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Lots of physical phenomena can be expressed by the FC equation, including, inter alia, dissipative and dispersive partial differential equations (PDEs). In this paper, we consider the FC equation
∂ϕ(t,s)∂t=−μ0 0C1−α1tϕ(t,s)+ 0C1−α2t∂2ϕ(t,s)∂s2+f(t,s),0≤s≤1,0≤t≤T, | (1.1) |
ϕ(0,s)=0,ϕ(1,s)=0,s∈[0,T], | (1.2) |
ϕ(t,0)=φ(t),t∈[0,1] | (1.3) |
where μ0∈R,0 <α1,α2<1 are constants. There are some definitions of fractional derivatives, such as the Caputo type, Riemann-Liouville type and so on. In the following, we adopt the Caputo type time fractional-order partial derivative as
0Cαtϕ(t)=1Γ(1−α)∫t0ϕ′(t)(t−τ)αdτ, | (1.4) |
and Γ(α) is the Γ function.
In [1], a scheme combining the finite difference method in the time direction and a spectral method in the space direction was proposed. In [2], two implicit compact difference schemes for the FC equation were studied, this scheme was proved to be stable, and the convergence order O(τ+h4) was given. In [3], a two-dimensional FC equation was solved by orthogonal spline collocation (OSC) methods for space discretization and finite difference method for time, which was proved to be unconditionally stable. In[4], the FC equation with two time Riemann-Liouville derivatives was solved by an explicit numerical method; and the accuracy, stability and convergence of this method were studied. In [5], FC equation with two fractional time derivatives were considered, and two new implicit numerical methods for the FC equation were proposed, respectively. The stability and convergence of these methods were also investigated. In [6], nonlinear FC equation was solved by a two-grid algorithm with the finite element (FE) method. A time second-order fully discrete two-grid FE scheme and the space direction were approximated. In [7], the discrete Crank-Nicolson (CN) finite element method was obtained by the finite difference in time and the finite element in space to approximate the FC equation, the stability and error estimate were analyzed in detail and the optimal convergence rate was obtained. In [8], the FC equation involving two integro-differential operators was solved by semi-discrete finite difference approximation, and the scheme was proved unconditionally stable. In reference [9], numerical integration with the reproducing kernel gradient smoothing integration are constructed. In reference [10], recursive moving least squares (MLS) approximation was constructed.
Like the above methods to solve the FC equation by finite difference approach or finite element method, the time direction and space direction were solved separatively. In the following, we presented the BRIM to solve the time direction and space direction of FC equation at the same time. Lagrange interpolation has been presented by mathematician Lagrange to fitting data to be a certain function. When the number n increases, there are Runge phenomenon that the interpolation result deviates from the original function. In order to avoid the Runge phenomenon, among them, barycentric interpolation was developed in 1960s to overcome it. In recent years, linear rational interpolation (LRI) was proposed by Floater [14,15,16] and error of linear rational interpolation [11,12,13] is also proved. The barycentric interpolation collocation method (BICM) has developed by Wang et al.[25,26] and the algorithm of BICM has used for linear/non-linear problems [27,28]. In recent research, Volterra integro-differential equation (VIDE) [17,21], heat equation (HE) [18], biharmonic equation (BE) [19], telegraph equation (TE) [20], generalized Poisson equations [22], fractional reaction-diffusion equation [23] and KPP equation [24] have been studied by the linear barycentric rational interpolation method (LBRIM) and their convergence rate are also proved.
In this paper, BRIM has been used to solve the FC equation. As the fractional derivative is the nonlocal operator, the spectral method is developed to solve the FC equation and the coefficient matrix is the full matrix. The fractional derivative of the FC equation is changed to nonsingular integral by the order of density function plus one. New Gauss formula is constructed to compute it simply and matrix equation of discrete FC equation is obtained by the unknown function replaced by barycentric rational interpolation basis function. Then, the convergence rate of BRIM is proved.
As there is singularity in Eq (1.1), the numerical methods cannot get high accuracy, by fractional integration to second part of (1.1) to overcome the difficulty of singularity. We get
0Cαtϕ(t,s)=1Γ(ξ−α)∫t0∂ξϕ(τ,s)∂τξdτ(τ−t)α+1−ξ=1(ξ−α)Γ(ξ−α)[∂ξϕ(0,s)∂tξtξ−α+∫t0∂ξ+1ϕ(τ,s)∂τξ+1dτ(t−τ)α−ξ]=Γξα[∂ξϕ(0,s)∂tξtξ−α+∫t0∂ξ+1ϕ(τ,s)∂τξ+1dτ(t−τ)α−ξ], | (2.1) |
where Γξα=1(ξ−α)Γ(ξ−α).
Combining (2.1) and (1.1), we have
∂ϕ∂t+μ0Γξα1[∂ξϕ(0,s)∂tξtξ−α1+∫t0∂ξ+1ϕ(τ,s)∂τξ+1dτ(t−τ)α1−ξ]=Γξα2[∂ξ+2ϕ(0,s)∂tξ∂s2sξ−α2+∫t0∂ξ+3ϕ(τ,s)∂τξ+1∂s2dτ(s−τ)α2−ξ]+f(t,s). | (2.2) |
In the following, we give the discrete formula of FC equation and to get the matrix equation from BRIM.
Let
ϕ(t,s)=m∑j=1Rj(t)ϕj(s) | (2.3) |
where
ϕ(ti,s)=ϕi(s),i=1,2,⋯,m |
and
Rj(t)=λjt−tjn∑k=1λkt−tk | (2.4) |
where
λk=∑j∈Jk(−1)jj+dt∏i=j,j≠k1tk−ti, Jk={j∈{0,1,⋯,l−dt}:k−dt≤j≤k} |
is the basis function [18]. Taking (2.3) into Eq (2.2),
m∑j=1R′j(t)ϕj(s)+μ0Γξα1m∑j=1[R(ξ)j(0)ϕj(s)tξ−α1+∫t0ϕj(s)R(ξ+1)j(τ)dτ(t−τ)α1−ξ]=Γξα2m∑j=1[R(ξ)j(0)ϕ(2)j(s)tξ−α2+∫t0ϕ(2)j(s)R(ξ+1)j(τ)dτ(t−τ)α2−ξ]+f(t,s). | (2.5) |
By taking 0=t1<t2<⋯<tm=T,a=s1<s2<⋯<sn=b with ht=T/m,hs=(b−s)/n or uninform as Chebychev point s=cos((0:m)′π/m),t=cos((0:n)′π/n), we get
m∑j=1R′j(ti)ϕj(s)+μ0Γξα1m∑j=1[R(ξ)j(0)ϕj(s)tξ−α1i+∫ti0ϕj(s)R(ξ+1)j(τ)dτ(ti−τ)α1−ξ]=Γξα2m∑j=1[R(ξ)j(0)ϕ(2)j(s)tξ−α2i+∫ti0ϕ(2)j(s)R(ξ+1)j(τ)dτ(ti−τ)α2−ξ]+f(ti,s), | (2.6) |
by noting the notation, Rj(ti)=δij,R′j(ti)=R(1,0)ij, where R(1,0)ij is the first order derivative of barycentric matrix. Equation (2.6) can be written as
m∑j=1R(1,0)ijϕj(s)+μ0Γξα1m∑j=1[R(ξ)j(0)ϕj(s)tξ−α1i+∫ti0ϕj(s)R(ξ+1)j(τ)dτ(ti−τ)α1−ξ]=Γξα2m∑j=1[R(ξ)j(0)ϕ(2)j(s)tξ−α2i+∫ti0ϕ(2)j(s)R(ξ+1)j(τ)dτ(ti−τ)α2−ξ]+f(ti,s). | (2.7) |
Similarly as the discrete t for s, we get
ϕj(s)=n∑k=1Rk(s)ϕik | (2.8) |
where ϕi(sj)=ϕ(ti,sj)=ϕij,i=1,⋯,m;j=1,⋯,n and
Ri(s)=wis−sim∑k=1wks−sk | (2.9) |
where
wi=∑j∈Ji(−1)jj+ds∏k=j,j≠i1si−sk, Ji={j∈{0,1,⋯,m−ds}:i−ds≤j≤i}, |
is the basis function [18].
Taking (2.8) into Eq (2.7), we get
m∑j=1n∑k=1R(1,0)ijRk(s)ϕik+μ0Γξα1m∑j=1n∑k=1[R(ξ)j(0)Rk(s)tξ−α1i+∫ti0Rk(s)R(ξ+1)j(τ)dτ(ti−τ)α1−ξ]ϕik=Γξα2m∑j=1n∑k=1[R(ξ)j(0)R(2)k(s)tξ−α2i+∫ti0R(2)k(s)R(ξ+1)j(τ)dτ(ti−τ)α2−ξ]ϕik+f(ti,s). | (2.10) |
By taking s1,s2,⋯,sn at the mesh-point, we get
m∑j=1n∑k=1R(1,0)ijRk(sl)ϕik+μ0Γξα1m∑j=1n∑k=1[R(ξ)j(0)Rk(sl)tξ−α1i+∫ti0Rk(sl)R(ξ+1)j(τ)dτ(ti−τ)α1−ξ]ϕik=Γξα2m∑j=1n∑k=1[R(ξ)j(0)R(2)k(sl)tξ−α2i+∫ti0R(2)k(sl)R(ξ+1)j(τ)dτ(ti−τ)α2−ξ]ϕik+f(ti,sl). | (2.11) |
By noting the notation, Rk(sl)=δkl,R″k(sl)=R(0,2)ij, where R(0,2)ij is the second order derivative of barycentrix matrix.
m∑j=1n∑k=1R(1,0)ijδklϕik+μ0Γξα1m∑j=1n∑k=1[R(ξ)j(0)δkltξ−α1i+δkl∫ti0R(ξ+1)j(τ)dτ(ti−τ)α1−ξ]ϕik=Γξα2m∑j=1n∑k=1[R(ξ)j(0)R(0,2)ijtξ−α2i+R(0,2)ij∫ti0R(ξ+1)j(τ)dτ(ti−τ)α2−ξ]ϕik+f(ti,sl), | (2.12) |
where
Rk(τ)=λkτ−τkn∑k=0λkτ−τk |
and
{R′i(τ)=Ri(τ)[−1τ−τk+l∑s=0λk(τ−τk)2l∑s=0λkτ−τk],⋮R(ξ+1)i(τ)=[R(ξ)i(τ)]′,ξ∈N+. |
The integral term of (2.12) can be written as
∫ti0R(ξ+1)j(τ)dτ(ti−τ)α1−ξ=Qα1j(ti)=Qα1ji, | (2.13) |
∫ti0R(ξ+1)j(τ)dτ(ti−τ)α2−ξ=Qα2j(ti)=Qα2ji, | (2.14) |
then we get
m∑j=1n∑k=1R(1,0)ijδklϕik+μ0Γξα1m∑j=1n∑k=1[R(ξ)j(0)δkltξ−α1i+δklQα2j(ti)]ϕik=Γξα2m∑j=1n∑k=1[R(ξ)j(0)R(0,2)ijtξ−α2i+R(0,2)ijQα1j(ti)]ϕik+f(ti,sl). | (2.15) |
The integral (2.12) is calculated by
Qα1j(ti)=∫ti0R(ξ+1)j(τ)dτ(ti−τ)α1−ξ:=g∑i=1R(ξ+1)i(τθ,α1i)Gθ,α1i, | (2.16) |
and
Qα2j(ti)=∫ti0R(ξ+1)j(τ)dτ(ti−τ)α2−ξ:=g∑i=1R(ξ+1)i(τθ,α2i)Gθ,α2i, | (2.17) |
where Gθ,α1i,Gθ,α2i are Gauss weights and τθ,α1i,τθ,α2i are Gauss points with weights (ti−τ)ξ−α1,(ti−τ)ξ−α2, see reference [22].
Equation systems (2.15) can be written as
[R(01)⊗In+Γξα2(M(ξ0)1⊗In+Im⊗Qα2)][ϕ11⋮ϕ1nϕn1⋮ϕmn]−[μ0Γξα1(M(ξ0)1⊗In+Im⊗Qα1)][ϕ11⋮ϕ1nϕn1⋮ϕmn]=[f11⋮f1nfn1⋮fmn], | (2.18) |
Im and In are identity matrices, ⊗ is Kronecker product.
Then Eq (2.18) can be noted as
[R(01)⊗In+Γξα2(M(ξ0)1⊗In+Im⊗Qα2)−μ0Γξα1(M(ξ0)1⊗In+Im⊗Qα1)]Φ=F | (2.19) |
and
RΦ=F, | (2.20) |
with R=R(01)⊗In+Γξα2(M(ξ0)1⊗In+Im⊗Qα2)−μ0Γξα1(M(ξ0)1⊗In+Im⊗Qα1) and Φ=[ϕ11…ϕ1n…ϕn1…ϕmn]T,F=[f11…f1n…fn1…fmn]T.
The boundary condition can be solved by substitution method, additional method or elimination method, see [26]. We adopt substitution method and additional method to deal with boundary condition.
In this part, error estimate of the FC equation is given with rn(s)=n∑i=1ri(s)ϕi to replace ϕ(s), where ri(s) is defined as (2.9) and ϕi=ϕ(si). We also define
e(s):=ϕ(s)−rn(s)=(s−si)⋯(s−si+d)ϕ[si,si+1,…,si+d,s], | (3.1) |
see reference [18].
Then we have
Lemma 1. For e(s) be defined by (3.1) and ϕ(s)∈Cd+2[a,b],d=1,2,⋯, there
|e(k)(s)|≤Chd−k+1,k=0,1,⋯. | (3.2) |
For the FC equation, rational interpolation function of ϕ(t,s) is defined as rmn(t,s)
rmn(t,s)=m+ds∑i=1n+dt∑j=1wi,j(s−si)(t−tj)ϕi,jm+ds∑i=1n+dt∑j=1wi,j(s−si)(t−tj) | (3.3) |
where
wi,j=(−1)i−ds+j−dt∑k1∈Jik1+ds∏h1=k1,h1≠j1|si−sh1|∑k2∈Jik2+dt∏h2=k2,h2≠j1|tj−th2|. | (3.4) |
We define e(t,s) be the error of ϕ(t,s) as
e(t,s):=ϕ(t,s)−rmn(t,s)=(s−si)⋯(s−si+ds)ϕ[si,si+1,…,si+d1,s;t]+(t−tj)⋯(t−tj+dt)ϕ[s;tj,tj+1,…,tj+d2,t]−(s−si)⋯(s−si+ds)(t−tj)⋯(t−tj+dt)ϕ[si,si+1,…,si+d1,s;tj,tj+1,…,tj+d2,t]. | (3.5) |
With similar analysis of Lemma 1, we have
Theorem 1. For e(t,s) defined as (3.5) and ϕ(t,s)∈Cds+2[a,b]×Cdt+2[0,T], then we have
|e(k1,k2)(s,t)|≤C(hds−k1+1s+hdt−k2+1t),k1,k2=0,1,⋯. | (3.6) |
Let ϕ(sm,tn) be the approximate function of ϕ(t,s) and L to be bounded operator, there holds
Lϕ(tm,sn)=f(tm,sn) | (3.7) |
and
limm,n→∞Lϕ(tm,sn)=ϕ(t,s). | (3.8) |
Then we get
Theorem 2. For ϕ(tm,sn):Lϕ(tm,sn)=ϕ(t,s) and L defined as (3.7), there
|ϕ(t,s)−ϕ(tm,sn)|≤C(hds−1+τdt−1). |
Proof. By
Lϕ(t,s)−Lϕ(tm,sn)=∂ϕ(t,s)∂t− 0C1−α1t∂2ϕ(t,s)∂s2+μ0 0C1−α2tϕ(t,s)−f(t,s)−[∂ϕ(tm,sn)∂t− 0C1−α1t∂2ϕ(tm,sn)∂s2+μ0 0C1−α2tϕ(tm,sn)−f(tm,sn)]=∂ϕ∂t−∂ϕ∂t(tm,sn)−[0C1−α1t∂2ϕ∂s2− 0C1−α1t∂2ϕ∂s2(sm,tn)]+μ0[0C1−α2tϕ(t,s)− 0C1−α2t(tm,sn))]−[f(t,s)−f(tm,sn)]:=E1(t,s)+E2(t,s)+E3(t,s)+E4(t,s), | (3.9) |
here
E1(t,s)=∂ϕ∂t−∂ϕ∂t(tm,sn), |
E2(t,s)=0C1−α1t∂2ϕ∂s2− 0C1−α1t∂2ϕ∂s2(tm,sn), |
E3(t,s)=μ0[0C1−α2tϕ(t,s)− 0C1−α2t(tm,sn))], |
E4(t,s)=f(t,s)−f(tm,sn). |
As for E1(t,s), we get
E1(t,s)=|∂ϕ∂t(t,s)−∂ϕ∂t(tm,sn)|=|∂ϕ∂t(t,s)−∂ϕ∂t(tm,s)+∂ϕ∂t(tm,s)−∂ϕ∂t(tm,sn)|≤|∂ϕ∂t(t,s)−∂ϕ∂t(tm,s)|+|∂ϕ∂t(tm,s)−∂ϕ∂t(tm,sn)|=|m−ds∑i=1(−1)i∂ϕ∂t[si,si+1,…,si+d1,sn,t]m−ds∑i=1λi(s)|+|n−dt∑j=1(−1)j∂ϕ∂t[tj,tj+1,…,tj+d2,sn,tm]n−dt∑j=1λj(t)|=|∂e∂t(tm,s)|+|∂e∂t(tm,sn)|, |
we get
|E1(t,s)|≤C(hds+τdt). | (3.10) |
As E2(t,s), we have
E2(t,s)=0C1−α1t∂2ϕ∂s2− 0C1−α1t∂2ϕ∂s2(tm,sn)=Γξα2[∂ξ+2ϕ(0,s)∂tξ∂s2sξ−α2+∫t0∂ξ+3ϕ(τ,s)∂τξ+1∂s2dτ(t−τ)α2−ξ]−Γξα2[∂ξ+2ϕ(0,sn)∂tξ∂s2sξ−α2n+∫tm0∂ξ+3ϕ(τ,sn)∂τξ+1∂s2dτ(tm−τ)α2−ξ]=Γξα2[∂ξ+2ϕ(0,s)∂tξ∂s2sξ−α2−∂ξ+2ϕ(0,sn)∂tξ∂s2sξ−α2n]+Γξα2[∫t0∂ξ+3ϕ(τ,s)∂τξ+1∂s2dτ(t−τ)α2−ξ−∫tm0∂ξ+3ϕ(τ,sn)∂τξ+1∂s2dτ(tm−τ)α2−ξ] | (3.11) |
and
|E2(t,s)|≤|Γξα2[∂ξ+2ϕ(0,s)∂tξ∂s2sξ−α2−∂ξ+2ϕ(0,sn)∂tξ∂s2sξ−α2n]|+|Γξα2[∫t0∂ξ+3ϕ(τ,s)∂τξ+1∂s2dτ(t−τ)α2−ξ−∫tm0∂ξ+3ϕ(τ,sn)∂τξ+1∂s2dτ(tm−τ)α2−ξ]|≤|Γξα2||∂ξ+2ϕ∂tξ∂s2(0,s)−∂ξ+2ϕ∂tξ∂s2(0,sn)|+|Γξα2||∂ξ+3ϕ∂tξ+1∂s2(t,s)−∂ξ+3ϕ∂tξ+1∂s2(tm,sn)|:=E21(t,s)+E22(t,s) | (3.12) |
where
E21(t,s)=|Γξα2||∂ξ+2ϕ∂tξ∂s2(0,s)−∂ξ+2ϕ∂tξ∂s2(0,sn)|,E22(t,s)=|Γξα2||∂ξ+3ϕ∂tξ+1∂s2(t,s)−∂ξ+3ϕ∂tξ+1∂s2(tm,sn)|. | (3.13) |
Now we estimate E21(t,s) and E22(t,s) part by part, for the second part we have
E22(t,s)=|Γξα2||∂ξ+3ϕ∂tξ+1∂s2(t,s)−∂ξ+3ϕ∂tξ+1∂s2(tm,sn)|=|Γξα2||∂ξ+3ϕ∂tξ+1∂s2(t,s)−∂ξ+3ϕ∂tξ+1∂s2(tm,s)+∂ξ+3ϕ∂tξ+1∂s2(tm,s)−∂ξ+3ϕ∂tξ+1∂s2(tm,sn)|≤|Γξα2||∂ξ+3ϕ∂tξ+1∂s2(t,s)−∂ξ+3ϕ∂tξ+1∂s2(tm,s)|+|Γξα2||∂ξ+3ϕ∂tξ+1∂s2(tm,s)−∂ξ+3ϕ∂tξ+1∂s2(tm,sn)|=|Γξα2||m−ds∑i=1(−1)i∂ξ+3ϕ∂tξ+1∂s2[si,si+1,…,si+d1,sn,t]m−ds∑i=1λi(s)|+|Γξα2||n−dt∑j=1(−1)j∂ξ+3ϕ∂tξ+1∂s2[tj,tj+1,…,tj+d2,sn,tm]n−dt∑j=1λj(t)|=|Γξα2||∂ξ+3e∂tξ+1∂s2(tm,s)|+|Γξα2||∂ξ+3e∂tξ+1∂s2(tm,sn)|, |
then we have
|E22(t,s)|≤|∂ξ+3e∂tξ+1∂s2(tm,s)|+|∂ξ+3e∂tξ+1∂s2(tm,sn)|≤C(hds−ξ+τdt−1). | (3.14) |
For E21(t,s), we get
|E21(t,s)|≤C(hds+1−ξ+τdt−1). | (3.15) |
Similarly as E2(t,s), for E3(t,s) we have
|E3(t,s)|≤C(hds+τdt). | (3.16) |
Combining (3.9), (3.14), (3.16) together, proof of Theorem 2 is completed.
In this part, one example is presented to test the theorem. The nonuniform partition in this experiment defined as second kind of Chybechev point s=cos((0:m)′π/m),t=cos((0:n)′π/n).
Example 1. Consider the FC equation
∂ϕ∂t= 0C1−α1t∂2ϕ∂s2ϕ(t,s)−μ0 0C1−α2tϕ(t,s)+f(t,s),0≤s≤1,0≤t≤T, |
with the analysis solutions is
ϕ(t,s)=t2sin(πs), |
with the initial condition
ϕ(s,0)=0, |
and boundary condition
ϕ(0,t)=ϕ(1,t)=0, |
and
f(t,s)=2(t+π2t1+α1Γ(2+α1)+t1+α2Γ(2+α2))sin(πs). |
In Figures 1 and 2, errors of m=n=10, [a,b]=[0,1] and m=n=10,dt=ds=7, [a,b]=[0,1] in Example 1. (a) uniform; (b) nonuniform for FC equation by rational interpolation collocation methods are presented, respectively. From the figure, we know that the precision can reach to 10−6 for both uniform and nonuniform partition.
In Table 1, errors of the FC equation with m=n=10,α1=α2=0.2 for substitution methods and additional methods are presented, there are nearly no difference for the two methods. Additional method is more simple than substitution methods to add the boundary condition. In the following, we choosing the substitution method to deal with the boundary condition.
method of substitution | method of additional | |||
uniform | nonuniform | uniform | nonuniform | |
Larange | 1.4662e-06 | 2.1919e-08 | 2.7900e-07 | 1.4310e-07 |
Rational | 1.3038e-05 | 2.4541e-07 | 4.9788e-06 | 1.4310e-07 |
Errors of the FC equation for α1=0.4,α2=0.6,dt=ds=5 with t=0.1,0.9,1,5,10,15 are presented under the uniform and nonuniform in Table 2. As the time variable become from 0.5 to 15, there are high accuracy for our methods. We can improve the accuracy by increasing m,n or choosing the parameter dt,ds approximately which means our methods is useful.
uniform | nonuniform | uniform | nonuniform | |
t | (12,12) | (12,12) | (12,12)dt=ds=5 | (12,12)dt=ds=5 |
0.5 | 2.1021e-11 | 3.8250e-09 | 6.8506e-06 | 1.6436e-06 |
1 | 9.0394e-13 | 4.4206e-10 | 4.6667e-06 | 7.8141e-07 |
5 | 6.1833e-12 | 5.6655e-08 | 2.3777e-04 | 4.2230e-05 |
10 | 1.0094e-12 | 8.5622e-07 | 1.9813e-04 | 1.5634e-05 |
15 | 3.5397e-12 | 1.8827e-05 | 8.5498e-04 | 8.2551e-05 |
In Table 3, errors of α1=0.01,0.1,0.3,0.5,0.9,0.99 under uniform with m=n=10,dt=5,ds=5 with α2=0.1,0.4,0.6,0.8,0.99 are presented under the uniform partition. From the table, we know that for different α1,α2 our methods have high accuracy with little number m and n. In the following table, numerical results are presented to test our theorem. From Tables 4 and 5, error of uniform for α1=α2=0.2,ds=5 with different dt are given, the convergence rate is O(hdt). From Table 5, with space variable uniform for α1=α2=0.2,dt=5, the convergence rate is O(h7), we will investigate in future paper. For Tables 6 and 7, the errors of Chebyshev partition for s and t are presented. For dt=5, the convergence rate is O(hds) in Table 6, while in Table 7, the convergence rate is O(hdt) which agrees with our theorem.
α1 | α2=0.1 | α2=0.4 | α2=0.6 | α2=0.8 | α2=0.99 |
0.01 | 1.0153e-04 | 1.0246e-04 | 1.0300e-04 | 1.0346e-04 | 1.0384e-04 |
0.1 | 1.2753e-05 | 1.2865e-05 | 1.2930e-05 | 1.2987e-05 | 1.3033e-05 |
0.3 | 2.7464e-05 | 2.7704e-05 | 2.7845e-05 | 2.7971e-05 | 2.8074e-05 |
0.5 | 4.5746e-06 | 4.6152e-06 | 4.6399e-06 | 4.6609e-06 | 4.6794e-06 |
0.9 | 9.0295e-06 | 9.1193e-06 | 9.1240e-06 | 9.2142e-06 | 9.2479e-06 |
0.99 | 1.8981e-06 | 1.8247e-06 | 1.5293e-06 | 1.9193e-06 | 2.0670e-06 |
m,n | dt=2 | dt=3 | dt=4 | dt=5 | ||||
8 | 1.3626e-02 | 6.9619e-03 | 2.0708e-03 | 9.8232e-04 | ||||
10 | 9.6780e-03 | 1.5332 | 3.4354e-03 | 3.1653 | 6.9542e-04 | 4.8900 | 3.2829e-04 | 4.9117 |
12 | 7.0485e-03 | 1.7389 | 1.9408e-03 | 3.1320 | 2.9186e-04 | 4.7621 | 1.3132e-04 | 5.0255 |
14 | 5.4466e-03 | 1.6725 | 1.2017e-03 | 3.1097 | 1.4211e-04 | 4.6686 | 6.0148e-05 | 5.0654 |
m,n | ds=2 | ds=3 | ds=4 | |||
8 | 4.9495e-04 | 4.9492e-04 | 4.9486e-04 | |||
10 | 1.0051e-04 | 7.1443 | 1.0053e-04 | 7.1431 | 1.0053e-04 | 7.1426 |
12 | 2.7700e-05 | 7.0690 | 2.7711e-05 | 7.0679 | 2.7714e-05 | 7.0673 |
14 | 9.4272e-06 | 6.9921 | 9.4315e-06 | 6.9917 | 9.4314e-06 | 6.9925 |
m,n | ds=2 | ds=3 | ds=4 | |||
8 | 2.8113e-05 | 2.8110e-05 | 2.8108e-05 | |||
10 | 2.1197e-05 | 1.2654 | 2.1196e-05 | 1.2652 | 2.1195e-05 | 1.2651 |
12 | 6.6990e-06 | 6.3180 | 6.6989e-06 | 6.3178 | 6.6988e-06 | 6.3176 |
14 | 1.6712e-06 | 9.0069 | 1.6712e-06 | 9.0068 | 1.6712e-06 | 9.0067 |
m,n | dt=2 | dt=3 | dt=4 | dt=5 | ||||
8 | 3.1539e-02 | 8.7995e-03 | 2.1930e-03 | 3.3004e-04 | ||||
10 | 2.4329e-02 | 1.1632 | 4.0288e-03 | 3.5010 | 2.7133e-04 | 9.3648 | 2.2278e-04 | 1.7613 |
12 | 1.5223e-02 | 2.5716 | 1.9127e-03 | 4.0859 | 9.5194e-05 | 5.7449 | 5.1702e-05 | 8.0116 |
14 | 1.1407e-02 | 1.8721 | 1.1143e-03 | 3.5049 | 3.5772e-05 | 6.3493 | 1.1369e-05 | 9.8255 |
In the following table, α1=0.4,α2=0.6 is chosen to present numerical results. From Tables 8 and 9, error of uniform partition dt=5 with different ds are given, the convergence rate is O(h7). From Table 8, with space variable s,ds=5, the convergence rate is O(hdt) which agrees with our theorem.
m,n | ds=2 | ds=3 | ds=4 | |||
8 | 4.9427e-04 | 4.9426e-04 | 4.9414e-04 | |||
10 | 1.0035e-04 | 7.1455 | 1.0041e-04 | 7.1427 | 1.0041e-04 | 7.1413 |
12 | 2.7639e-05 | 7.0720 | 2.7674e-05 | 7.0684 | 2.7684e-05 | 7.0669 |
14 | 9.3984e-06 | 6.9977 | 9.4153e-06 | 6.9942 | 9.4254e-06 | 6.9895 |
m,n | dt=1 | dt=2 | dt=3 | dt=4 | ||||
8 | 1.3587e-02 | 6.9513e-03 | 2.0677e-03 | 9.8084e-04 | ||||
10 | 9.6497e-03 | 1.5334 | 3.4314e-03 | 3.1637 | 6.9462e-04 | 4.8884 | 3.2791e-04 | 4.9102 |
12 | 7.0259e-03 | 1.7404 | 1.9389e-03 | 3.1311 | 2.9157e-04 | 4.7613 | 1.3118e-04 | 5.0249 |
14 | 5.4269e-03 | 1.6752 | 1.2005e-03 | 3.1096 | 1.4198e-04 | 4.6682 | 6.0090e-05 | 5.0648 |
For Tables 10 and 11, the errors of Chebyshev partition for non-uniform with α1=0.4,α2=0.6 are presented. For dt=5, the convergence rate is O(h7) in Table 11, while in Table 10, the convergence rate is O(hdt) which agrees with our theorem.
m,n | dt=1 | dt=2 | dt=3 | dt=4 | ||||
8 | 3.1481e-02 | 8.7825e-03 | 2.1876e-03 | 3.2930e-04 | ||||
10 | 2.4263e-02 | 1.1671 | 4.0219e-03 | 3.5000 | 2.7124e-04 | 9.3553 | 2.2231e-04 | 1.7606 |
12 | 1.5185e-02 | 2.5704 | 1.9076e-03 | 4.0912 | 9.5106e-05 | 5.7481 | 5.1649e-05 | 8.0057 |
14 | 1.1373e-02 | 1.8751 | 1.1117e-03 | 3.5026 | 3.5733e-05 | 6.3504 | 1.1365e-05 | 9.8211 |
m,n | ds=2 | ds=3 | ds=4 | |||
8 | 2.8065e-05 | 2.8059e-05 | 2.8056e-05 | |||
10 | 2.1156e-05 | 1.2665 | 2.1154e-05 | 1.2660 | 2.1153e-05 | 1.2656 |
12 | 6.6875e-06 | 6.3168 | 6.6874e-06 | 6.3164 | 6.6873e-06 | 6.3161 |
14 | 1.6693e-06 | 9.0033 | 1.6693e-06 | 9.0031 | 1.6693e-06 | 9.0030 |
In this paper, BRIM was used to solve the (1+1) dimensional FC equation that is presented. For fractional-order PDEs, the convergence order is seriously affected by the orders of fractional derivatives. By fractional integration, the singularity of the fractional derivative of the FC equation can be changed to nonsingular integral, with adding one order to the derivatives of density function. So there are no effects on the orders of fractional derivatives. The singularity of fractional derivative is overcome by the integral to density function from the singular kernel. For the arbitrary fractional derivative, the new Gauss formula is constructed to calculated it simply. For the Diriclet boundary condition, the FC equation is changed to the discrete FC equation and the matrix equation of it is given. In the future, the FC equation with Nuemann condition can be solved by BRIM, and high dimensional FC equation can also be studied by our methods.
The work of Jin Li was supported by Natural Science Foundation of Shandong Province (Grant No. ZR2022MA003).
The authors declare that they have no conflicts of interest.
[1] | Bureau of Transportation Statistics, National Transportation Statistics. Available from: www.bts.gov/topics/national-transportation-statistics. |
[2] | Rogelj JD, Shindell K, Jiang S, et al. (2018) Mitigation Pathways Compatible with 1.5 ℃ in the Context of Sustainable Development. In: Global Warming of 1.5 ℃. An IPCC Special Report on the impacts of global warming of 1.5 ℃ above pre-industrial levels and related global greenhouse gas emission pathways, in the context of strengthening the global response to the threat of climate change, sustainable development, and efforts to eradicate poverty [Masson-Delmotte V, Zhai P, Pörtner HO, et al. (eds.)]. In Press. |
[3] | US Environmental Protection Agency, The 2018 EPA Automotive Trends Report: Greenhouse Gas Emissions, Fuel Economy, and Technology since 1975. EPA-420-R-19-002, Available from: nepis.epa.gov/Exe/ZyPDF.cgi/P100W5C2.PDF?Dockey=P100W5C2.PDF. |
[4] |
Yuksel T, Tamayao MAM, Hendrickson C, et al. (2016) Effect of regional grid mix, driving patterns and climate on the comparative carbon footprint of gasoline and plug-in electric vehicles in the United States. Environ Res Lett 11:044007. doi: 10.1088/1748-9326/11/4/044007
![]() |
[5] |
Jones CM, Kammen DM (2011) Quantifying carbon footprint reduction opportunities for US households and communities. Environ Sci Technol 45: 4088-4095. doi: 10.1021/es102221h
![]() |
[6] |
Ivanova D, Stadler K, Steen-Olsen K, et al. (2016) Environmental impact assessment of household consumption. J Ind Ecol 20: 526-536. doi: 10.1111/jiec.12371
![]() |
[7] |
Wiedenhofer D, Smetschka B, Akenji L, et al. (2018) Household time use, carbon footprints, and urban form: a review of the potential contributions of everyday living to the 1.5 C climate target. Curr Opin Environ Sustain 30: 7-17. doi: 10.1016/j.cosust.2018.02.007
![]() |
[8] |
Dietz T, Gardner GT, Gilligan J, et al. (2009). Household actions can provide a behavioral wedge to rapidly reduce US carbon emissions. PNAS 106: 18452-18456. doi: 10.1073/pnas.0908738106
![]() |
[9] |
Jansen KH, Brown TM, Samuelsen GS (2010) Emissions impacts of plug-in hybrid electric vehicle deployment on the US western grid. J Power Sources 195: 5409-5416. doi: 10.1016/j.jpowsour.2010.03.013
![]() |
[10] |
Zivin JSG, Kotchen MJ, Mansur ET (2014) Spatial and temporal heterogeneity of marginal emissions: Implications for electric cars and other electricity-shifting policies. J Econ Behav Organ 107: 248-268. doi: 10.1016/j.jebo.2014.03.010
![]() |
[11] | Tamayao MAM, Michalek JJ, Hendrickson C, et al. (2015) Regional variability and uncertainty of electric vehicle life cycle CO2 emissions across the United States. Environ Sci Technol 49: 8844-8855. |
[12] |
Axsen J, Kurani KS, McCarthy R, et al. (2011) Plug-in hybrid vehicle GHG impacts in California: Integrating consumer-informed recharge profiles with an electricity-dispatch model. Energy Policy 39: 1617-1629. doi: 10.1016/j.enpol.2010.12.038
![]() |
[13] |
Yuksel T, Michalek JJ (2015) Effects of regional temperature on electric vehicle efficiency, range, and emissions in the United States. Environ Sci Technol 49: 3974-3980. doi: 10.1021/es505621s
![]() |
[14] |
Miotti M, Supran GJ, Kim EJ, et al. (2016) Personal vehicles evaluated against climate change mitigation targets. Environ Sci Technol 50: 10795-10804. doi: 10.1021/acs.est.6b00177
![]() |
[15] | Yang F, Xie Y, Deng Y, et, al. (2018) Considering Battery Degradation in Life Cycle Greenhouse Gas Emission Analysis of Electric Vehicles. Procedia CIRP 69, 505-510. |
[16] |
Ellingsen LAW, Singh B, Strømman AH (2016) The size and range effect: lifecycle greenhouse gas emissions of electric vehicles. Environ Res Lett 11: 054010. doi: 10.1088/1748-9326/11/5/054010
![]() |
[17] |
Manjunath A, Gross G (2017) Towards a meaningful metric for the quantification of GHG emissions of electric vehicles (EVs). Energy Policy 102: 423-429. doi: 10.1016/j.enpol.2016.12.003
![]() |
[18] | Bicer Y, Dincer I (2017) Comparative life cycle assessment of hydrogen, methanol and electric vehicles from well to wheel. Int J Hydrog Energy 42: 3767-3777. |
[19] |
Peng T, Ou X, Yan X (2018) Development and application of an electric vehicles life-cycle energy consumption and greenhouse gas emissions analysis model. Chem Eng Res Des 131: 699-708. doi: 10.1016/j.cherd.2017.12.018
![]() |
[20] |
Küng L, Bütler T, Georges G, et al. (2018) Decarbonizing passenger cars using different powertrain technologies: Optimal fleet composition under evolving electricity supply. Transp Res Part C Emerg Technol 95: 785-801. doi: 10.1016/j.trc.2018.09.003
![]() |
[21] |
Wu Z, Wang M, Zheng J, et al. (2018) Life cycle greenhouse gas emission reduction potential of battery electric vehicle. J Clean Prod 190: 462-470. doi: 10.1016/j.jclepro.2018.04.036
![]() |
[22] |
Kamiya G, Axsen J, Crawford C (2019) Modeling the GHG emissions intensity of plug- in electric vehicles using short-term and long-term perspectives. Transp Res D 69: 209-223. doi: 10.1016/j.trd.2019.01.027
![]() |
[23] |
Onat NC, Noori M, Kucukvar M, et al. (2017) Exploring the suitability of electric vehicles in the United States. Energy 121: 631-642. doi: 10.1016/j.energy.2017.01.035
![]() |
[24] | US Department of Transportation, Federal Highway Administration, 2017 National Household Travel Survey. Available from: nhts.ornl.gov/. |
[25] | US Energy Information Administration, Annual Energy Outlook 2019 with projections to 2050. January 24, 2019. Available from: www.eia.gov/aeo/. |
[26] | US Census Bureau, County Population Totals and Components of Change: 2010-2018. Available from: www.census.gov/data/datasets/time-series/demo/popest/2010s-counties-total.html. |
[27] | IPCC, 2013: Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovern-mental Panel on Climate Change [Stocker TF, Qin D. Plattner GK, et al. (eds.)]. NY, USA, Cambridge University Press, 1535. |
[28] | Argonne National Laboratory, GREET 1 Model. Available from: greet.es.anl.gov/. |
[29] |
Venkatesh A, Jaramillo P, Griffin WM, et al. (2012) Uncertainty in life cycle greenhouse gas emissions from United States coal. Energy Fuels 26: 4917-4923. doi: 10.1021/ef300693x
![]() |
[30] |
Howarth RW, Santoro R, Ingraffea A (2011) Methane and the greenhouse-gas footprint of natural gas from shale formations. Clim Change 106: 679-690. doi: 10.1007/s10584-011-0061-5
![]() |
[31] |
Alvarez RA, Pacala SW, Winebrake JJ, et al. (2012) Greater focus needed on methane leakage from natural gas infrastructure. PNAS 109: 6435-6440. doi: 10.1073/pnas.1202407109
![]() |
[32] | Alvarez RA, Zavala-Araiza D, Lyon DR, et al. (2018) Assessment of methane emissions from the US oil and gas supply chain. Science 361: 186-188. |
[33] |
Caulton DR, Shepson PB, Santoro RL, et al. (2014) Toward a better understanding and quantification of methane emissions from shale gas development. PNAS 111: 6237-6242. doi: 10.1073/pnas.1316546111
![]() |
[34] |
Schneising O, Burrows JP, Dickerson RR, et al. (2014) Remote sensing of fugitive methane emissions from oil and gas production in North American tight geologic formations. Earths Future 2: 548-558. doi: 10.1002/2014EF000265
![]() |
[35] |
Miller SM, Wofsy SC, Michalak AM, et al. (2013) Anthropogenic emissions of methane in the United States. PNAS 110: 20018-20022. doi: 10.1073/pnas.1314392110
![]() |
[36] |
Brandt AR, Heath GA, Kort EA, et al. (2014) Methane leaks from North American natural gas systems. Science 343: 733-735. doi: 10.1126/science.1247045
![]() |
[37] |
Norgate T, Haque N, Koltun P (2014) The impact of uranium ore grade on the greenhouse gas footprint of nuclear power. J Clean Prod 84: 360-367. doi: 10.1016/j.jclepro.2013.11.034
![]() |
[38] |
Warner ES, Heath GA (2012) Life cycle greenhouse gas emissions of nuclear electricity generation. J Ind Ecol 16: S73-S92. doi: 10.1111/j.1530-9290.2012.00472.x
![]() |
[39] |
Lenzen, M (2008) Life cycle energy and greenhouse gas emissions of nuclear energy: A review. Energy Convers Manag 49: 2178-2199. doi: 10.1016/j.enconman.2008.01.033
![]() |
[40] |
Sovacool BK (2008) Valuing the greenhouse gas emissions from nuclear power: A critical survey. Energy Policy 36: 2950-2963. doi: 10.1016/j.enpol.2008.04.017
![]() |
[41] |
Beerten J, Laes E, Meskens G, et al. (2009) Greenhouse gas emissions in the nuclear life cycle: A balanced appraisal. Energy Policy 37: 5056-5068. doi: 10.1016/j.enpol.2009.06.073
![]() |
[42] |
Kadiyala A, Kommalapati R, Huque Z (2016) Quantification of the lifecycle greenhouse gas emissions from nuclear power generation systems. Energies 9: 863. doi: 10.3390/en9110863
![]() |
[43] |
Song C, Gardner KH, Klein SJ, et al. (2018) Cradle-to-grave greenhouse gas emissions from dams in the United States of America. Renew Sust Energ Rev 90: 945-956. doi: 10.1016/j.rser.2018.04.014
![]() |
[44] |
Hertwich EG (2013) Addressing biogenic greenhouse gas emissions from hydropower in LCA. Environ Sci Technol 47: 9604-9611. doi: 10.1021/es401820p
![]() |
[45] | Teodoru CR, Bastien J, Bonneville MC, et al. (2012) The net carbon footprint of a newly created boreal hydroelectric reservoir. Global Biogeochem Cy 26: GB2016. |
[46] |
Pacca S (2007) Impacts from decommissioning of hydroelectric dams: a life cycle perspective. Clim Change 84: 281-294. doi: 10.1007/s10584-007-9261-4
![]() |
[47] |
Dolan SL, Heath GA (2012) Life cycle greenhouse gas emissions of utility-scale wind power. J Ind Ecol 16: S136-S154. doi: 10.1111/j.1530-9290.2012.00464.x
![]() |
[48] |
Arvesen A, Hertwich EG (2012) Assessing the life cycle environmental impacts of wind power: A review of present knowledge and research needs. Renew Sust Energ Rev 16: 5994-6006. doi: 10.1016/j.rser.2012.06.023
![]() |
[49] |
Nugent D, Sovacool BK (2014) Assessing the lifecycle greenhouse gas emissions from solar PV and wind energy: a critical meta-survey. Energy Policy 65: 229-244. doi: 10.1016/j.enpol.2013.10.048
![]() |
[50] |
Sullivan JL, Clark C, Han J, et al. (2013) Cumulative energy, emissions, and water consumption for geothermal electric power production. J Renew Sustain Energy 5: 023127. doi: 10.1063/1.4798315
![]() |
[51] |
Bayer P, Rybach L, Blum P, et al. (2013) Review on life cycle environmental effects of geothermal power generation. Renew Sust Energ Rev 26: 446-463. doi: 10.1016/j.rser.2013.05.039
![]() |
[52] |
Fthenakis VM, Kim HC (2011) Photovoltaics: Life-cycle analyses. Solar Energy 85: 1609-1628. doi: 10.1016/j.solener.2009.10.002
![]() |
[53] |
Wong JH, Royapoor M, Chan CW (2016) Review of life cycle analyses and embodied energy requirements of single-crystalline and multi-crystalline silicon photovoltaic systems. Renew Sust Energ Rev 58: 608-618. doi: 10.1016/j.rser.2015.12.241
![]() |
[54] | Fthenakis V, Raugei M (2017) Environmental life-cycle assessment of photovoltaic systems. In The Performance of Photovoltaic (PV) Systems (209-232). Woodhead Publishing. |
[55] |
Milousi M, Souliotis M, Arampatzis G, et al. (2019) Evaluating the Environmental Performance of Solar Energy Systems Through a Combined Life Cycle Assessment and Cost Analysis. Sustainability 11: 2539. doi: 10.3390/su11092539
![]() |
[56] |
Djomo SN, Kasmioui OE, Ceulemans R (2011) Energy and greenhouse gas balance of bioenergy production from poplar and willow: a review. GCB Bioenergy 3: 181-197. doi: 10.1111/j.1757-1707.2010.01073.x
![]() |
[57] |
Kadiyala A, Kommalapati R, Huque Z (2016) Evaluation of the life cycle greenhouse gas emissions from different biomass feedstock electricity generation systems. Sustainability 8: 1181. doi: 10.3390/su8111181
![]() |
[58] |
Fargione J, Hill J, Tilman D (2008) Land clearing and the biofuel carbon debt. Science 319: 1235-1238. doi: 10.1126/science.1152747
![]() |
[59] |
Repo A, Tuovinen JP, Liski J (2015) Can we produce carbon and climate neutral forest bioenergy?. GCB Bioenergy 7: 253-262. doi: 10.1111/gcbb.12134
![]() |
[60] |
Holtsmark B (2015) Quantifying the global warming potential of CO2 emissions from wood fuels. GCB Bioenergy 7: 195-206. doi: 10.1111/gcbb.12110
![]() |
[61] |
Hudiburg TW, Law BE, Wirth C, et al. (2011) Regional carbon dioxide implications of forest bioenergy production. Nat Clim Change 1: 419-423. doi: 10.1038/nclimate1264
![]() |
[62] |
Hudiburg TW, Luyssaert S, Thornton PE, et al. (2013) Interactive effects of envi- ronmental change and management strategies on regional forest carbon emissions. Environ Sci Technol 47: 13132-13140. doi: 10.1021/es402903u
![]() |
[63] |
Schulze ED, Körner C, Law BE, et al. (2012) Large-scale bioenergy from additional harvest of forest biomass is neither sustainable nor greenhouse gas neutral. GCB Bioenergy 4: 611-616. doi: 10.1111/j.1757-1707.2012.01169.x
![]() |
[64] |
Cherubini F, Huijbregts M, Kindermann G, et al. (2016) Global spatially explicit CO2 emission metrics for forest bioenergy. Sci Rep 6: 20186. doi: 10.1038/srep20186
![]() |
[65] |
Booth MS (2018) Not carbon neutral: Assessing the net emissions impact of residues burned for bioenergy. Environ Res Lett 13: 035001. doi: 10.1088/1748-9326/aaac88
![]() |
[66] | US Environmental Protection Agency. Emissions & Generation Resource Integrated Database (eGRID). Available from: www.epa.gov/energy/emissions-generation-resource-integrated-database-. |
[67] |
Casals LC, Martinez-Laserna E, García BA, et al. (2016) Sustainability analysis of the electric vehicle use in Europe for CO2 emissions reduction. J Clean Prod 127: 425-437. doi: 10.1016/j.jclepro.2016.03.120
![]() |
[68] |
Van Vliet O, Brouwer AS, Kuramochi T, et al. (2011) Energy use, cost and CO2 emissions of electric cars. J Power Source 196: 2298-2310. doi: 10.1016/j.jpowsour.2010.09.119
![]() |
[69] | US Environmental Protection Agency. Air Markets Program Data. Website ampd.epa.gov/ampd/ |
[70] |
Siler-Evans K, Azevedo IL, Morgan MG (2012) Marginal emissions factors for the US electricity system. Environ Sci Technol 46: 4742-4748. doi: 10.1021/es300145v
![]() |
[71] | Argonne National Laboratory, GREET 2 Model. Available from: greet.es.anl.gov/ |
[72] |
Ellingsen LAW, Majeau-Bettez G, Singh B, et al. (2014) Life cycle assessment of a lithium-ion battery vehicle pack. J Ind Ecol 18: 113-124. doi: 10.1111/jiec.12072
![]() |
[73] |
Kim HC, Wallington TJ, Arsenault R, et al. (2016) Cradle-to-Gate Emissions from a Commercial Electric Vehicle Li-Ion Battery: A Comparative Analysis. Environ Sci Technol 50: 7715-7722. doi: 10.1021/acs.est.6b00830
![]() |
[74] |
Peters JF, Baumann M, Zimmermann B, et al (2017) The environmental impact of Li-Ion batteries and the role of key parameters-A review. Renew Sust Energ Rev 67: 491-506. doi: 10.1016/j.rser.2016.08.039
![]() |
[75] |
Ellingsen LAW, Hung CR, Strømman AH (2017) Identifying key assumptions and differences in life cycle assessment studies of lithium-ion traction batteries with focus on greenhouse gas emissions. Transp Res D 55: 82-90. doi: 10.1016/j.trd.2017.06.028
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1. | Jin Li, Yongling Cheng, Barycentric rational interpolation method for solving 3 dimensional convection–diffusion equation, 2024, 304, 00219045, 106106, 10.1016/j.jat.2024.106106 | |
2. | Jin Li, Yongling Cheng, Spectral collocation method for convection-diffusion equation, 2024, 57, 2391-4661, 10.1515/dema-2023-0110 |
method of substitution | method of additional | |||
uniform | nonuniform | uniform | nonuniform | |
Larange | 1.4662e-06 | 2.1919e-08 | 2.7900e-07 | 1.4310e-07 |
Rational | 1.3038e-05 | 2.4541e-07 | 4.9788e-06 | 1.4310e-07 |
uniform | nonuniform | uniform | nonuniform | |
t | (12,12) | (12,12) | (12,12)dt=ds=5 | (12,12)dt=ds=5 |
0.5 | 2.1021e-11 | 3.8250e-09 | 6.8506e-06 | 1.6436e-06 |
1 | 9.0394e-13 | 4.4206e-10 | 4.6667e-06 | 7.8141e-07 |
5 | 6.1833e-12 | 5.6655e-08 | 2.3777e-04 | 4.2230e-05 |
10 | 1.0094e-12 | 8.5622e-07 | 1.9813e-04 | 1.5634e-05 |
15 | 3.5397e-12 | 1.8827e-05 | 8.5498e-04 | 8.2551e-05 |
α1 | α2=0.1 | α2=0.4 | α2=0.6 | α2=0.8 | α2=0.99 |
0.01 | 1.0153e-04 | 1.0246e-04 | 1.0300e-04 | 1.0346e-04 | 1.0384e-04 |
0.1 | 1.2753e-05 | 1.2865e-05 | 1.2930e-05 | 1.2987e-05 | 1.3033e-05 |
0.3 | 2.7464e-05 | 2.7704e-05 | 2.7845e-05 | 2.7971e-05 | 2.8074e-05 |
0.5 | 4.5746e-06 | 4.6152e-06 | 4.6399e-06 | 4.6609e-06 | 4.6794e-06 |
0.9 | 9.0295e-06 | 9.1193e-06 | 9.1240e-06 | 9.2142e-06 | 9.2479e-06 |
0.99 | 1.8981e-06 | 1.8247e-06 | 1.5293e-06 | 1.9193e-06 | 2.0670e-06 |
m,n | dt=2 | dt=3 | dt=4 | dt=5 | ||||
8 | 1.3626e-02 | 6.9619e-03 | 2.0708e-03 | 9.8232e-04 | ||||
10 | 9.6780e-03 | 1.5332 | 3.4354e-03 | 3.1653 | 6.9542e-04 | 4.8900 | 3.2829e-04 | 4.9117 |
12 | 7.0485e-03 | 1.7389 | 1.9408e-03 | 3.1320 | 2.9186e-04 | 4.7621 | 1.3132e-04 | 5.0255 |
14 | 5.4466e-03 | 1.6725 | 1.2017e-03 | 3.1097 | 1.4211e-04 | 4.6686 | 6.0148e-05 | 5.0654 |
m,n | ds=2 | ds=3 | ds=4 | |||
8 | 4.9495e-04 | 4.9492e-04 | 4.9486e-04 | |||
10 | 1.0051e-04 | 7.1443 | 1.0053e-04 | 7.1431 | 1.0053e-04 | 7.1426 |
12 | 2.7700e-05 | 7.0690 | 2.7711e-05 | 7.0679 | 2.7714e-05 | 7.0673 |
14 | 9.4272e-06 | 6.9921 | 9.4315e-06 | 6.9917 | 9.4314e-06 | 6.9925 |
m,n | ds=2 | ds=3 | ds=4 | |||
8 | 2.8113e-05 | 2.8110e-05 | 2.8108e-05 | |||
10 | 2.1197e-05 | 1.2654 | 2.1196e-05 | 1.2652 | 2.1195e-05 | 1.2651 |
12 | 6.6990e-06 | 6.3180 | 6.6989e-06 | 6.3178 | 6.6988e-06 | 6.3176 |
14 | 1.6712e-06 | 9.0069 | 1.6712e-06 | 9.0068 | 1.6712e-06 | 9.0067 |
m,n | dt=2 | dt=3 | dt=4 | dt=5 | ||||
8 | 3.1539e-02 | 8.7995e-03 | 2.1930e-03 | 3.3004e-04 | ||||
10 | 2.4329e-02 | 1.1632 | 4.0288e-03 | 3.5010 | 2.7133e-04 | 9.3648 | 2.2278e-04 | 1.7613 |
12 | 1.5223e-02 | 2.5716 | 1.9127e-03 | 4.0859 | 9.5194e-05 | 5.7449 | 5.1702e-05 | 8.0116 |
14 | 1.1407e-02 | 1.8721 | 1.1143e-03 | 3.5049 | 3.5772e-05 | 6.3493 | 1.1369e-05 | 9.8255 |
m,n | ds=2 | ds=3 | ds=4 | |||
8 | 4.9427e-04 | 4.9426e-04 | 4.9414e-04 | |||
10 | 1.0035e-04 | 7.1455 | 1.0041e-04 | 7.1427 | 1.0041e-04 | 7.1413 |
12 | 2.7639e-05 | 7.0720 | 2.7674e-05 | 7.0684 | 2.7684e-05 | 7.0669 |
14 | 9.3984e-06 | 6.9977 | 9.4153e-06 | 6.9942 | 9.4254e-06 | 6.9895 |
m,n | dt=1 | dt=2 | dt=3 | dt=4 | ||||
8 | 1.3587e-02 | 6.9513e-03 | 2.0677e-03 | 9.8084e-04 | ||||
10 | 9.6497e-03 | 1.5334 | 3.4314e-03 | 3.1637 | 6.9462e-04 | 4.8884 | 3.2791e-04 | 4.9102 |
12 | 7.0259e-03 | 1.7404 | 1.9389e-03 | 3.1311 | 2.9157e-04 | 4.7613 | 1.3118e-04 | 5.0249 |
14 | 5.4269e-03 | 1.6752 | 1.2005e-03 | 3.1096 | 1.4198e-04 | 4.6682 | 6.0090e-05 | 5.0648 |
m,n | dt=1 | dt=2 | dt=3 | dt=4 | ||||
8 | 3.1481e-02 | 8.7825e-03 | 2.1876e-03 | 3.2930e-04 | ||||
10 | 2.4263e-02 | 1.1671 | 4.0219e-03 | 3.5000 | 2.7124e-04 | 9.3553 | 2.2231e-04 | 1.7606 |
12 | 1.5185e-02 | 2.5704 | 1.9076e-03 | 4.0912 | 9.5106e-05 | 5.7481 | 5.1649e-05 | 8.0057 |
14 | 1.1373e-02 | 1.8751 | 1.1117e-03 | 3.5026 | 3.5733e-05 | 6.3504 | 1.1365e-05 | 9.8211 |
m,n | ds=2 | ds=3 | ds=4 | |||
8 | 2.8065e-05 | 2.8059e-05 | 2.8056e-05 | |||
10 | 2.1156e-05 | 1.2665 | 2.1154e-05 | 1.2660 | 2.1153e-05 | 1.2656 |
12 | 6.6875e-06 | 6.3168 | 6.6874e-06 | 6.3164 | 6.6873e-06 | 6.3161 |
14 | 1.6693e-06 | 9.0033 | 1.6693e-06 | 9.0031 | 1.6693e-06 | 9.0030 |
method of substitution | method of additional | |||
uniform | nonuniform | uniform | nonuniform | |
Larange | 1.4662e-06 | 2.1919e-08 | 2.7900e-07 | 1.4310e-07 |
Rational | 1.3038e-05 | 2.4541e-07 | 4.9788e-06 | 1.4310e-07 |
uniform | nonuniform | uniform | nonuniform | |
t | (12,12) | (12,12) | (12,12)dt=ds=5 | (12,12)dt=ds=5 |
0.5 | 2.1021e-11 | 3.8250e-09 | 6.8506e-06 | 1.6436e-06 |
1 | 9.0394e-13 | 4.4206e-10 | 4.6667e-06 | 7.8141e-07 |
5 | 6.1833e-12 | 5.6655e-08 | 2.3777e-04 | 4.2230e-05 |
10 | 1.0094e-12 | 8.5622e-07 | 1.9813e-04 | 1.5634e-05 |
15 | 3.5397e-12 | 1.8827e-05 | 8.5498e-04 | 8.2551e-05 |
α1 | α2=0.1 | α2=0.4 | α2=0.6 | α2=0.8 | α2=0.99 |
0.01 | 1.0153e-04 | 1.0246e-04 | 1.0300e-04 | 1.0346e-04 | 1.0384e-04 |
0.1 | 1.2753e-05 | 1.2865e-05 | 1.2930e-05 | 1.2987e-05 | 1.3033e-05 |
0.3 | 2.7464e-05 | 2.7704e-05 | 2.7845e-05 | 2.7971e-05 | 2.8074e-05 |
0.5 | 4.5746e-06 | 4.6152e-06 | 4.6399e-06 | 4.6609e-06 | 4.6794e-06 |
0.9 | 9.0295e-06 | 9.1193e-06 | 9.1240e-06 | 9.2142e-06 | 9.2479e-06 |
0.99 | 1.8981e-06 | 1.8247e-06 | 1.5293e-06 | 1.9193e-06 | 2.0670e-06 |
m,n | dt=2 | dt=3 | dt=4 | dt=5 | ||||
8 | 1.3626e-02 | 6.9619e-03 | 2.0708e-03 | 9.8232e-04 | ||||
10 | 9.6780e-03 | 1.5332 | 3.4354e-03 | 3.1653 | 6.9542e-04 | 4.8900 | 3.2829e-04 | 4.9117 |
12 | 7.0485e-03 | 1.7389 | 1.9408e-03 | 3.1320 | 2.9186e-04 | 4.7621 | 1.3132e-04 | 5.0255 |
14 | 5.4466e-03 | 1.6725 | 1.2017e-03 | 3.1097 | 1.4211e-04 | 4.6686 | 6.0148e-05 | 5.0654 |
m,n | ds=2 | ds=3 | ds=4 | |||
8 | 4.9495e-04 | 4.9492e-04 | 4.9486e-04 | |||
10 | 1.0051e-04 | 7.1443 | 1.0053e-04 | 7.1431 | 1.0053e-04 | 7.1426 |
12 | 2.7700e-05 | 7.0690 | 2.7711e-05 | 7.0679 | 2.7714e-05 | 7.0673 |
14 | 9.4272e-06 | 6.9921 | 9.4315e-06 | 6.9917 | 9.4314e-06 | 6.9925 |
m,n | ds=2 | ds=3 | ds=4 | |||
8 | 2.8113e-05 | 2.8110e-05 | 2.8108e-05 | |||
10 | 2.1197e-05 | 1.2654 | 2.1196e-05 | 1.2652 | 2.1195e-05 | 1.2651 |
12 | 6.6990e-06 | 6.3180 | 6.6989e-06 | 6.3178 | 6.6988e-06 | 6.3176 |
14 | 1.6712e-06 | 9.0069 | 1.6712e-06 | 9.0068 | 1.6712e-06 | 9.0067 |
m,n | dt=2 | dt=3 | dt=4 | dt=5 | ||||
8 | 3.1539e-02 | 8.7995e-03 | 2.1930e-03 | 3.3004e-04 | ||||
10 | 2.4329e-02 | 1.1632 | 4.0288e-03 | 3.5010 | 2.7133e-04 | 9.3648 | 2.2278e-04 | 1.7613 |
12 | 1.5223e-02 | 2.5716 | 1.9127e-03 | 4.0859 | 9.5194e-05 | 5.7449 | 5.1702e-05 | 8.0116 |
14 | 1.1407e-02 | 1.8721 | 1.1143e-03 | 3.5049 | 3.5772e-05 | 6.3493 | 1.1369e-05 | 9.8255 |
m,n | ds=2 | ds=3 | ds=4 | |||
8 | 4.9427e-04 | 4.9426e-04 | 4.9414e-04 | |||
10 | 1.0035e-04 | 7.1455 | 1.0041e-04 | 7.1427 | 1.0041e-04 | 7.1413 |
12 | 2.7639e-05 | 7.0720 | 2.7674e-05 | 7.0684 | 2.7684e-05 | 7.0669 |
14 | 9.3984e-06 | 6.9977 | 9.4153e-06 | 6.9942 | 9.4254e-06 | 6.9895 |
m,n | dt=1 | dt=2 | dt=3 | dt=4 | ||||
8 | 1.3587e-02 | 6.9513e-03 | 2.0677e-03 | 9.8084e-04 | ||||
10 | 9.6497e-03 | 1.5334 | 3.4314e-03 | 3.1637 | 6.9462e-04 | 4.8884 | 3.2791e-04 | 4.9102 |
12 | 7.0259e-03 | 1.7404 | 1.9389e-03 | 3.1311 | 2.9157e-04 | 4.7613 | 1.3118e-04 | 5.0249 |
14 | 5.4269e-03 | 1.6752 | 1.2005e-03 | 3.1096 | 1.4198e-04 | 4.6682 | 6.0090e-05 | 5.0648 |
m,n | dt=1 | dt=2 | dt=3 | dt=4 | ||||
8 | 3.1481e-02 | 8.7825e-03 | 2.1876e-03 | 3.2930e-04 | ||||
10 | 2.4263e-02 | 1.1671 | 4.0219e-03 | 3.5000 | 2.7124e-04 | 9.3553 | 2.2231e-04 | 1.7606 |
12 | 1.5185e-02 | 2.5704 | 1.9076e-03 | 4.0912 | 9.5106e-05 | 5.7481 | 5.1649e-05 | 8.0057 |
14 | 1.1373e-02 | 1.8751 | 1.1117e-03 | 3.5026 | 3.5733e-05 | 6.3504 | 1.1365e-05 | 9.8211 |
m,n | ds=2 | ds=3 | ds=4 | |||
8 | 2.8065e-05 | 2.8059e-05 | 2.8056e-05 | |||
10 | 2.1156e-05 | 1.2665 | 2.1154e-05 | 1.2660 | 2.1153e-05 | 1.2656 |
12 | 6.6875e-06 | 6.3168 | 6.6874e-06 | 6.3164 | 6.6873e-06 | 6.3161 |
14 | 1.6693e-06 | 9.0033 | 1.6693e-06 | 9.0031 | 1.6693e-06 | 9.0030 |