ-derivative of n+1 order in some field of point x0. A Taylor-like formula with integral remainder is given by
f(x)=f(x0)+δf(x0)(logx−logx0)+δ2f(x0)2!(logx−logx0)2+⋯+δnf(x0)n!(logx−logx0)n+1n!∫xx0δn+1f(s)(logx−logs)ndss. |
Lemma 3.2. For any function u∈C3((a,T]), the local truncation errors (Rt)ki satisfy
(Rt)ki≤A(k)0Gk+k−1∑j=1(A(k)k−j−1−A(k)k−j)Gj. | (3.20) |
where
Gk=∫tktk−1(logs−logtk−1)|δ2u(xi,s)|dss,1≤k≤n. | (3.21) |
Lemma 3.3. Assume that A3 holds, and u(x,t) satisfies the regularity assumption (1.5), then
n∑k=1P(n)n−k‖(Rt)k‖≤C(τσ1σ+n∑j=2τjmaxj≤k≤n(logtka)α−1(logtk−1a)σ−2τ2−αk). |
where C=(1+ρ)Γ(2−α).
The proofs of Lemmas 3.2 and 3.3 are left to Appendix 7.1 and 7.2 for brevity.
Theorem 3.2. Let u(x,t) be the exact solution of (1.1)–(1.3), and let {uni|0≤i≤M,0≤n≤N} be the solution of (3.4)–(3.6). Suppose the regularity assumptions (1.4)–(1.5) hold. Set
eni=u(xi,tn)−uni, 0≤i≤M, 0≤n≤N. |
If the maximum step size is τ≤B1, then the discrete solution is convergent in the L2-norm with ˉC=2Eα(4κ+max{1,ρ}(logtn)α) such that
‖en‖≤CˉC(‖e0‖+(logtna)αh2+(1+ρ)Γ(2−α)max1≤k≤nEkt), | (3.22) |
where
Ekt=τσ1σ+k∑j=2τjmaxj≤l≤k(logtla)α−1(logtl−1a)σ−2τ2−αl. | (3.23) |
In particular, if graded mesh is used, then it holds
‖en‖≤2Eα(4κ+(logtn)α)(‖e0‖+(logtna)αh2+2Γ(2−α)(1σ+4rr3−αϑ)N−min{rσ,2−α}), |
with
ϑ={lnn,σr≥2−α,12−α−rσ,σr<2−α. | (3.24) |
Proof. By combining (3.18), (3.19) and Lemma 3.3, we arrive at (3.22). We now proceed to examine the global approximation error on the graded mesh. It is easy to verify τk≤rlogTaN−rkr−1 as follows:
τk=logtk−logtk−1=loga+logTa(logkN)r−loga−logTa(logk−1N)r=logTaN−r(kr−(k−1)r)≤logTaN−rrkr−1. | (3.25) |
Note that σ>0, it follows from (3.23) and (3.25) that
Ent≤((logTa)σ(1N)rσσ)+n∑j=2τjmaxj≤k≤n(logTa)σ−2(k−1N)r(σ−2)×(logTa)α−1(kN)r(α−1)(logTa)2−αN−r(2−α)r2−αk(2−α)(r−1)=(logTa)σ(1N)rσσ+n∑j=2τjmaxj≤k≤n(logTa)σ−1r2−α(kN)−r×(kk−1)r(2−σ)N−σrkrσ−(2−α). | (3.26) |
Taking into account that
(kk−1)r(2−σ)≤(1+1k−1)2r≤22r=4r, |
(3.26) can be simplified to
Ent≤(logTa)σN−rσ(1σ+r2−αlogTan∑j=2τjmaxj≤k≤n(kN)−r4rkσr−(2−α))≤(logTa)σN−rσ(1σ+4rr2−α(logTa)−1n∑j=2τjmaxj≤k≤n(kN)−rkσr−(2−α)). | (3.27) |
For σr≥2−α, we have from (3.27) that
Ent≤(logTa)σN−rσ(1σ+4rr2−α(logTa)−1n∑j=2τj(jN)−rnσr−(2−α))=(logTa)σN−rσ(1σ+4rr2−α(logTa)−1n∑j=2(logTa)N−rrjr−1j−rNrnσr−(2−α))=(logTa)σN−rσ(1σ+4rr3−αnσr−(2−α)n∑j=2j−1)≤(logTa)σN−rσ(1σ+4rr3−αNσr−(2−α)lnn)≤(logTa)σ(1σ+4rr3−αlnn)N−min{rσ,2−α}. | (3.28) |
For σr<2−α, it follows from (3.27) that
Ent≤(logTa)σN−rσ(1σ+4rr2−α(logTa)−1n∑j=2τj(jN)−rjσr−(2−α))≤(logTa)σN−rσ(1σ+4rr2−α(logTa)−1n∑j=2r(logTa)N−rjr−1(jN)−rjσr−(2−α))=(logTa)σN−rσ(1σ+4rr3−αn∑j=2jσr−(3−α)). | (3.29) |
Note that
n∑j=2jσr−(3−α)≤∫n1sσr−(3−α)ds=1rσ−(2−α)(nrσ−(2−α)−1)≤12−α−rσ, |
then we get from (3.29) that
Ent≤(logTa)σ(1σ+4rr3−α2−α−rσ)N−rσ. | (3.30) |
By utilizing (3.28), (3.30), and (3.23), we achieve the desired global approximation error on the graded mesh.
In this section, we delve into the α-robust error estimate pertaining to the Alikhanov scheme. Referring to [28], the Alikhanov scheme for the Caputo-Hadamard derivative is expressed as follows:
CHDαa,tv(tn−θ)≈CHDαa,τvn−θ=n∑k=1A(n)n−k∇τvkfor1≤n≤N, | (4.1) |
with θ=α2. Here the discrete convolution kernel A(n)n−k is defined as: A(n)0=a(n)0 if n=1 and, for n≥2,
A(n)n−k={a(n)n−1−b(n)n−1,k=1,a(n)n−k+ρk−1b(n)n−k+1−b(n)n−k,k=2,...,n−1,a(n)0+ρn−1b(n)1,k=n. |
Moreover, the discrete coefficients a(n)n−k and b(n)n−k are determined by:
a(n)0=1τn∫tn−θtn−1δϖn(s)dss,a(n)n−k=1τk∫tktk−1δϖn(s)dss,1≤k≤n−1,b(n)n−k=2τk(τk+τk+1)∫tktk−1(logs−logtk−1/2)δϖn(s)dss,1≤k≤n−1, |
where
ϖn(s)=−ω2−α(logtn−θ−logs), logtk−1/2=12(logtk−1+logtk). |
As proved in [29], one can verify that Assumptions A1 and A2 hold with πA=11/4.
Next, the difference scheme is established. Considering (1.1) at mesh point (xi,tn−θ), the fully discrete scheme for problem (1.1) is given as
CHDαa,τun−θi−δ2xun−θi+κun−θi=fn−θi,1≤i≤M−1,1≤k≤N, | (4.2) |
u0i=φ(xi),0≤i≤M, | (4.3) |
un0=unM=0,1≤k≤N. | (4.4) |
We proceed to assess the stability and convergence of Alikhanov's scheme.
Theorem 4.1. Assume that the assumptions (1.5) and A3 hold. Let κ+:=max{−κ,0} and vni be the solution of the following differential equation:
(CHDαa,τ−δ2x+κ)vn−θi=gn−θi,1≤i≤M−1,1≤n≤N, | (4.5) |
v0i=φ(xi),0≤i≤M, | (4.6) |
vn0=vnM=0,1≤n≤N. | (4.7) |
If the maximum step size is τ≤B2 with
B2=1α√11κ+max{1,ρ}Γ(2−α), | (4.8) |
we have
‖vn‖≤Eα(11κ+max{1,ρ}(logtn)α)(‖v0‖+2max1≤k≤nk∑j=1P(k)k−j‖gj−θ‖). | (4.9) |
Proof. Taking the inner products with 2vn−θi on both sides of (4.5), we obtain
2(CHDαa,τvn−θ,vn−θ)−2(δ2xvn−θ,vn−θ)+2κ(vn−θ,vn−θ)=2(fn−θ,vn−θ), |
Utilizing (3.12), we can further derive
n∑k=1A(n)n−k∇τ‖vk‖2−2(δ2xvn−θ,vn−θ)+2κ(vn−θ,vn−θ)≤2(fn−θ,vn−θ). |
Taking into account (2.1) and applying the Schwarz inequality, we arrive at
n∑k=1A(n)n−k∇τ‖vk‖2≤2κ+‖vn−θ‖2+2‖fn−θ‖‖vn−θ‖. |
By applying Lemma 2.1 to the aforementioned inequality, we successfully derive the desired estimate (4.9), thus, conclusively completing the proof.
We now consider the convergence of the scheme. To this end, let u(x,t) be the exact solution of (1.1)–(1.3), and let {un−θi|0≤i≤M,0≤n≤N} be the solution of problem (4.2)–(4.4). Set
en−θi=u(xi,tn−θ)−un−θi. |
It is straightforward to obtain the following error equation:
(CHDαa,τ−δ2x+κ)en−θi=Rni,1≤i≤M−1,1≤n≤N, | (4.10) |
e0i=0,0≤i≤M, | (4.11) |
en0=enM=0,1≤n≤N, | (4.12) |
where Rni=(Rt)ni+(Rs)ni+(Rl)ni with
(Rt)ni=CHDαa,tu(xi,tn−θ)−CHDαa,τun−θi, | (4.13) |
(Rs)ni=∂2xu(xi,tn−θ)−δ2xu(xi,tn−θ), | (4.14) |
(Rl)ni=δ2x(u(xi,tn−θ)−un−θi)−κ(u(xi,tn−θ)−un−θi). | (4.15) |
By the stability analysis in Theorem 4.1, if the maximum step size fulfills the condition stated in (4.8), one can derive the following inequality:
‖en‖≤˜C(‖e0‖+2max1≤k≤nk∑j=1P(k)k−j(‖(Rt)j‖+‖(Rs)j‖+‖(Rl)j‖)) | (4.16) |
with ˜C=Eα(11κ+max{1,ρ}(logtn)α).
Now, our sole task is to estimate the terms ∑nj=1P(n)n−j‖(Rl)n‖ and ∑nj=1P(n)n−j‖(Rt)n‖, which we will accomplish by utilizing the lemmas outlined below.
Lemma 4.1. Assume that A3 holds, and u(x,t) satisfies the regularity assumption (1.5), then
n∑j=1P(n)n−j‖(Rl)j‖≤C(τσ+α1σ+(logtna)αmax2≤k≤n(logtk−1a)σ−2τ2k). | (4.17) |
Proof. Set v(t)=(δ2x−κ)u(xi,t). Using the Taylor-like formula given in Lemma 3.1, we derive
v(tj)=v(tj−θ)+δv(tj−θ)(logtj−logtj−θ)+∫tjtj−θδ2v(s)(logtj−logs)dss, | (4.18) |
v(tj−1)=v(tj−θ)+δv(tj−θ)(logtj−1−logtj−θ)+∫tj−1tj−θδ2v(s)(logtj−1−logs)dss. | (4.19) |
By combining (4.18) and (4.19), we obtain
θv(tj−1)+θ(1−θ)v(tj)=v(tj−θ)+θ∫tj−θtj−1δ2v(s)(logs−logtj−1)dss+(1−θ)∫tjtj−θδ2v(s)(logtj−logs)dss. |
This leads to the following expression for (Rl)ji:
(Rl)ji=−θ∫tj−θtj−1δ2v(s)(logs−logtj−1)dss−(1−θ)∫tjtj−θδ2v(s)(logtj−logs)dss. | (4.20) |
For j=1, utilizing the regularity assumption (1.5), we deduce from (4.20) that
‖(Rl)1‖≤θ∫t1−θt0|δ2v(s)|(logs−logt0)dss+(1−θ)∫t1t1−θ|δ2v(s)|(logt1−logs)dss≤C∫t1−θt0(logsa)σ−1dss+C∫t1t1−θ(logt1−logs)(logsa)σ−2dss≤C∫t1−θt0(logsa)σ−1dss≤Cτσ1σ. | (4.21) |
Analogously, for 2≤j≤N, we have
‖(Rl)j‖≤θ∫tj−θtj−1|δ2v(s)|(logs−logtj−1)dss+(1−θ)∫tjtj−θ|δ2v(s)|(logtj−logs)dss≤C(logtj−1a)σ−2(∫tj−θtj−1(logs−logtj−1)dss+∫tjtj−θ(logtj−logs)dss)≤C(logtj−1a)σ−2τ2j. | (4.22) |
It follows from (2.4), A1, and A2 that
P(n)n−1≤1/A(1)0≤114Γ(2−α)τα1. | (4.23) |
By integrating the information from (4.21) to (4.23) along with Lemma 2.4, we arrive at the conclusion stated in (4.17).
Lemma 4.2. Assume that A3 holds and u(x,t) satisfies the regularity assumption (1.5), then
n∑j=1P(n)n−j‖(Rt)j‖≤C(τσ1σ+τσ−31τ32)+Cn∑j=2τjmaxj≤k≤n((logtk−1a)σ−3(logtka)α−1τ3−αk+(logtka)σ+α−4τ3k+1ταk). | (4.24) |
Proof. It follows from [29] that the local truncation error defined in (4.13) can be bounded by
‖(Rt)k‖≤A(k)0Gkloc+k−1∑j=1(A(k)k−j−1−A(k)k−j)Gjhis, | (4.25) |
where
Gkloc=32∫tk−1/2tk−1(logs−logtk−1)2|δ3u(s)|dss+3τk2∫tktk−1/2(logtk−logs)|δ3u(s)|dss,%Gkhis=52∫tktk−1(logs−logtk−1)2|δ3u(s)|dss+52∫tk+1tk(logtk+1−logs)|δ3u(s)|dss. |
By the regularity assumption (1.5), it is easy to obtain
G1loc≤Cτσ1σ,Gkloc≤C(logtk−1a)σ−3τ3k, k≥2, | (4.26) |
and
G1his≤C(τσ1σ+(logt1a)σ−3τ32), | (4.27) |
Gkhis≤C((logtk−1a)σ−3τ3k+(logtka)σ−3τ3k+1), k≥2. | (4.28) |
Similar to (7.8), we have
n∑k=1P(n)n−k‖(Rt)k‖≤n∑k=1P(n)n−kA(k)0(Gkloc+Gkhis):=n∑k=1P(n)n−kGk. | (4.29) |
Applying Lemma 2.5 by taking υk=Gk in (2.11) to (4.29), we get
n∑k=1P(n)n−k‖(Rt)k‖≤11Γ(2−α)4n∑j=1τjmaxj≤k≤n(logtka)α−1Gk=11Γ(2−α)4[n∑j=2τjmaxj≤k≤n(logtka)α−1Gk+max{τ1max2≤k≤n(logtka)α−1Gk,τα1G1}]≤11Γ(2−α)4(n∑j=2τjmaxj≤k≤n(logtka)α−1Gk+ρτ2max2≤k≤n(logtka)α−1Gk+τα1G1)≤114(1+ρ)Γ(2−α)(n∑j=2τjmaxj≤k≤n((logtka)α−1⋅Gk)+τα1G1). | (4.30) |
According to (4.26), (4.27), and (4.28), we have
G1≤CA(1)0(τσ1σ+τσ1σ+(logt1a)σ−3τ32)≤Cτ−α1Γ(2−α)(τσ1σ+(logt1a)σ−3τ32), | (4.31) |
and
Gk≤CA(k)0(logtk−1a)σ−3τ3k+CA(k)0((logtk−1a)σ−3τ3k+(logtka)σ−3τ3k+1)≤Cτ−α1Γ(2−α)((logtk−1a)σ−3τ3−αk+(logtka)σ−3τ3k+1τ−αk). | (4.32) |
Therefore, combining (4.30), (4.31), and (3.21) we obtain the desired result.
Theorem 4.2. Let u(x,t) be the exact solution of (1.1)–(1.3), and let {uni|0≤i≤M,0≤n≤N} be the solution of (4.2)–(4.4). Suppose the regularity assumptions (1.4)–(1.5) hold. Let
eni=u(xi,tn)−uni, 0≤i≤M, 0≤n≤N. |
If the maximum time step is τ≤B2, then the discrete solution is convergent in the L2-norm with ˜C=Eα(11κ+max{1,ρ}(logtn)α) such that
‖en‖≤˜CC(τσ+α1σ+τσ−31τ32+(logtna)αmax2≤k≤n(logtk−1a)σ−2τ2k+(logtna)αh2+Ent), | (4.33) |
where
Ent=n∑j=2τjmaxj≤k≤n((logtk−1a)σ−3(logtka)α−1τ3−αk+(logtka)σ+α−4τ3k+1ταk). | (4.34) |
In particular, if graded mesh is used, then it holds that
‖en‖≤CEα(11κ+(logtn)α)(N−min{σr,2}+(logtna)αh2). | (4.35) |
Proof. By combining (3.19), (4.16), Lemma 4.1 and Lemma 4.2, we arrive at (4.33). We now proceed to examine the global approximation error on the graded mesh. Due to (2.2) and (3.25), we can estimate the righthand side of (4.33) term by term as follows. It easy to check that
τσ+α1σ=1σ(logTa)σ+α(1N)r(σ+α)≤CσN−r(σ+α). | (4.36) |
Further, it holds that
τ32τσ−31=τ32(logt1a)σ−3≤(logTa)σ−3(1N)r(σ−3)(logTa)3N−3rr323(r−1)≤CN−rσ. | (4.37) |
For the third term in the righthand side of (4.33), we have
(logtna)αmax2≤k≤n(logtk−1a)σ−2τ2k≤Cmax2≤k≤n((logTa)σ(k−1N)r(σ−2)N−2rr2k2(r−1))≤CN−rσmax2≤k≤n(kk−1)r(2−σ)krσ−2≤CN−rσmax2≤k≤nkrσ−2. |
If rσ≥2, then max2≤k≤nkrσ−2=nrσ−2≤Nrσ−2. If rσ<2, then max2≤k≤nkrσ−2=2rσ−2<1. Thus, we obtain
(logtna)αmax2≤k≤n(logtk−1a)σ−2τ2k≤CN−min{2,rσ}. | (4.38) |
Moreover, we have
(logtk−1a)σ−3(logtka)α−1τ3−αk+(logtka)σ+α−4τ3k+1ταk≤(logTa)σ−3(k−1N)r(σ−3)(logTa)α−1(kN)r(α−1)(rlogTaN−rkr−1)3τ−α1+(logTa)σ+α−4(kN)r(σ+α−4)(rlogTaN−r(k+1)r−1)3τ−α1=τ−α1r3(logTa)σ+α−1N−r(σ+α−1)kr(σ−1)−(3−α)((kk−1)r(3−σ)+(k+1k)3(r−1))≤Cτ−α1N−r(σ+α−1)kr(σ−1)−(3−α)=C(logTa)−α(1N)−rαN−r(σ+α−1)kr(σ−1)−(3−α)≤Ckr(σ−1)−(3−α)N−r(σ−1). | (4.39) |
Combining (4.34) and (4.39), we obtain
Ent≤Cn∑j=2τjmaxj≤k≤nk−rkrσ−(3−α)N−r(σ−1)≤Cn∑j=2N−rjr−1maxj≤k≤nk−rkrσ−(3−α)N−r(σ−1). | (4.40) |
If rσ≥3−α, we have from (4.40) that
Ent≤Cn∑j=2N−rj−1nrσ−(3−α)N−r(σ−1)≤CN−(3−α)n∑j=2j−1≤ClnnN−(3−α). | (4.41) |
If rσ<3−α, we get from (4.40) that
Ent≤Cn∑j=2N−rjr−1jr(σ−1)−(3−α)N−r(σ−1)≤CN−rσn∑j=2jrσ−(4−α). | (4.42) |
Note that
n∑j=2jrσ−(4−α)≤∫n1srσ−(4−α)ds=1rσ−(3−α)(nrσ−(3−α)−1)≤13−α−rσ, | (4.43) |
then we have from (4.42) that
Ent≤13−α−rσN−(3−α). | (4.44) |
By combining (4.36), (4.37), (4.38), (4.41), and (4.44), we achieve the desired global approximation error (4.35) on the graded mesh.
In this section, we present several numerical examples to validate the theoretical result stated in Theorem 3.2 and Theorem 4.2. In our computations, the spatial domain Ω=[0,π] is uniformly divided into M parts, and the time interval Λ is divided into N subintervals using graded meshes tk=a(Ta)(k/N)r. The grading constant r≥1 controls the extent to which the time levels are concentrated near t=a. As r increases, the initial step sizes become smaller than the later ones, which can be visually observed in Figure 1.
Example 5.1. Consider the problem (1.1)–(1.3) with
a=1,T=2,κ=2,f(x,t)=(sinx)(Γ(1+α)+(logt)α+κ⋅(logt)α). |
It can be verified that the corresponding exact solution is u(x,t)=(sinx)(logt)α and σ=α.
Since the spacial accuracy is standard for the second order central difference scheme, we only explore the convergence rate of the time stepping scheme. To this end, we calculate the L2 errors between the exact and numerical solutions
Error(M,N)=max1≤k≤N‖ek‖. |
In Tables 1–3, we list the temporal L2 errors for the L1 scheme by taking fixed M and increasing N for different α. Based on the obtained numerical errors, we further estimate the order of temporal convergence using the formula as
Order=log2Error(M,N/2)Error(M,N). |
Results are also listed in Tables 1–3. As can be observed, the temporal convergence order for the L1 scheme is close to min{rσ,2−α} for all cases, aligning with the theoretical analysis presented in Theorem 3.2. In Tables 4–7, we list the temporal L2 errors for Alikhanov's scheme as a function of N for different α. Also shown are the corresponding decay rates based on graded meshes about Alikhanov's scheme. From Tables 4–7, it is observed that the convergence rate is close to min{rσ,2} in time.
Example 5.2. Consider the problem (1.1) with
a=2,T=3,κ=−1,f(x,t)=sin(x)⋅(Γ(1+α)+(logt2)α+κ⋅(logt2)α). |
N | α=0.3 | α=0.5 | α=0.7 | |||
Error(M,N) | Order | Error(M,N) | Order | Error(M,N) | Order | |
64 | 1.1134×10−2 | − | 2.6984×10−3 | − | 9.4560×10−4 | − |
128 | 7.9395×10−3 | 0.5153 | 1.3721×10−3 | 0.9757 | 4.0801×10−4 | 1.2126 |
256 | 5.4559×10−3 | 0.5412 | 6.9778×10−4 | 0.9756 | 1.7224×10−4 | 1.2442 |
512 | 3.7010×10−4 | 0.5599 | 3.5227×10−4 | 0.9861 | 7.1677×10−5 | 1.2648 |
min{rσ,2−α} | 0.6 | 1.0 | 1.3 |
N | α=0.3 | α=0.5 | α=0.7 | |||
Error(M, N) | Order | Error(M, N) | Order | Error(M, N) | Order | |
64 | 3.7010×10−3 | - | 7.0310×10−4 | - | 6.2060×10−4 | - |
128 | 2.0350×10−3 | 0.8629 | 2.6509×10−4 | 1.4072 | 2.5489×10−4 | 1.2838 |
256 | 1.1093×10−3 | 0.8797 | 9.7726×10−5 | 1.4397 | 1.0412×10−4 | 1.2917 |
512 | 5.9718×10−4 | 0.8890 | 3.5547×10−5 | 1.4590 | 4.2399×10−5 | 1.2961 |
min{rσ,2−α} | 0.9 | 1.5 | 1.3 |
N | α=0.5 | α=0.7 | α=0.9 | |||
Error(M, N) | Order | Error(M, N) | Order | Error(M, N) | Order | |
32 | 1.7972×10−3 | - | 2.4367×10−3 | - | 1.9017×10−3 | - |
64 | 7.0306×10−4 | 1.3541 | 1.1229×10−3 | 1.1176 | 1.0393×10−3 | 0.8716 |
128 | 2.6507×10−4 | 1.4072 | 4.9963×10−4 | 1.1683 | 5.5426×10−4 | 0.9070 |
256 | 9.7719×10−5 | 1.4397 | 2.1701×10−4 | 1.2031 | 2.8970×10−4 | 0.9360 |
min{rσ,2−α} | 1.5 | 1.3 | 1.1 |
N | α=0.3 | α=0.5 | α=0.7 | |||
Error(M,N) | Order | Error(M,N) | Order | Error(M,N) | Order | |
32 | 2.7134×10−3 | − | 2.7984×10−3 | − | 2.4560×10−3 | − |
64 | 1.4395×10−3 | 0.9746 | 1.4721×10−3 | 0.9757 | 1.2811×10−3 | 0.9426 |
128 | 7.4559×10−4 | 0.9871 | 6.9776×10−4 | 0.9839 | 6.7224×10−4 | 0.9622 |
256 | 3.7010×10−4 | 0.9935 | 4.5227×10−4 | 0.9860 | 3.1677×10−4 | 0.9809 |
min{rσ,2} | 1.0 | 1.0 | 1.0 |
N | α=0.3 | α=0.5 | α=0.8 | |||
Error(M, N) | Order | Error(M, N) | Order | Error(M, N) | Order | |
64 | 2.2791×10−2 | - | 1.7493×10−2 | - | 3.7540×10−3 | - |
128 | 2.4622×10−2 | 0.1838 | 1.3208×10−2 | 0.4053 | 2.2841×10−3 | 0.7240 |
256 | 2.1453×10−2 | 0.1982 | 9.8096×10−3 | 0.4291 | 1.3224×10−3 | 0.7552 |
512 | 1.8563×10−2 | 0.2117 | 7.1924×10−4 | 0.4477 | 8.1627×10−4 | 0.7658 |
min{rσ,2} | 0.3 | 0.5 | 0.8 |
N | α=0.3 | α=0.5 | α=0.8 | |||
Error(M, N) | Order | Error(M, N) | Order | Error(M, N) | Order | |
16 | 5.2002×10−3 | - | 1.1646×10−3 | - | 2.2565×10−4 | - |
32 | 3.0471×10−3 | 0.7711 | 4.3955×10−4 | 1.4057 | 5.8962×10−5 | 1.9362 |
64 | 1.7190×10−3 | 0.8259 | 1.5959×10−4 | 1.4616 | 1.4430×10−5 | 2.0308 |
128 | 9.4796×10−4 | 0.8587 | 5.6967×10−5 | 1.4862 | 3.3768E-06 | 2.0953 |
min{rσ,2} | 0.9 | 1.5 | 2 |
N | α=0.5 | α=0.7 | α=0.9 | |||
Error(M, N) | Order | Error(M, N) | Order | Error(M, N) | Order | |
16 | 5.9243×10−4 | - | 4.1200×10−4 | - | 1.6522×10−4 | - |
32 | 1.6398×10−4 | 1.8531 | 1.1720×10−4 | 1.8136 | 5.0273×10−5 | 1.7165 |
64 | 4.2850×10−5 | 1.9361 | 3.3138×10−5 | 1.9008 | 1.4495×10−5 | 1.7942 |
128 | 1.0862×10−5 | 1.9801 | 8.0773E-06 | 1.9582 | 4.0215E-06 | 1.8497 |
min{rσ,2} | 2 | 2 | 2 |
Tables 8 and 9 display the temporal L2 errors by taking fixed M and increasing N for the L1 scheme and Alikhanov's scheme with κ=−1, respectively. Additionally, the maximum time step size τ, as well as the conditions B1 and B2 defined in (3.10) and (4.8), respectively, are also presented. The convergent results displayed show that the rate of convergence in time is in agreement with Theorem 3.2 and Theorem 4.2. It is worth noting that when α approaches 1−, the convergence order in the time direction is also in accordance with the theoretical analysis.
N | α=0.3 | α=0.5 | α=0.99 | τ | |||
Error(M,N) | Order | Error(M,N) | Order | Error(M,N) | Order | ||
32 | 6.2201×10−3 | − | 1.8010×10−3 | − | 2.6758×10−4 | − | 3.6837×10−2 |
64 | 3.3333×10−3 | 0.8999 | 6.4494×10−4 | 1.4816 | 1.3421×10−4 | 0.9954 | 1.8711×10−2 |
128 | 1.7863×10−3 | 0.9000 | 2.2925×10−4 | .4922 | 6.5942×10−5 | 1.0252 | 9.4290×10−3 |
256 | 9.5724×10−4 | 0.9000 | 8.1375×10−5 | 1.49427 | 3.2376×10−5 | 1.0262 | 4.7330×10−3 |
min{rσ,2−α} | 0.9 | 1.5 | 1.01 | B1=0.2479 |
N | α=0.2 | α=0.5 | α=0.99 | τ | |||
Error(M,N) | Order | Error(M,N) | Order | Error(M,N) | Order | ||
32 | 2.9061×10−2 | − | 1.0479×10−2 | − | 4.3943×10−5 | − | 3.6836×10−2 |
64 | 2.5299×10−2 | 0.1999 | 7.4103×10−2 | 0.4999 | 2.2143×10−5 | 0.9887 | 1.8711×10−2 |
128 | 2.2025×10−2 | 0.1999 | 5.2399×10−3 | 0.4999 | 1.1154×10−5 | 0.9892 | 9.4290×10−3 |
256 | 1.9174×10−2 | 0.1999 | 3.7052×10−3 | 0.4999 | 5.6176E-06 | 0.9896 | 4.7330×10−3 |
min{rσ,2} | 0.2 | 0.5 | 0.99 | B2=0.0892 |
In this paper, we have proposed L1 and Alikhanov schemes with nonuniform time steps for solving Caputo-Hadamard fractional reaction sub-diffusion equations. We conduct a rigorous analysis of the stability and convergence of these two schemes, and further derive α-robust error estimates under specific regularity conditions imposed on the exact solution. The derivation of these regularity assumptions is currently under active investigation and will be the subject of our forthcoming research.
Proof. Let v(t)=u(xi,t), for 1≤k≤n, and using the Taylor-like formula given in Lemma 3.1, we derive
δv(t)−▽τvk/τk=1τk∫tk−1tδ2v(y)logtk−1ydyy−1τk∫tktδ2v(y)logtkydyy. |
Thus the truncation error at time t=tn is given as
(Rt)ni=CHDαa,tu(xi,tn)−CHDαa,τuni=n∑k=1∫tktk−1ω1−α(logtn−logs)(δv(s)−▽τvk/τk)dss=n∑k=11τk∫tktk−1ω1−α(logtn−logs)∫stk−1δ2v(y)logytk−1dyydss−n∑k=11τk∫tktk−1ω1−α(logtn−logs))∫tksδ2v(y)logtkydyydss. |
Exchanging the order of integration, we have
(Rt)ni=n∑k=11τk∫tktk−1δ2v(y)logytk−1∫tkyω1−α(logtn−logs)dssdyy−n∑k=11τk∫tktk−1δ2v(y)logtky∫ytk−1ω1−α(logtn−logs)dssdyy=n∑k=11τk∫tktk−1δ2v(y)logytk−1ω2−α(logtn−logy)dyy+n∑k=11τk∫tktk−1δ2v(y)logtkyω2−α(logtn−logy)dyy−n∑k=11τk∫tktk−1δ2v(y)logytk−1ω2−α(logtn−logtk)dyy−n∑k=11τk∫tktk−1δ2v(y)logtkyω2−α(logtn−logtk−1)dyy. | (7.1) |
For the sake of simplicity, we denote ϖn(y)=ω2−α(logtn−logy). Note that the linear logarithmic interpolation function of ϖn(y) with respect to the nodes tk−1 and tk is given by
Lklog,1ϖn(y)=1τklogytk−1ϖn(tk)+1τklogtkyϖn(tk−1). |
Let
˜Lklog,1ϖn(y)=ϖn(y)−Lklog,1ϖn(y), |
then (7.1) can be rewritten as follows:
(Rt)ni=n∑k=11τk∫tktk−1δ2v(y)logytk−1ϖn(y)dyy+n∑k=11τk∫tktk−1δ2v(y)logtktk−1ϖn(y)dyy−n∑k=11τk∫tktk−1δ2v(y)logytk−1ϖn(y)dyy−n∑k=1∫tktk−1δ2v(y)Lklog,1ϖn(y)dyy=n∑k=1∫tktk−1δ2v(y)˜Lklog,1ϖn(y)dyy≜n∑k=1(~Rt)nk,n≥1. | (7.2) |
Similar to the proof of Theorem 3.1 in [29], ˜Lklog,1ϖn(y) can be rewritten as
˜Lklog,1ϖn(y)=∫tktk−1ξk(y,s)δ2ϖn(s)dss, | (7.3) |
where ξk(y,s)=max{logy−logs,0}−(logy−logtk−1)(logtk−logs)/τk such that
−logy−logtk−1τk(logy−logs)≤ξk(y,s)≤0,∀y,s∈(tk−1,tk). | (7.4) |
For k=n, since the function ϖn(y) is decreasing with respect to y and δ2ϖn(y)<0, one has
0≤˜Lnlog,1ϖn(y)≤ϖn(tn−1)−Lnlog,1ϖn(y)=ω2−α(logtn−logtn−1)−1τnlogtnyω2−α(logtn−logtn−1)=(logy−logtn−1)A(n)0. |
Thus,
(~Rt)nn≤∫tntn−1|δ2v(y)|˜Lnlog,1ϖn(y)dyy≤A(n)0∫tntn−1(logy−logtn−1)|δ2v(y)|dyy. | (7.5) |
For 1≤k≤n−1, using (7.3) and (7.4), one has
˜Lklog,1ϖn(y)≤(logtk−1−logy)∫tktk−1δ2ω2−α(logtn−logy)dyy≤(logy−logtk−1)(A(n)n−k−1−A(n)n−k). | (7.6) |
Then we obtain for 1≤k≤n−1 that
(~Rt)nk≤∫tktk−1|δ2v(y)|˜Lklog,1ϖn(y)dyy≤∫tktk−1|δ2v(y)|(A(n)n−k−1−A(n)n−k)(logy−logtk−1)dyy. | (7.7) |
Combining (7.2), (7.5), and (7.7), the proof is complete.
Proof. After multiplying the inequality (3.20) by P(n)n−k and summing the index k from 1 to n, it is possible to switch the order of summation and apply the definition (2.5) of P(n)n−k to obtain
n∑k=1P(n)n−k‖(Rt)k‖≤n∑k=1P(n)n−kA(k)0Gk+n∑k=2P(n)n−kk−1∑j=1(A(k)k−j−1−A(k)k−j)Gj=n∑k=1P(n)n−kA(k)0Gk+n−1∑j=1Gjn∑k=j+1P(n)n−k(A(k)k−j−1−A(k)k−j)=n∑k=1P(n)n−kA(k)0Gk+n−1∑j=1P(n)n−kA(k)0Gj≤2n∑k=1P(n)n−kA(k)0Gk:=n∑k=1P(n)n−kGk. | (7.8) |
Applying Lemma 2.5 by taking υk=Gk in (2.11) to (7.8), one has
n∑k=1P(n)n−k‖(Rt)k‖≤k∑j=1P(k)k−jGk≤Γ(2−α)n∑j=1τjmaxj≤k≤n((logtka)α−1⋅Gk)=Γ(2−α)n∑j=2τjmaxj≤k≤n((logtka)α−1⋅Gk)+Γ(2−α)max{τ1max2≤k≤n((logtka)α−1⋅Gk),τα1G1}≤Γ(2−α)(n∑j=2τjmaxj≤k≤n((logtka)α−1⋅Gk))+Γ(2−α)(ρτ2max2≤k≤n((logtka)α−1⋅Gk)+τα1G1)≤(1+ρ)Γ(2−α)(n∑j=2τjmaxj≤k≤n((logtka)α−1⋅Gk)+τα1G1). | (7.9) |
On one hand, it follows from (3.2) that
A(k)0=1τk∫tktk−1ω1−α(logtk−logs)dss=1Γ(2−α)ταk, k=1,...,n. | (7.10) |
On the other hand, the assumption of regularity (1.5) leads to the conclusions that
G1≤∫t1t0(logs−logt0)⋅C(1+(logsa)σ−2)dss≤Cτσ1σ, | (7.11) |
and
Gk≤∫tktk−1(logs−logtk−1)dss⋅C(1+(logtk−1a)σ−2)≤Cτ2k(logtk−1a)σ−2 | (7.12) |
for 2≤k≤n. By combining (7.9), (7.10), and (7.11) with (7.12), we obtain the desired result.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work is supported by the Natural Science Foundation of Fujian Province of China (2022J01338, 2024J01119) and Fujian Alliance of Mathematics of China (2024SXLMMS03).
The authors declare that they have no competing interests.
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N | α=0.3 | α=0.5 | α=0.7 | |||
Error(M,N) | Order | Error(M,N) | Order | Error(M,N) | Order | |
64 | 1.1134×10−2 | − | 2.6984×10−3 | − | 9.4560×10−4 | − |
128 | 7.9395×10−3 | 0.5153 | 1.3721×10−3 | 0.9757 | 4.0801×10−4 | 1.2126 |
256 | 5.4559×10−3 | 0.5412 | 6.9778×10−4 | 0.9756 | 1.7224×10−4 | 1.2442 |
512 | 3.7010×10−4 | 0.5599 | 3.5227×10−4 | 0.9861 | 7.1677×10−5 | 1.2648 |
min{rσ,2−α} | 0.6 | 1.0 | 1.3 |
N | α=0.3 | α=0.5 | α=0.7 | |||
Error(M, N) | Order | Error(M, N) | Order | Error(M, N) | Order | |
64 | 3.7010×10−3 | - | 7.0310×10−4 | - | 6.2060×10−4 | - |
128 | 2.0350×10−3 | 0.8629 | 2.6509×10−4 | 1.4072 | 2.5489×10−4 | 1.2838 |
256 | 1.1093×10−3 | 0.8797 | 9.7726×10−5 | 1.4397 | 1.0412×10−4 | 1.2917 |
512 | 5.9718×10−4 | 0.8890 | 3.5547×10−5 | 1.4590 | 4.2399×10−5 | 1.2961 |
min{rσ,2−α} | 0.9 | 1.5 | 1.3 |
N | α=0.5 | α=0.7 | α=0.9 | |||
Error(M, N) | Order | Error(M, N) | Order | Error(M, N) | Order | |
32 | 1.7972×10−3 | - | 2.4367×10−3 | - | 1.9017×10−3 | - |
64 | 7.0306×10−4 | 1.3541 | 1.1229×10−3 | 1.1176 | 1.0393×10−3 | 0.8716 |
128 | 2.6507×10−4 | 1.4072 | 4.9963×10−4 | 1.1683 | 5.5426×10−4 | 0.9070 |
256 | 9.7719×10−5 | 1.4397 | 2.1701×10−4 | 1.2031 | 2.8970×10−4 | 0.9360 |
min{rσ,2−α} | 1.5 | 1.3 | 1.1 |
N | α=0.3 | α=0.5 | α=0.7 | |||
Error(M,N) | Order | Error(M,N) | Order | Error(M,N) | Order | |
32 | 2.7134×10−3 | − | 2.7984×10−3 | − | 2.4560×10−3 | − |
64 | 1.4395×10−3 | 0.9746 | 1.4721×10−3 | 0.9757 | 1.2811×10−3 | 0.9426 |
128 | 7.4559×10−4 | 0.9871 | 6.9776×10−4 | 0.9839 | 6.7224×10−4 | 0.9622 |
256 | 3.7010×10−4 | 0.9935 | 4.5227×10−4 | 0.9860 | 3.1677×10−4 | 0.9809 |
min{rσ,2} | 1.0 | 1.0 | 1.0 |
N | α=0.3 | α=0.5 | α=0.8 | |||
Error(M, N) | Order | Error(M, N) | Order | Error(M, N) | Order | |
64 | 2.2791×10−2 | - | 1.7493×10−2 | - | 3.7540×10−3 | - |
128 | 2.4622×10−2 | 0.1838 | 1.3208×10−2 | 0.4053 | 2.2841×10−3 | 0.7240 |
256 | 2.1453×10−2 | 0.1982 | 9.8096×10−3 | 0.4291 | 1.3224×10−3 | 0.7552 |
512 | 1.8563×10−2 | 0.2117 | 7.1924×10−4 | 0.4477 | 8.1627×10−4 | 0.7658 |
min{rσ,2} | 0.3 | 0.5 | 0.8 |
N | α=0.3 | α=0.5 | α=0.8 | |||
Error(M, N) | Order | Error(M, N) | Order | Error(M, N) | Order | |
16 | 5.2002×10−3 | - | 1.1646×10−3 | - | 2.2565×10−4 | - |
32 | 3.0471×10−3 | 0.7711 | 4.3955×10−4 | 1.4057 | 5.8962×10−5 | 1.9362 |
64 | 1.7190×10−3 | 0.8259 | 1.5959×10−4 | 1.4616 | 1.4430×10−5 | 2.0308 |
128 | 9.4796×10−4 | 0.8587 | 5.6967×10−5 | 1.4862 | 3.3768E-06 | 2.0953 |
min{rσ,2} | 0.9 | 1.5 | 2 |
N | α=0.5 | α=0.7 | α=0.9 | |||
Error(M, N) | Order | Error(M, N) | Order | Error(M, N) | Order | |
16 | 5.9243×10−4 | - | 4.1200×10−4 | - | 1.6522×10−4 | - |
32 | 1.6398×10−4 | 1.8531 | 1.1720×10−4 | 1.8136 | 5.0273×10−5 | 1.7165 |
64 | 4.2850×10−5 | 1.9361 | 3.3138×10−5 | 1.9008 | 1.4495×10−5 | 1.7942 |
128 | 1.0862×10−5 | 1.9801 | 8.0773E-06 | 1.9582 | 4.0215E-06 | 1.8497 |
min{rσ,2} | 2 | 2 | 2 |
N | α=0.3 | α=0.5 | α=0.99 | τ | |||
Error(M,N) | Order | Error(M,N) | Order | Error(M,N) | Order | ||
32 | 6.2201×10−3 | − | 1.8010×10−3 | − | 2.6758×10−4 | − | 3.6837×10−2 |
64 | 3.3333×10−3 | 0.8999 | 6.4494×10−4 | 1.4816 | 1.3421×10−4 | 0.9954 | 1.8711×10−2 |
128 | 1.7863×10−3 | 0.9000 | 2.2925×10−4 | .4922 | 6.5942×10−5 | 1.0252 | 9.4290×10−3 |
256 | 9.5724×10−4 | 0.9000 | 8.1375×10−5 | 1.49427 | 3.2376×10−5 | 1.0262 | 4.7330×10−3 |
min{rσ,2−α} | 0.9 | 1.5 | 1.01 | B1=0.2479 |
N | α=0.2 | α=0.5 | α=0.99 | τ | |||
Error(M,N) | Order | Error(M,N) | Order | Error(M,N) | Order | ||
32 | 2.9061×10−2 | − | 1.0479×10−2 | − | 4.3943×10−5 | − | 3.6836×10−2 |
64 | 2.5299×10−2 | 0.1999 | 7.4103×10−2 | 0.4999 | 2.2143×10−5 | 0.9887 | 1.8711×10−2 |
128 | 2.2025×10−2 | 0.1999 | 5.2399×10−3 | 0.4999 | 1.1154×10−5 | 0.9892 | 9.4290×10−3 |
256 | 1.9174×10−2 | 0.1999 | 3.7052×10−3 | 0.4999 | 5.6176E-06 | 0.9896 | 4.7330×10−3 |
min{rσ,2} | 0.2 | 0.5 | 0.99 | B2=0.0892 |
N | α=0.3 | α=0.5 | α=0.7 | |||
Error(M,N) | Order | Error(M,N) | Order | Error(M,N) | Order | |
64 | 1.1134×10−2 | − | 2.6984×10−3 | − | 9.4560×10−4 | − |
128 | 7.9395×10−3 | 0.5153 | 1.3721×10−3 | 0.9757 | 4.0801×10−4 | 1.2126 |
256 | 5.4559×10−3 | 0.5412 | 6.9778×10−4 | 0.9756 | 1.7224×10−4 | 1.2442 |
512 | 3.7010×10−4 | 0.5599 | 3.5227×10−4 | 0.9861 | 7.1677×10−5 | 1.2648 |
min{rσ,2−α} | 0.6 | 1.0 | 1.3 |
N | α=0.3 | α=0.5 | α=0.7 | |||
Error(M, N) | Order | Error(M, N) | Order | Error(M, N) | Order | |
64 | 3.7010×10−3 | - | 7.0310×10−4 | - | 6.2060×10−4 | - |
128 | 2.0350×10−3 | 0.8629 | 2.6509×10−4 | 1.4072 | 2.5489×10−4 | 1.2838 |
256 | 1.1093×10−3 | 0.8797 | 9.7726×10−5 | 1.4397 | 1.0412×10−4 | 1.2917 |
512 | 5.9718×10−4 | 0.8890 | 3.5547×10−5 | 1.4590 | 4.2399×10−5 | 1.2961 |
min{rσ,2−α} | 0.9 | 1.5 | 1.3 |
N | α=0.5 | α=0.7 | α=0.9 | |||
Error(M, N) | Order | Error(M, N) | Order | Error(M, N) | Order | |
32 | 1.7972×10−3 | - | 2.4367×10−3 | - | 1.9017×10−3 | - |
64 | 7.0306×10−4 | 1.3541 | 1.1229×10−3 | 1.1176 | 1.0393×10−3 | 0.8716 |
128 | 2.6507×10−4 | 1.4072 | 4.9963×10−4 | 1.1683 | 5.5426×10−4 | 0.9070 |
256 | 9.7719×10−5 | 1.4397 | 2.1701×10−4 | 1.2031 | 2.8970×10−4 | 0.9360 |
min{rσ,2−α} | 1.5 | 1.3 | 1.1 |
N | α=0.3 | α=0.5 | α=0.7 | |||
Error(M,N) | Order | Error(M,N) | Order | Error(M,N) | Order | |
32 | 2.7134×10−3 | − | 2.7984×10−3 | − | 2.4560×10−3 | − |
64 | 1.4395×10−3 | 0.9746 | 1.4721×10−3 | 0.9757 | 1.2811×10−3 | 0.9426 |
128 | 7.4559×10−4 | 0.9871 | 6.9776×10−4 | 0.9839 | 6.7224×10−4 | 0.9622 |
256 | 3.7010×10−4 | 0.9935 | 4.5227×10−4 | 0.9860 | 3.1677×10−4 | 0.9809 |
min{rσ,2} | 1.0 | 1.0 | 1.0 |
N | α=0.3 | α=0.5 | α=0.8 | |||
Error(M, N) | Order | Error(M, N) | Order | Error(M, N) | Order | |
64 | 2.2791×10−2 | - | 1.7493×10−2 | - | 3.7540×10−3 | - |
128 | 2.4622×10−2 | 0.1838 | 1.3208×10−2 | 0.4053 | 2.2841×10−3 | 0.7240 |
256 | 2.1453×10−2 | 0.1982 | 9.8096×10−3 | 0.4291 | 1.3224×10−3 | 0.7552 |
512 | 1.8563×10−2 | 0.2117 | 7.1924×10−4 | 0.4477 | 8.1627×10−4 | 0.7658 |
min{rσ,2} | 0.3 | 0.5 | 0.8 |
N | α=0.3 | α=0.5 | α=0.8 | |||
Error(M, N) | Order | Error(M, N) | Order | Error(M, N) | Order | |
16 | 5.2002×10−3 | - | 1.1646×10−3 | - | 2.2565×10−4 | - |
32 | 3.0471×10−3 | 0.7711 | 4.3955×10−4 | 1.4057 | 5.8962×10−5 | 1.9362 |
64 | 1.7190×10−3 | 0.8259 | 1.5959×10−4 | 1.4616 | 1.4430×10−5 | 2.0308 |
128 | 9.4796×10−4 | 0.8587 | 5.6967×10−5 | 1.4862 | 3.3768E-06 | 2.0953 |
min{rσ,2} | 0.9 | 1.5 | 2 |
N | α=0.5 | α=0.7 | α=0.9 | |||
Error(M, N) | Order | Error(M, N) | Order | Error(M, N) | Order | |
16 | 5.9243×10−4 | - | 4.1200×10−4 | - | 1.6522×10−4 | - |
32 | 1.6398×10−4 | 1.8531 | 1.1720×10−4 | 1.8136 | 5.0273×10−5 | 1.7165 |
64 | 4.2850×10−5 | 1.9361 | 3.3138×10−5 | 1.9008 | 1.4495×10−5 | 1.7942 |
128 | 1.0862×10−5 | 1.9801 | 8.0773E-06 | 1.9582 | 4.0215E-06 | 1.8497 |
min{rσ,2} | 2 | 2 | 2 |
N | α=0.3 | α=0.5 | α=0.99 | τ | |||
Error(M,N) | Order | Error(M,N) | Order | Error(M,N) | Order | ||
32 | 6.2201×10−3 | − | 1.8010×10−3 | − | 2.6758×10−4 | − | 3.6837×10−2 |
64 | 3.3333×10−3 | 0.8999 | 6.4494×10−4 | 1.4816 | 1.3421×10−4 | 0.9954 | 1.8711×10−2 |
128 | 1.7863×10−3 | 0.9000 | 2.2925×10−4 | .4922 | 6.5942×10−5 | 1.0252 | 9.4290×10−3 |
256 | 9.5724×10−4 | 0.9000 | 8.1375×10−5 | 1.49427 | 3.2376×10−5 | 1.0262 | 4.7330×10−3 |
min{rσ,2−α} | 0.9 | 1.5 | 1.01 | B1=0.2479 |
N | α=0.2 | α=0.5 | α=0.99 | τ | |||
Error(M,N) | Order | Error(M,N) | Order | Error(M,N) | Order | ||
32 | 2.9061×10−2 | − | 1.0479×10−2 | − | 4.3943×10−5 | − | 3.6836×10−2 |
64 | 2.5299×10−2 | 0.1999 | 7.4103×10−2 | 0.4999 | 2.2143×10−5 | 0.9887 | 1.8711×10−2 |
128 | 2.2025×10−2 | 0.1999 | 5.2399×10−3 | 0.4999 | 1.1154×10−5 | 0.9892 | 9.4290×10−3 |
256 | 1.9174×10−2 | 0.1999 | 3.7052×10−3 | 0.4999 | 5.6176E-06 | 0.9896 | 4.7330×10−3 |
min{rσ,2} | 0.2 | 0.5 | 0.99 | B2=0.0892 |