Loading [MathJax]/jax/output/SVG/jax.js
Research article Topical Sections

Helium Ion Microscopy of proton exchange membrane fuel cell electrode structures

  • Received: 02 October 2017 Accepted: 27 November 2017 Published: 05 December 2017
  • Characterization of composite materials with microscopy techniques is an essential route to understanding their properties and degradation mechanisms, though the observation with a suitable type of microscopy is not always possible. In this work, we present proton exchange membrane fuel cell electrode interface structure dependence on ionomer content, systematically studied by Helium Ion Microscopy (HIM). A special focus was on acquiring high resolution images of the electrode structure and avoiding interface damage from irradiation and tedious sample preparation. HIM demonstrated its advantages in surface imaging, which is paramount in studies of the interface morphology of ionomer covered or absorbed catalyst structures in a combination with electrochemical characterization and accelerated stress test. The electrode porosity was found to depend on the ionomer content. The stressed electrodes demonstrated higher porosity in comparison to the unstressed ones on the condition of no external mechanical pressure. Moreover, formation of additional small grains was observed for the electrodes with the low ionomer content, indicating Pt redeposition through Ostwald ripening. Polymer nanofiber structures were found in the crack regions of the catalyst layer, which appear due to the internal stress originated from the solvent evaporation. These fibers have fairly uniform diameters of a few tens of nanometers, and their density increases with the increasing ionomer content in the electrodes. In the hot-pressed electrodes, we found more closed contact between the electrode components, reduced particle size, polymer coalescence and formation of nano-sized polymer fiber architecture between the particles.

    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

    Related Papers:

    [1] Chang Hou, Hu Chen . Stability and pointwise-in-time convergence analysis of a finite difference scheme for a 2D nonlinear multi-term subdiffusion equation. Electronic Research Archive, 2025, 33(3): 1476-1489. doi: 10.3934/era.2025069
    [2] Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao . A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28(4): 1439-1457. doi: 10.3934/era.2020076
    [3] Lijie Liu, Xiaojing Wei, Leilei Wei . A fully discrete local discontinuous Galerkin method based on generalized numerical fluxes to variable-order time-fractional reaction-diffusion problem with the Caputo fractional derivative. Electronic Research Archive, 2023, 31(9): 5701-5715. doi: 10.3934/era.2023289
    [4] Jun Liu, Yue Liu, Xiaoge Yu, Xiao Ye . An efficient numerical method based on QSC for multi-term variable-order time fractional mobile-immobile diffusion equation with Neumann boundary condition. Electronic Research Archive, 2025, 33(2): 642-666. doi: 10.3934/era.2025030
    [5] Li Tian, Ziqiang Wang, Junying Cao . A high-order numerical scheme for right Caputo fractional differential equations with uniform accuracy. Electronic Research Archive, 2022, 30(10): 3825-3854. doi: 10.3934/era.2022195
    [6] Jun Pan, Yuelong Tang . Two-grid H1-Galerkin mixed finite elements combined with L1 scheme for nonlinear time fractional parabolic equations. Electronic Research Archive, 2023, 31(12): 7207-7223. doi: 10.3934/era.2023365
    [7] Akeel A. AL-saedi, Jalil Rashidinia . Application of the B-spline Galerkin approach for approximating the time-fractional Burger's equation. Electronic Research Archive, 2023, 31(7): 4248-4265. doi: 10.3934/era.2023216
    [8] Yining Yang, Yang Liu, Cao Wen, Hong Li, Jinfeng Wang . Efficient time second-order SCQ formula combined with a mixed element method for a nonlinear time fractional wave model. Electronic Research Archive, 2022, 30(2): 440-458. doi: 10.3934/era.2022023
    [9] Jinheng Liu, Kemei Zhang, Xue-Jun Xie . The existence of solutions of Hadamard fractional differential equations with integral and discrete boundary conditions on infinite interval. Electronic Research Archive, 2024, 32(4): 2286-2309. doi: 10.3934/era.2024104
    [10] Wenjing An, Xingdong Zhang . An implicit fully discrete compact finite difference scheme for time fractional diffusion-wave equation. Electronic Research Archive, 2024, 32(1): 354-369. doi: 10.3934/era.2024017
  • Characterization of composite materials with microscopy techniques is an essential route to understanding their properties and degradation mechanisms, though the observation with a suitable type of microscopy is not always possible. In this work, we present proton exchange membrane fuel cell electrode interface structure dependence on ionomer content, systematically studied by Helium Ion Microscopy (HIM). A special focus was on acquiring high resolution images of the electrode structure and avoiding interface damage from irradiation and tedious sample preparation. HIM demonstrated its advantages in surface imaging, which is paramount in studies of the interface morphology of ionomer covered or absorbed catalyst structures in a combination with electrochemical characterization and accelerated stress test. The electrode porosity was found to depend on the ionomer content. The stressed electrodes demonstrated higher porosity in comparison to the unstressed ones on the condition of no external mechanical pressure. Moreover, formation of additional small grains was observed for the electrodes with the low ionomer content, indicating Pt redeposition through Ostwald ripening. Polymer nanofiber structures were found in the crack regions of the catalyst layer, which appear due to the internal stress originated from the solvent evaporation. These fibers have fairly uniform diameters of a few tens of nanometers, and their density increases with the increasing ionomer content in the electrodes. In the hot-pressed electrodes, we found more closed contact between the electrode components, reduced particle size, polymer coalescence and formation of nano-sized polymer fiber architecture between the particles.


    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)=taω1α(logtlogs)δv(x,s)dss, 0<a<t,

    where 0<α<1,ωβ(t)=tβ1Γ(β), δv(x,s)=(ss)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),             xΩ, (1.2)
    u(xl,t)=u(xr,t)=0,    tΛ. (1.3)

    Here, the real constant κR 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(1α) or 1/(1α) 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 α1.

    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)σl),   l=0,1,2, (1.5)

    with the regularity parameter σ(0,1)(1,2), and δlu(x,t)=(tt)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=xrxlM. Set discrete grid Ωh={xi|0iM} with xi=xl+ih. For any grid function w={wi|wi=w(xi), xiΩh}, let δ2xwi be the standard second-order approximation of 2xw(xi), i.e.,

    2xw(xi)δ2xwi:=wi12wi+wi+1h2.

    Denote the space of grid functions W={w|w0=wM=0}. For any w,vW, the discrete L2 inner product and the associate L2 norm are given as

    w,v:=hMi=0wivi,  w:=w,w.

    Denote

    xwi=wiwi1h, 1iM,

    for any grid functions w,vW, 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<<tk1<tk<<tN=T, and set

    τk=logtklogtk1,   1kN,ρk=τkτk+1,                 1kN1.

    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,r1,

    and, correspondingly,

    logtk=loga+(logTa)(kN)r,   k=0,1,,N. (2.2)

    Define vn:=v(x,tn),tnθ:=θtn1+(1θ)tn, and vnθ:=θvn1+(1θ)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,tnθ)CHDαa,τvnθ=nk=1A(n)nkτvk, (2.3)

    where the difference operator is τvk=vkvk1 for k1. To conduct our error analysis, we require the following three assumptions.

    A1. The discrete kernels are monotone, meaning that

    A(n)0A(n)1A(n)2...A(n)n1for1nN.

    A2. There exists a constant πA>0 such that the discrete kernels satisfy a lower bound

    A(n)nk1πAτktktk1ω1α(logtnlogs)dssfor1knN.

    A3. There exists a constant ρ>0 for which the step size ratios ρk satisfy

    ρkρfor1kN1.

    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=logtklogtk1logtk+1logtk=kr(k1)r(k+1)rkr=ηr11ηr12<1,

    with η1(k1,k),η2(k,k+1).

    Next, to derive the global consistency error, we introduce the discrete complementary convolution (DCC) kernels [16]

    P(n)nk=1A(k)0{1,k=n,nj=k+1(A(j)jk1A(i)jk)P(n)nj,1kn1, (2.4)

    which are specifically chosen to enforce the identity

    nj=mP(n)njA(j)jm1,for1mnN. (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)N1l=0,(vk)Nk=0 satisfy

    nk=1A(n)nkτ(vk)2nk=1λnk(vkθ)2+vnθgn,1nN. (2.6)

    If a constant Π satisfies ΠN1l=0λl and the maximum step size satisfies

    max1nNτn1α2max{1,ρ}πAΓ(2α)Π, (2.7)

    then it holds that

    vnEα(2max{1,ρ}πAΠ(logtn)α)(v0+max1knkj=1P(k)kjgn), (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)ν1(logsa)β1dss=Γ(ν)Γ(β)Γ(ν+β)(logba)ν+β1.

    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

    (ba)bag(s)h(s)dsbag(t)dtbah(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

    nj=1P(n)njπAω1+α(logtnloga).

    Proof. Denote h(t)=ω1+α(logtloga), then it holds δh(t)=ωα(logtloga). In one side, it follows from the definition of the Caputo-Hadamard derivative that

    CHDαa,th(tj)=jk=1tktk1ω1α(logtjlogs)ωα(logsloga)dss=jk=1logtklogtk1ω1α(logtjτ)ωα(τloga)dτ.

    It is easy to check that ω1α(logtjτ) is monotone increasing and ωα(τloga) is monotone decreasing on the interval [logtk1,logtk]. Thus, by Lemma 2.3 and A2, we obtain

    CHDαa,th(tj)jk=11τklogtklogtk1ω1α(logtjτ)dτlogtklogtk1ωα(τloga)dτ=jk=11τktktk1ω1α(logtjlogt)dtttktk1ωα(logsloga)dssπAjk=1A(j)jktktk1ωα(logsloga)dss=πAjk=1A(j)jktktk1δh(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)=tjaω1α(logtjlogs)ωα(logsloga)dss=1Γ(1α)Γ(α)tja(logtjs)α(logsa)α1dss=1. (2.10)

    Using (2.9), (2.10), and (2.5), we have

    nj=1P(n)nj=nj=1P(n)njCHDαa,th(tj)nj=1P(n)njπAjk=1A(j)jktktk1δh(s)dss=πAnk=1tktk1δh(s)dssnj=kP(n)njA(j)jk=πAtnt0δh(s)dss=πAω1+α(logtnloga).

    Lemma 2.5. Assume that A1A2 hold, and for any positive sequence (υk)nk=1, one has

    nk=1P(n)nkυkΓ(2α)πAnj=1τjmaxjkn(υk(logtka)α1). (2.11)

    Proof. It follows from A2 that

    kj=1A(k)kjτj1πAtkt0ω1α(logtklogs)dss=(logtkloga)1απAΓ(2α). (2.12)

    Thus, using (2.12) and (2.5), we have

    nk=1P(n)nkυk1nk=1P(n)nkυkπAΓ(2α)kj=1(logtkloga)α1A(k)kjτjπAΓ(2α)nj=1τjnk=jP(n)nkA(k)kj(logtkloga)α1υkΓ(2α)πAnj=1τjmaxjkn(υk(logtka)α1)nk=jP(n)nkA(k)kj=Γ(2α)πAnj=1τjmaxjkn(υk(logtka)α1).

    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=nk=1A(n)nkτvk, (3.1)

    where the discrete coefficients A(n)nk are defined by

    A(n)nk=1τktktk1ω1α(logtnlogs)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)nk1A(n)nk=1Γ(1α)(1τk+1tk+1tk(logtns)αdss1τktktk1(logtns)αdss)=1Γ(1α)[(logtnξk+1)α(logtnξk)α]>0, (3.3)

    with ξk+1(tk,tk+1),ξk(tk1,tk). Thus Assumption A1 holds.

    Let uni be the discrete approximation of solution u(xi,tn) for xiΩh,0nN. The fully discrete scheme for problem (1.1)–(1.3) is given as

    CHDαa,τuniδ2xuni+κuni=fni,1iM1,1nN, (3.4)
    u0i=φ(xi),0iM, (3.5)
    un0=unM=0,1nN, (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,τδ2x+κ)vni=gni,1iM1,1nN, (3.7)
    v0i=φ(xi),0iM, (3.8)
    vn0=vnM=0,1nN. (3.9)

    If the maximum step size is τB1 with

    B1=1α4κ+max{1,ρ}Γ(2α), (3.10)

    we have

    vn2Eα(4κ+max{1,ρ}(logtn)α)(v0+2max1knkj=1P(k)kjgj),1nN. (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)0vn22n1k=1(A(n)nk1A(n)nk)(vk,vn)2A(n)n1(v0,vn)2A(n)0vn2n1k=1(A(n)nk1A(n)nk)vn2A(n)n1vn2n1k=1(A(n)nk1A(n)nk)vk2A(n)n1vk2=nk=1A(n)nkτvk2. (3.12)

    It follows that

    nk=1A(n)nkτvk22(δ2xvn,vn)+2κ(vn,vn)2(gn,vn),

    Noting (2.1), and using the Schwarz inequality, we obtain

    nk=1A(n)nkτvk22κ+vn2+2gnvn.

    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|0iM,0nN} be the solution of problem (3.4)–(3.6). Set

    eni=u(xi,tn)uni,  0iM, 0nN.

    It is straightforward to obtain the following error equation:

    CHDαa,τeniδ2xeni+κeni=(Rt)ni+(Rs)ni,1iM1,1nN, (3.13)
    e0i=0,0iM, (3.14)
    en0=enM=0,1nN, (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),

    enˉC(e0+2max1knkj=1P(k)kj((Rt)j+(Rs)j)),  1nN. (3.18)

    Under the assumption of spatial regularity given by (1.4), the Taylor expansion provides a direct demonstration that (Rs)nCh2. When combined with Lemma 2.4, we can further deduce that

    nj=1P(n)nj(Rs)nC(logtna)αh2. (3.19)

    Now, we only need to estimate the term nj=1P(n)nj(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)(logxlogx0)+δ2f(x0)2!(logxlogx0)2++δnf(x0)n!(logxlogx0)n+1n!xx0δn+1f(s)(logxlogs)ndss.

    Lemma 3.2. For any function uC3((a,T]), the local truncation errors (Rt)ki satisfy

    (Rt)kiA(k)0Gk+k1j=1(A(k)kj1A(k)kj)Gj. (3.20)

    where

    Gk=tktk1(logslogtk1)|δ2u(xi,s)|dss,1kn. (3.21)

    Lemma 3.3. Assume that A3 holds, and u(x,t) satisfies the regularity assumption (1.5), then

    nk=1P(n)nk(Rt)kC(τσ1σ+nj=2τjmaxjkn(logtka)α1(logtk1a)σ2τ2αk).

    where C=(1+ρ)Γ(2α).

    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|0iM,0nN} be the solution of (3.4)–(3.6). Suppose the regularity assumptions (1.4)–(1.5) hold. Set

    eni=u(xi,tn)uni,  0iM, 0nN.

    If the maximum step size is τB1, then the discrete solution is convergent in the L2-norm with ˉC=2Eα(4κ+max{1,ρ}(logtn)α) such that

    enCˉC(e0+(logtna)αh2+(1+ρ)Γ(2α)max1knEkt), (3.22)

    where

    Ekt=τσ1σ+kj=2τjmaxjlk(logtla)α1(logtl1a)σ2τ2αl. (3.23)

    In particular, if graded mesh is used, then it holds

    en2Eα(4κ+(logtn)α)(e0+(logtna)αh2+2Γ(2α)(1σ+4rr3αϑ)Nmin{rσ,2α}),

    with

    ϑ={lnn,σr2α,12αrσ,σr<2α. (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 τkrlogTaNrkr1 as follows:

    τk=logtklogtk1=loga+logTa(logkN)rlogalogTa(logk1N)r=logTaNr(kr(k1)r)logTaNrrkr1. (3.25)

    Note that σ>0, it follows from (3.23) and (3.25) that

    Ent((logTa)σ(1N)rσσ)+nj=2τjmaxjkn(logTa)σ2(k1N)r(σ2)×(logTa)α1(kN)r(α1)(logTa)2αNr(2α)r2αk(2α)(r1)=(logTa)σ(1N)rσσ+nj=2τjmaxjkn(logTa)σ1r2α(kN)r×(kk1)r(2σ)Nσrkrσ(2α). (3.26)

    Taking into account that

    (kk1)r(2σ)(1+1k1)2r22r=4r,

    (3.26) can be simplified to

    Ent(logTa)σNrσ(1σ+r2αlogTanj=2τjmaxjkn(kN)r4rkσr(2α))(logTa)σNrσ(1σ+4rr2α(logTa)1nj=2τjmaxjkn(kN)rkσr(2α)). (3.27)

    For σr2α, we have from (3.27) that

    Ent(logTa)σNrσ(1σ+4rr2α(logTa)1nj=2τj(jN)rnσr(2α))=(logTa)σNrσ(1σ+4rr2α(logTa)1nj=2(logTa)Nrrjr1jrNrnσr(2α))=(logTa)σNrσ(1σ+4rr3αnσr(2α)nj=2j1)(logTa)σNrσ(1σ+4rr3αNσr(2α)lnn)(logTa)σ(1σ+4rr3αlnn)Nmin{rσ,2α}. (3.28)

    For σr<2α, it follows from (3.27) that

    Ent(logTa)σNrσ(1σ+4rr2α(logTa)1nj=2τj(jN)rjσr(2α))(logTa)σNrσ(1σ+4rr2α(logTa)1nj=2r(logTa)Nrjr1(jN)rjσr(2α))=(logTa)σNrσ(1σ+4rr3αnj=2jσr(3α)). (3.29)

    Note that

    nj=2jσr(3α)n1sσr(3α)ds=1rσ(2α)(nrσ(2α)1)12αrσ,

    then we get from (3.29) that

    Ent(logTa)σ(1σ+4rr3α2αrσ)Nrσ. (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(tnθ)CHDαa,τvnθ=nk=1A(n)nkτvkfor1nN, (4.1)

    with θ=α2. Here the discrete convolution kernel A(n)nk is defined as: A(n)0=a(n)0 if n=1 and, for n2,

    A(n)nk={a(n)n1b(n)n1,k=1,a(n)nk+ρk1b(n)nk+1b(n)nk,k=2,...,n1,a(n)0+ρn1b(n)1,k=n.

    Moreover, the discrete coefficients a(n)nk and b(n)nk are determined by:

    a(n)0=1τntnθtn1δϖn(s)dss,a(n)nk=1τktktk1δϖn(s)dss,1kn1,b(n)nk=2τk(τk+τk+1)tktk1(logslogtk1/2)δϖn(s)dss,1kn1,

    where

    ϖn(s)=ω2α(logtnθlogs), logtk1/2=12(logtk1+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,tnθ), the fully discrete scheme for problem (1.1) is given as

    CHDαa,τunθiδ2xunθi+κunθi=fnθi,1iM1,1kN, (4.2)
    u0i=φ(xi),0iM, (4.3)
    un0=unM=0,1kN. (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,τδ2x+κ)vnθi=gnθi,1iM1,1nN, (4.5)
    v0i=φ(xi),0iM, (4.6)
    vn0=vnM=0,1nN. (4.7)

    If the maximum step size is τB2 with

    B2=1α11κ+max{1,ρ}Γ(2α), (4.8)

    we have

    vnEα(11κ+max{1,ρ}(logtn)α)(v0+2max1knkj=1P(k)kjgjθ). (4.9)

    Proof. Taking the inner products with 2vnθi on both sides of (4.5), we obtain

    2(CHDαa,τvnθ,vnθ)2(δ2xvnθ,vnθ)+2κ(vnθ,vnθ)=2(fnθ,vnθ),

    Utilizing (3.12), we can further derive

    nk=1A(n)nkτvk22(δ2xvnθ,vnθ)+2κ(vnθ,vnθ)2(fnθ,vnθ).

    Taking into account (2.1) and applying the Schwarz inequality, we arrive at

    nk=1A(n)nkτvk22κ+vnθ2+2fnθvnθ.

    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 {unθi|0iM,0nN} be the solution of problem (4.2)–(4.4). Set

    enθi=u(xi,tnθ)unθi.

    It is straightforward to obtain the following error equation:

    (CHDαa,τδ2x+κ)enθi=Rni,1iM1,1nN, (4.10)
    e0i=0,0iM, (4.11)
    en0=enM=0,1nN, (4.12)

    where Rni=(Rt)ni+(Rs)ni+(Rl)ni with

    (Rt)ni=CHDαa,tu(xi,tnθ)CHDαa,τunθi, (4.13)
    (Rs)ni=2xu(xi,tnθ)δ2xu(xi,tnθ), (4.14)
    (Rl)ni=δ2x(u(xi,tnθ)unθi)κ(u(xi,tnθ)unθi). (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+2max1knkj=1P(k)kj((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)nj(Rl)n and nj=1P(n)nj(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

    nj=1P(n)nj(Rl)jC(τσ+α1σ+(logtna)αmax2kn(logtk1a)σ2τ2k). (4.17)

    Proof. Set v(t)=(δ2xκ)u(xi,t). Using the Taylor-like formula given in Lemma 3.1, we derive

    v(tj)=v(tjθ)+δv(tjθ)(logtjlogtjθ)+tjtjθδ2v(s)(logtjlogs)dss, (4.18)
    v(tj1)=v(tjθ)+δv(tjθ)(logtj1logtjθ)+tj1tjθδ2v(s)(logtj1logs)dss. (4.19)

    By combining (4.18) and (4.19), we obtain

    θv(tj1)+θ(1θ)v(tj)=v(tjθ)+θtjθtj1δ2v(s)(logslogtj1)dss+(1θ)tjtjθδ2v(s)(logtjlogs)dss.

    This leads to the following expression for (Rl)ji:

    (Rl)ji=θtjθtj1δ2v(s)(logslogtj1)dss(1θ)tjtjθδ2v(s)(logtjlogs)dss. (4.20)

    For j=1, utilizing the regularity assumption (1.5), we deduce from (4.20) that

    (Rl)1θt1θt0|δ2v(s)|(logslogt0)dss+(1θ)t1t1θ|δ2v(s)|(logt1logs)dssCt1θt0(logsa)σ1dss+Ct1t1θ(logt1logs)(logsa)σ2dssCt1θt0(logsa)σ1dssCτσ1σ. (4.21)

    Analogously, for 2jN, we have

    (Rl)jθtjθtj1|δ2v(s)|(logslogtj1)dss+(1θ)tjtjθ|δ2v(s)|(logtjlogs)dssC(logtj1a)σ2(tjθtj1(logslogtj1)dss+tjtjθ(logtjlogs)dss)C(logtj1a)σ2τ2j. (4.22)

    It follows from (2.4), A1, and A2 that

    P(n)n11/A(1)0114Γ(2α)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

    nj=1P(n)nj(Rt)jC(τσ1σ+τσ31τ32)+Cnj=2τjmaxjkn((logtk1a)σ3(logtka)α1τ3αk+(logtka)σ+α4τ3k+1ταk). (4.24)

    Proof. It follows from [29] that the local truncation error defined in (4.13) can be bounded by

    (Rt)kA(k)0Gkloc+k1j=1(A(k)kj1A(k)kj)Gjhis, (4.25)

    where

    Gkloc=32tk1/2tk1(logslogtk1)2|δ3u(s)|dss+3τk2tktk1/2(logtklogs)|δ3u(s)|dss,%Gkhis=52tktk1(logslogtk1)2|δ3u(s)|dss+52tk+1tk(logtk+1logs)|δ3u(s)|dss.

    By the regularity assumption (1.5), it is easy to obtain

    G1locCτσ1σ,GklocC(logtk1a)σ3τ3k, k2, (4.26)

    and

    G1hisC(τσ1σ+(logt1a)σ3τ32), (4.27)
    GkhisC((logtk1a)σ3τ3k+(logtka)σ3τ3k+1), k2. (4.28)

    Similar to (7.8), we have

    nk=1P(n)nk(Rt)knk=1P(n)nkA(k)0(Gkloc+Gkhis):=nk=1P(n)nkGk. (4.29)

    Applying Lemma 2.5 by taking υk=Gk in (2.11) to (4.29), we get

    nk=1P(n)nk(Rt)k11Γ(2α)4nj=1τjmaxjkn(logtka)α1Gk=11Γ(2α)4[nj=2τjmaxjkn(logtka)α1Gk+max{τ1max2kn(logtka)α1Gk,τα1G1}]11Γ(2α)4(nj=2τjmaxjkn(logtka)α1Gk+ρτ2max2kn(logtka)α1Gk+τα1G1)114(1+ρ)Γ(2α)(nj=2τjmaxjkn((logtka)α1Gk)+τα1G1). (4.30)

    According to (4.26), (4.27), and (4.28), we have

    G1CA(1)0(τσ1σ+τσ1σ+(logt1a)σ3τ32)Cτα1Γ(2α)(τσ1σ+(logt1a)σ3τ32), (4.31)

    and

    GkCA(k)0(logtk1a)σ3τ3k+CA(k)0((logtk1a)σ3τ3k+(logtka)σ3τ3k+1)Cτα1Γ(2α)((logtk1a)σ3τ3αk+(logtka)σ3τ3k+1ταk). (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|0iM,0nN} be the solution of (4.2)–(4.4). Suppose the regularity assumptions (1.4)–(1.5) hold. Let

    eni=u(xi,tn)uni,  0iM, 0nN.

    If the maximum time step is τB2, 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+τσ31τ32+(logtna)αmax2kn(logtk1a)σ2τ2k+(logtna)αh2+Ent), (4.33)

    where

    Ent=nj=2τjmaxjkn((logtk1a)σ3(logtka)α1τ3αk+(logtka)σ+α4τ3k+1ταk). (4.34)

    In particular, if graded mesh is used, then it holds that

    enCEα(11κ+(logtn)α)(Nmin{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)CσNr(σ+α). (4.36)

    Further, it holds that

    τ32τσ31=τ32(logt1a)σ3(logTa)σ3(1N)r(σ3)(logTa)3N3rr323(r1)CNrσ. (4.37)

    For the third term in the righthand side of (4.33), we have

    (logtna)αmax2kn(logtk1a)σ2τ2kCmax2kn((logTa)σ(k1N)r(σ2)N2rr2k2(r1))CNrσmax2kn(kk1)r(2σ)krσ2CNrσmax2knkrσ2.

    If rσ2, then max2knkrσ2=nrσ2Nrσ2. If rσ<2, then max2knkrσ2=2rσ2<1. Thus, we obtain

    (logtna)αmax2kn(logtk1a)σ2τ2kCNmin{2,rσ}. (4.38)

    Moreover, we have

    (logtk1a)σ3(logtka)α1τ3αk+(logtka)σ+α4τ3k+1ταk(logTa)σ3(k1N)r(σ3)(logTa)α1(kN)r(α1)(rlogTaNrkr1)3τα1+(logTa)σ+α4(kN)r(σ+α4)(rlogTaNr(k+1)r1)3τα1=τα1r3(logTa)σ+α1Nr(σ+α1)kr(σ1)(3α)((kk1)r(3σ)+(k+1k)3(r1))Cτα1Nr(σ+α1)kr(σ1)(3α)=C(logTa)α(1N)rαNr(σ+α1)kr(σ1)(3α)Ckr(σ1)(3α)Nr(σ1). (4.39)

    Combining (4.34) and (4.39), we obtain

    EntCnj=2τjmaxjknkrkrσ(3α)Nr(σ1)Cnj=2Nrjr1maxjknkrkrσ(3α)Nr(σ1). (4.40)

    If rσ3α, we have from (4.40) that

    EntCnj=2Nrj1nrσ(3α)Nr(σ1)CN(3α)nj=2j1ClnnN(3α). (4.41)

    If rσ<3α, we get from (4.40) that

    EntCnj=2Nrjr1jr(σ1)(3α)Nr(σ1)CNrσnj=2jrσ(4α). (4.42)

    Note that

    nj=2jrσ(4α)n1srσ(4α)ds=1rσ(3α)(nrσ(3α)1)13αrσ, (4.43)

    then we have from (4.42) that

    Ent13αrσN(3α). (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 r1 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.

    Figure 1.  The temporal meshes on [a,T] with N=5.

    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)=max1kNek.

    In Tables 13, 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 13. As can be observed, the temporal convergence order for the L1 scheme is close to min{rσ,2α} for all cases, aligning with the theoretical analysis presented in Theorem 3.2. In Tables 47, 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 47, 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).
    Table 1.  Numerical results for Example 5.1 with r=2,M=500 (L1 scheme).
    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,2α} 0.6 1.0 1.3

     | Show Table
    DownLoad: CSV
    Table 2.  Numerical results for Example 5.1 with r=3,M=500 (L1 scheme).
    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,2α} 0.9 1.5 1.3

     | Show Table
    DownLoad: CSV
    Table 3.  Numerical results for Example 5.1 with r=2αα,M=100 (L1 scheme).
    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,2α} 1.5 1.3 1.1

     | Show Table
    DownLoad: CSV
    Table 4.  Numerical results for Example 5.1 with r=1/α,M=500 (Alikhanov's scheme).
    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

     | Show Table
    DownLoad: CSV
    Table 5.  Numerical results for Example 5.1 with r=1,M=500 (Alikhanov's scheme).
    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

     | Show Table
    DownLoad: CSV
    Table 6.  Numerical results for Example 5.1 with r=3,M=100 (Alikhanov's scheme).
    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

     | Show Table
    DownLoad: CSV
    Table 7.  Numerical results for Example 5.1 with r=2/α,M=300 (Alikhanov's scheme).
    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

     | Show Table
    DownLoad: CSV

    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.

    Table 8.  Numerical results for Example 5.2 with r=3,M=500 (L1 scheme).
    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,2α} 0.9 1.5 1.01 B1=0.2479

     | Show Table
    DownLoad: CSV
    Table 9.  Numerical results for Example 5.2 with r=1,M=300 (Alikhanov's scheme).
    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

     | Show Table
    DownLoad: CSV

    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 1kn, and using the Taylor-like formula given in Lemma 3.1, we derive

    δv(t)τvk/τk=1τktk1tδ2v(y)logtk1ydyy1τktktδ2v(y)logtkydyy.

    Thus the truncation error at time t=tn is given as

    (Rt)ni=CHDαa,tu(xi,tn)CHDαa,τuni=nk=1tktk1ω1α(logtnlogs)(δv(s)τvk/τk)dss=nk=11τktktk1ω1α(logtnlogs)stk1δ2v(y)logytk1dyydssnk=11τktktk1ω1α(logtnlogs))tksδ2v(y)logtkydyydss.

    Exchanging the order of integration, we have

    (Rt)ni=nk=11τktktk1δ2v(y)logytk1tkyω1α(logtnlogs)dssdyynk=11τktktk1δ2v(y)logtkyytk1ω1α(logtnlogs)dssdyy=nk=11τktktk1δ2v(y)logytk1ω2α(logtnlogy)dyy+nk=11τktktk1δ2v(y)logtkyω2α(logtnlogy)dyynk=11τktktk1δ2v(y)logytk1ω2α(logtnlogtk)dyynk=11τktktk1δ2v(y)logtkyω2α(logtnlogtk1)dyy. (7.1)

    For the sake of simplicity, we denote ϖn(y)=ω2α(logtnlogy). Note that the linear logarithmic interpolation function of ϖn(y) with respect to the nodes tk1 and tk is given by

    Lklog,1ϖn(y)=1τklogytk1ϖn(tk)+1τklogtkyϖn(tk1).

    Let

    ˜Lklog,1ϖn(y)=ϖn(y)Lklog,1ϖn(y),

    then (7.1) can be rewritten as follows:

    (Rt)ni=nk=11τktktk1δ2v(y)logytk1ϖn(y)dyy+nk=11τktktk1δ2v(y)logtktk1ϖn(y)dyynk=11τktktk1δ2v(y)logytk1ϖn(y)dyynk=1tktk1δ2v(y)Lklog,1ϖn(y)dyy=nk=1tktk1δ2v(y)˜Lklog,1ϖn(y)dyynk=1(~Rt)nk,n1. (7.2)

    Similar to the proof of Theorem 3.1 in [29], ˜Lklog,1ϖn(y) can be rewritten as

    ˜Lklog,1ϖn(y)=tktk1ξk(y,s)δ2ϖn(s)dss, (7.3)

    where ξk(y,s)=max{logylogs,0}(logylogtk1)(logtklogs)/τk such that

    logylogtk1τk(logylogs)ξk(y,s)0,y,s(tk1,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(tn1)Lnlog,1ϖn(y)=ω2α(logtnlogtn1)1τnlogtnyω2α(logtnlogtn1)=(logylogtn1)A(n)0.

    Thus,

    (~Rt)nntntn1|δ2v(y)|˜Lnlog,1ϖn(y)dyyA(n)0tntn1(logylogtn1)|δ2v(y)|dyy. (7.5)

    For 1kn1, using (7.3) and (7.4), one has

    ˜Lklog,1ϖn(y)(logtk1logy)tktk1δ2ω2α(logtnlogy)dyy(logylogtk1)(A(n)nk1A(n)nk). (7.6)

    Then we obtain for 1kn1 that

    (~Rt)nktktk1|δ2v(y)|˜Lklog,1ϖn(y)dyytktk1|δ2v(y)|(A(n)nk1A(n)nk)(logylogtk1)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)nk 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)nk to obtain

    nk=1P(n)nk(Rt)knk=1P(n)nkA(k)0Gk+nk=2P(n)nkk1j=1(A(k)kj1A(k)kj)Gj=nk=1P(n)nkA(k)0Gk+n1j=1Gjnk=j+1P(n)nk(A(k)kj1A(k)kj)=nk=1P(n)nkA(k)0Gk+n1j=1P(n)nkA(k)0Gj2nk=1P(n)nkA(k)0Gk:=nk=1P(n)nkGk. (7.8)

    Applying Lemma 2.5 by taking υk=Gk in (2.11) to (7.8), one has

    nk=1P(n)nk(Rt)kkj=1P(k)kjGkΓ(2α)nj=1τjmaxjkn((logtka)α1Gk)=Γ(2α)nj=2τjmaxjkn((logtka)α1Gk)+Γ(2α)max{τ1max2kn((logtka)α1Gk),τα1G1}Γ(2α)(nj=2τjmaxjkn((logtka)α1Gk))+Γ(2α)(ρτ2max2kn((logtka)α1Gk)+τα1G1)(1+ρ)Γ(2α)(nj=2τjmaxjkn((logtka)α1Gk)+τα1G1). (7.9)

    On one hand, it follows from (3.2) that

    A(k)0=1τktktk1ω1α(logtklogs)dss=1Γ(2α)ταk,  k=1,...,n. (7.10)

    On the other hand, the assumption of regularity (1.5) leads to the conclusions that

    G1t1t0(logslogt0)C(1+(logsa)σ2)dssCτσ1σ, (7.11)

    and

    Gktktk1(logslogtk1)dssC(1+(logtk1a)σ2)Cτ2k(logtk1a)σ2 (7.12)

    for 2kn. 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.

    [1] Ren G, Ma G, Cong N (2015) Review of electrical energy storage system for vehicular applications. Renew Sust Energ Rev 41: 225–236. doi: 10.1016/j.rser.2014.08.003
    [2] Chang H, Wu H (2013) Graphene-based nanocomposites: preparation, functionalization, and energy and environmental applications. Energ Environ Sci 6: 3483–3507. doi: 10.1039/c3ee42518e
    [3] Guer TM (2013) Critical Review of Carbon Conversion in "Carbon Fuel Cells". Chem Rev 113: 6179–6206. doi: 10.1021/cr400072b
    [4] Chen Z, Higgins D, Yu A, et al. (2011) A review on non-precious metal electrocatalysts for PEM fuel cells. Energ Environ Sci 4: 3167–3192. doi: 10.1039/c0ee00558d
    [5] Gasteiger HA, Vielstich W, Yokokawa H (2009) Handbook of Fuel Cells, Chichester, England: John Wiley & Sons Ltd., 5–6.
    [6] Scofield ME, Liu H, Wong SS (2015) A concise guide to sustainable PEMFCs: recent advances in improving both oxygen reduction catalysts and proton exchange membranes. Chem Soc Rev 44: 5836–5860. doi: 10.1039/C5CS00302D
    [7] Suzuki T, Tsushim S, Hirai S (2011) Effects of Nafion® Ionomer and Carbon Particles on Structure Formation in a Proton-exchange Membrane Fuel Cell Catalyst Layer Fabricated by the Decal-transfer Method. Int J Hydrogen Energ 36: 12361–12369. doi: 10.1016/j.ijhydene.2011.06.090
    [8] Kim KH, Lee KY, Kim HJ, et al. (2010) The effects of Nafion® Ionomer Content in PEMFC MEAs Prepared by a Catalyst-coated Membrane (CCM) Spraying Method. Int J Hydrogen Energ 35: 2119–2126.
    [9] Zhao X, Li W, Fu Y, et al. (2012) Influence of Ionomer Content on the Proton Conduction and Oxygen Transport in the Carbon-supported Catalyst Layers in DMFC. Int J Hydrogen Energ 37: 9845–9852. doi: 10.1016/j.ijhydene.2012.03.107
    [10] Li W, Waje M, Chen Z, et al. (2010) Platinum Nanopaticles Supported on Stacked-cup Carbon Nanofibers as Electrocatalysts for Proton Exchange Membrane Fuel Cell. Carbon 48: 995–1003. doi: 10.1016/j.carbon.2009.11.017
    [11] Ma S, Chen Q, Jøgensen FH, et al. (2007) 19F NMR studies of NafionTM ionomer adsorption on PEMFC catalysts and supporting carbons. Solid State Ionics 178: 1568–1575. doi: 10.1016/j.ssi.2007.10.007
    [12] Andersen SM, Borghei M, Dhiman R, et al. (2014) Interaction of multi-walled carbon nanotubes with perfluorinated sulfonic acid ionomers and surface treatment studies. Carbon 71: 218–228. doi: 10.1016/j.carbon.2014.01.032
    [13] Andersen SM, Borghei M, Dhiman R, et al. (2014) Adsorption behavior of perfluorinated sulfonic acid ionomer on highly graphitized carbon nanofibers and their thermal stabilities. J Phys Chem C 118: 10814–10823. doi: 10.1021/jp501088d
    [14] Paul D, Fraser A, Pearce J, et al. (2011) Understanding the Ionomer Structure and the Proton Conduction Mechanism in PEFC Catalyst Layer: Adsorbed Nafion on Model Substrate. ECS Trans 41: 1393–1406.
    [15] Kongkanand A (2011) Interfacial Water Transport Measurements in Nafion Thin Films Using a Quartz-Crystal Microbalance. J Phys Chem C 115: 11318–11325.
    [16] Yu J, Jiang Z, Hou M, et al. (2014) Analysis of the Behavior and Degradation in Proton Exchange Membrane Fuel Cells with a Dead-ended Anode. J Power Sources 246: 90–94. doi: 10.1016/j.jpowsour.2013.06.163
    [17] Ma S, Solterbeck CH, Odgaard M, et al. (2009) Microscopy Studies on Pronton Exchange Membrane Fuel Cell Electrodes with Different Ionomer Contents. Appl Phys A-Mater 96: 581–589. doi: 10.1007/s00339-008-5050-9
    [18] Yakovlev S, Balsara NP, Downing KH (2013) Insights on the Study of Nafion Nanoscale Morphology byTransmission Electron Microscopy. Membranes 3: 424–439. doi: 10.3390/membranes3040424
    [19] Scheiba F, Benker N, Kunz U, et al. (2008) Electron Microscopy Techniques for the Analysis of the Polymer Electrolyte Distribution in Proton Exchange Membrane Fuel Cells. J Power Sources 177: 273–280. doi: 10.1016/j.jpowsour.2007.11.085
    [20] Radicea S, Oldani C, Merlo L, et al. (2013) Aquivion PerfluoroSulfonic Acid Ionomer Membranes: A Micro-Raman Spectroscopic Study of Ageing. Polym Degrad Stabil 98: 1138–1143. doi: 10.1016/j.polymdegradstab.2013.03.015
    [21] Eastcott JI, Easton EB (2014) Sulfonated Silica-based Fuel Cell Electrode Structures for Low Humidity Applications. J Power Sources 245: 487–494. doi: 10.1016/j.jpowsour.2013.07.005
    [22] Bautista-Rodríguez CM, Rosas-Paleta A, Rivera-Márquez JA, et al. (2009) Study of Electrical Resistance on the Surface of Nafion 115® Membrane Used as Electrolyte in PEMFC Technology Part I: Statistical Inference. Int J Electrochem Sci 4: 43–59.
    [23] Butler JH, Joy DC, Bradley G, et al. (1995) Low-voltage scanning electron microscopy of polymers. Polymer 36: 1781–1790. doi: 10.1016/0032-3861(95)90924-Q
    [24] Rodenburg C, Viswanathan P, Jepson MAE, et al. (2014) Helium ion microscopy based wall thickness and surface roughness analysis of polymer foams obtained from high internal phase emulsion. Ultramicroscopy 139: 13–19. doi: 10.1016/j.ultramic.2014.01.004
    [25] Notte J, Huang J (2016) The Helium Ion Microscope, In: Hlawacek G, Gölzhäuser A, Helium Ion Microscopy, Springer, 3–30.
    [26] Boden SA (2016) Introduction to Imaging Techniques in the HIM, In: Hlawacek G, Gölzhäuser A, Helium Ion Microscopy, Springer, 149–172.
    [27] Hlawacek G, Veligura V, Gastel R, et al. (2014) Helium Ion Microscopy. J Vac Sci Technol B 32: 020801.
    [28] Tintula KK, Jalajakshi A, Sahu AK, et al. (2012) Durability of Pt/C and Pt/MC-PEDOT Catalysts under Simulated Start-Stop Cycles in Polymer Electrolyte Fuel Cells. Fuel Cells 13: 158–166.
    [29] Egerton RF, Li P, Malac M (2004) Radiation damage in the TEM and SEM. Micron 35: 399–409. doi: 10.1016/j.micron.2004.02.003
    [30] Schneider R (2008) Scanning Electron Microscopy Studies of Nafion Deformation into Silicon Micro-Trenches for Fuel Cell Applications [PhD Thesis]. Princeton University, New Jersey.
    [31] Yakovlev S, Balsara NP, Downing KH (2013) Insights on the Study of Nafion Nanoscale Morphology by Transmission Electron Microscopy. Membranes 3: 424–439. doi: 10.3390/membranes3040424
    [32] Hoffman EA, Fekete ZA, Korugic-Karasz LS, et al. (2004) Theoretical and experimental X-ray photoelectron spectroscopy investigation of ion-implanted nafion. J Polym Sci Pol Chem 42: 551–556. doi: 10.1002/pola.10878
    [33] Lee J, Hwang I, Jung C, et al. (2016) Surface modification of Nafion membranes by ion implantation to reduce methanol crossover in direct methanol fuel cells. RSC Adv 6: 62467–62470. doi: 10.1039/C6RA12756H
    [34] Andersen SM, Skou E (2014) Electrochemical Performance and Durability of Carbon Supported Pt Catalyst in Contact with Aqueous and Polymeric Proton Conductors. ACS Appl Mater Interfaces 19: 16565–16576.
    [35] Andersen SM, Dhiman R, Larsen MJ, et al. (2015) Importance of Electrode Hot-Pressing Conditions for the Catalyst Performance of Proton Exchange Membrane Fuel Cells. Appl Catal B-Environ 172: 82–90.
    [36] Andersen SM, Grahl-Madsen L (2014) Understanding on Interface Contribution to the Electrode Performance of Proton Exchange Membrane Fuel Cells-Impact of the Ionomer Content. Int J Hydrogen Energ 41: 1892–1901.
    [37] Andersen SM (2016) Nano carbon supported platinum catalyst interaction behavior with perfluorosulfonic acid ionomer and their interface structures. Appl Catal B-Environ 181: 146–155. doi: 10.1016/j.apcatb.2015.07.049
    [38] Zhang S, Yuan XZ, Hin JNC, et al. (2009) A review of platinum-based catalyst layer degradation in proton exchange membrane fuel cells. J Power Sources 194: 588–600. doi: 10.1016/j.jpowsour.2009.06.073
    [39] Yu H, Roller JM, Mustain WE, et al. (2015) Influence of the ionomer/carbon ratio for low-Pt loading catalyst layer prepared by reactive spray deposition technology. J Power Sources 183: 84–94.
    [40] Reiser CA, Bregoli L, Patterson TW, et al. (2005) A reverse-current decay mechanism for fuel cells. Electrochem Solid-State Lett 8: A273–A276. doi: 10.1149/1.1896466
    [41] Vielstich W, Gasteiger HA, Yokokawa H (2009) Handbook of Fuel Cells: Advances in Electrocatalysis, Materials, Diagnostics and Durability: Part 2, Wiley-Blackwell.
    [42] Li G, Tan J, Gong J (2014) Effect of Compressive Pressure on the Contact Behavior Between Bipolar Plate and Gas Diffusion Layer in a Proton Exchange Membrane Fuel Cell. J Fuel Cell Sci Tech 11: 041009. doi: 10.1115/1.4027253
    [43] Diedrichs A, Rastedt M, Pinar FJ, et al. (2013) Effect of compression on the performance of a HT-PEM fuel cell. J Appl Electrochem 43: 1079–1099. doi: 10.1007/s10800-013-0597-3
    [44] Jeong B, Ocon JD, Lee J (2016) Electrode Architecture in Galvanic and Electrolytic Energy Cells. Angew Chem Int Ed 55: 4870–4880. doi: 10.1002/anie.201507780
    [45] Kim YS, Welch CF, Mack NH, et al. (2014) Highly durable fuel cell electrodes based on ionomers dispersed in glycerol. Phys Chem Chem Phys 16: 5927–5932. doi: 10.1039/C4CP00496E
    [46] Fultz DW, Chuang PYA (2011) The Property and Performance Differences Between Catalyst, Coated Membrane and Catalyst Coated Diffusion Media. J Fuel Cell Sci Tech 8: 041010. doi: 10.1115/1.4003632
    [47] Okur O, Karadag CL, San FJB, et al. (2013) Optimization of Parameters for Hot-Pressing Manufacture of Membrane Electrode Assembly for PEM (Polymer Electrolyte Membrane Fuel Cells) Fuel Cell. Energy 57: 574–580. doi: 10.1016/j.energy.2013.05.001
    [48] Jia S, Liu H (2012) Cold Pre-Compression of Membrane Electrode Assembly for PEM Fuel Cells. Int J Hydrogen Energ 37: 1367–13680.
  • Reader Comments
  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(6622) PDF downloads(956) Cited by(7)

Figures and Tables

Figures(7)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog