
A new fourth-order explicit grouping iterative method is constructed for the numerical solution of the fractional sub-diffusion equation. The discretization of the equation is based on fourth-order finite difference method. Captive fractional discretization having functions with a weak singularity at t=0 is used for time and similarly, the space derivative is approximated with the help of fourth-order approximation. Furthermore, the convergence and stability of the scheme are analyzed. Finally, the accuracy and validity are investigated by some numerical examples.
Citation: Muhammad Asim Khan, Norma Alias, Umair Ali. A new fourth-order grouping iterative method for the time fractional sub-diffusion equation having a weak singularity at initial time[J]. AIMS Mathematics, 2023, 8(6): 13725-13746. doi: 10.3934/math.2023697
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A new fourth-order explicit grouping iterative method is constructed for the numerical solution of the fractional sub-diffusion equation. The discretization of the equation is based on fourth-order finite difference method. Captive fractional discretization having functions with a weak singularity at t=0 is used for time and similarly, the space derivative is approximated with the help of fourth-order approximation. Furthermore, the convergence and stability of the scheme are analyzed. Finally, the accuracy and validity are investigated by some numerical examples.
Fractional-order differential equations (FDM) have many applications in various fields of engineering and science, such as chemical and physical phenomena [2,3,4,5]. For instance, the fractional-order diffusion equation is used to describe anomalous diffusion phenomena in the transport process through disordered and complex systems including fractal media, fractional kinetic equations regarding slow diffusion, and movement of small molecular along the concentration space [6].
In this article, the two-dimensional (2-D) time-fractional sub-diffusion equation (FSDE) with the weak singularity at initial time t=0 is considered as follows:
∂αV∂tα=∂2V∂x2+∂2V∂y2+G(x,y,t), | (1.1) |
with subject to conditions
V(x,y,0)=ζ0(x,y), |
and
V(0,y,t)=ζ1(x,y,t),V(L,y,t)=ζ2(x,y,t),V(x,0,t)=ζ3(x,y,t),V(x,L,t)=ζ4(x,y,t),0≤x,y≤M,0≤t≤N, |
where ζ0, ζ1, ζ2, ζ3, ζ4 are known functions and α∈(0,1).
The FSDE can be obtained from the anomalous diffusion system by replacing the time derivative with a fractional derivative α where 0<α<1. The FSDE is an important class of fractional partial differential equations (PDEs), which is mainly used in the modeling of fractional random walk, the phenomenon of wave propagation, diffusion unification, etc. [7,8].
The Caputo derivative of order α is
C0Dαtf(x)=1Γ(1−α)∫t0f′(x)(t−x)αdx. | (1.2) |
The Caputo derivative approximated using the L1 gives the accuracy 2−α [9,10], but the presence of kernel (t−x)α produces solutions for Eq (1.1) with the weak-singularity at initial time t=0, which increases the computation cost and give low convergence rate for the approximate methods on uniform meshes [11]. Therefore, to increase the convergence rate many researchers solved the FSDE using different high-order numerical methods. Based on the chronology, several numerical methods are proposed for the solution of 2-D time FSDE (1.1), for example, Cui [12] proposed high-order alternating direction implicit (ADI) method, and it is unconditionally stable and convergent with convergence order O(να+h40). Zhuang and Liu [13] proposed an unconditionally stable and convergent implicit difference scheme. Zhang et al. [14] used the Crank-Nicolson-type compact ADI scheme and proved the unconditional stability and convergence having convergence order O(νmin[2−α2,2α]+h014+h024) in H10 norm. Ji and Sun [15] used a high-order numerical scheme to solve (1.1), and proved convergence in L1(L∞)-norm and unconditional stability by the energy method. Wang et al. [16] solved 2-D FSDE using C-N alternating direction implicit finite difference method (FDM), where the fractional derivative is discretized using Riemann-Liouville fractional definition and to improve its temporal accuracy they used the Richardson extrapolation algorithm. Also, they proved its unique solvability, unconditional stability, and convergence O(ν2γ+h0x4+h0y4) of the scheme. Zhai and Feng [17] presented three different compact schemes for a 2-D time-fractional diffusion equation. The Caputo fractional definition is used for fractional derivative. All the schemes are fourth-order accurate for space and second-order accurate for the time variable. The stability of all the schemes is analyzed using Fourier analysis, which shows that two schemes are unconditionally stable, and the third one is conditionally stable.
The advantage of high-order schemes is that it produces more accurate results but at the same time increase the execution timings because of the escalated computational complexity of the scheme. Similarly, the advantage of explicit group methods over the standard point methods that the it considered quarter grid points of the solution domain and the points are considered as iterative points in the iterative process which reduces the computational complexity of the proposed method and hence reduce the execution time per iteration. Since, the computational complexity is greatly reduced using the explicit group method the 2-D time-fractional advection-diffusion, hyperbolic telegraph fractional differential, and fractional diffusion equations etc. [18,19,20,21,22] with second-order accuracy, therefore, we proposed the grouping strategy with uniform grids for the solution of the 2-D time FSDE with fourth-order accuracy. The purpose of this paper is to solve 2-D time FSDE with the fourth-order explicit group method (FEGM).
The paper is organized as follows: In Section 2, the derivation of the group explicit method from the finite difference method is presented. Section 3 discussed the stability of the proposed scheme, and the convergence of the proposed scheme is presented in Section 4. To show the efficiency of the proposed method, some numerical examples with discussion are presented in Section 5, and finally, Section 6 consists of the conclusion.
First, let us define some notations:
δ2xVki,j=Vki+1,j−2Vki,j+Vki−1,j,Vk+12i,j=Vk+1i,j+Vki,j2,xi=ih0,yj=jh0,i,j=0,1,2,3...,M,tk=kν,k=0,1,2,3...,N, |
where h0=Δx=Δy=LM represents the space step and ν=TN time step.
Since, the Taylor series expansion with respect to x is
Vki+1,j=Vki,j+h01!Vx|ki,j+h202!Vxx|ki,j+h303!V|ki,j+..., | (2.1) |
Vki−1,j=Vki,j−h01!Vx|ki,j+h202!Vxx|ki,j−h303!Vxxx|ki,j+... | (2.2) |
By adding Eqs (2.1) and (2.2) and after rearranging, we get
∂2Vki,j∂x2=Vki+1,j−2uki,j+Vki−1,jh20+2h204!∂4Vki,j∂x4+...=Vki+1,j−2Vki,j+Vki−1,jh20+O(h20). | (2.3) |
Therefore, the Taylor series expansions at points uki+1,j and uki,j+1 are
δ2xh20Vki,j=∂2V∂x2|ki,j−h2012∂4V∂x4|ki,j−h40360∂6V∂x6|ki,j+O(h60), | (2.4) |
δ2yh20Vki,j=∂2V∂y2|ki,j−h2012∂4V∂y4|ki,j+h40360∂6V∂y6|ki,j+O(h60). | (2.5) |
The difference operator δ2x, which maintain the three-point stencil is given by []
δ2xh20(1+112δ2x)Vki,j=∂2V∂x2|ki,j−h40240∂4V∂x4|ki,j+O(h60), | (2.6) |
and
δ2yh20(1+112δ2y)Vki,j=∂2V∂y2|ki,j−h40240∂4V∂y4|ki,j+O(h60). | (2.7) |
The fractional derivative is approximated using the Central difference formula as [24]
∂α∂tαV(x,y,t)|k+12i,j=1Γ(1−α)∫tk+120Vt(x,y,ε)(tk+12−ε)−α∂ε=1Γ(1−α)[∫tk+120Vt(x,y,ε)((k+12)ν−ε)−α∂ε+∫ttk+12tk(Vk+1i,j−Vki,jν+O(ν))((k+12)ν−ε)−α∂ε]=1Γ(1−α)k∑s0=1[∫sν(s0−1)νVs0i,j−Vs0−1i,jν+(ε−ts0−12)Vtt(xi,yj,cs0)×((k+12ν−ε)−α∂ε)]+∫(k+12)νkν(Vk+1i,j−Vki,jν+O(ν))((k+12)ν−ε)−α∂ε=1Γ(1−α)k∑s0=1Vs0i,j−Vs0−1i,jν∫sν(s0−1)ν((k+12)ν−ε)−α∂ε+1Γ(1−α)k∑s0=1∫sν(s−1)ν(ε−ts0−12)Vtt(xi,yj,cs0)((k+12)ν−ε)−α∂ε+1Γ(1−α)[Vk+1i,j−Vki,jν+O(ν)]∫(k+12)νkν[((k+12)ν−ε)−α−ε]∂ε=1να(1−α)Γ(1−α)k∑s0=1[Vs0i,j−Vs0−1i,j][(k−s0+32)1−α−(k−s0+12)1−α]+1να(1−α)Γ(1−α)(Vk+1i,j−Vki,j)121−α+1Γ(1−α)k∑s0=1∫sν(s−1)ν(ε−ts0−12)Vtt(xi,yj,cs0)((k+12)ν−ε)−α∂ε+1Γ(1−α)(1−α)21−αO(ν)2−α. |
Therefore, after some simplifications, the Crank-Nicolson (C-N) Caputo fractional derivative
∂α∂tαV(x,y,t)|k+12i,j=a1V|ki,j+k−1∑s=1(ak−s+1−ak−s)V|si,j−akV|0i,j+σVk+1i,j+Vki,j21−α+O(ν2−α),σ=1ναΓ(2−α),as=σ((s+12)1−α−(s−12)1−α),s=0,1,2,...,k. | (2.8) |
Now using Eqs (2.6)–(2.8) and C-N or standard point (SP) scheme at V(xi,yj,tk+12), the standard fourth-order finite difference scheme for Eq (1.1) is as follows:
a1Vki,j+k−1∑s=1(ak−s+1−ak−s)Vsi,j−akV0i,j+σVk+1i,j+Vki,j21−α=(1+112δ2x)−1×δ2xh20Vk+12i,j+(1+112δ2y)−1δ2yh20Vk+12i,j+fk+12i,j+O(ν2−α+h40). | (2.9) |
Substituting the values of δ2x, δ2y and Vk+12i,j into Eq (2.9), and after rearranging we get the standard point SP compact scheme:
λ1Vk+1i,j=λ2(Vk+1i+1,j+Vk+1i−1,j+Vk+1i,j+1+Vk+1i,j−1)+λ3(Vk+1i+1,j+1+Vk+1i−1,j+1+Vk+1i+1,j−1+Vk+1i−1,j−1)+λ4Vki,j+λ5(Vki+1,j+Vki−1,j+Vki,j+1+Vki,j−1)+λ6(Vki+1,j+1+Vki−1,j+1+Vki+1,j−1+Vki−1,j−1)+2518h20fk+12i,j+536h20(fk+1ti+1,j+fk+1ti−1,j+fk+1ti,j+1+fk+1ti,j−1)+h2072(fk+12i+1,j+1+fk+12i−1,j+1+fk+12i+1,j−1+fk+12i−1,j−1)−k−1∑s=1(ak−s+1−ak−s)(2518h20Vsi,j+536h20(Vsi+1,j+Vsi−1,j+Vsi,j+1+Vsi,j−1)+h2072(Vsi+1,j+1+Vsi−1,j+1+Vsi+1,j−1+Vsi−1,j−1))+O(ν2−α+h40), | (2.10) |
where
g0=σ21−α,g1=a1−g0,λ1=172(240+100h20g0),λ2=172(48−10h20g0),λ3=172(12−h20g0),λ4=−172(240+100h20g1),λ5=172(48−10h20g1),λ6=172(12−h20g1). |
Now, using Eq (2.10) will give the following system for the group of four points:
[λ1−λ2−λ3−λ2−λ2λ1−λ2−λ3−λ3−λ2λ1−λ2−λ2−λ3−λ2λ1][Vk+1i,jVk+1i+1,jVk+1i+1,j+1Vk+1i,j+1]=[rhsi,jrhsi+1,jrhsi+1,j+1rhsi,j+1], | (2.11) |
where
rhsi,j=λ2(Vk+1i−1,j+Vk+1i,j−1)+λ3(Vk+1i−1,j+1+Vk+1i+1,j−1+Vk+1i−1,j−1)+λ4Vki,j+λ5(Vki+1,j+Vki−1,j+Vki,j+1+Vki,j−1)+λ6(Vki+1,j+1+Vki−1,j+1+Vki+1,j−1+Vki−1,j−1)+2518h20fk+12i,j+536h20(fk+12i+1,j+fk+12i−1,j+fk+12i,j+1+fk+12i,j−1)+h2072(fk+12i+1,j+1+fk+12i−1,j+1+fk+12i+1,j−1+fk+12i−1,j−1)−Fi,j, |
rhsi+1,j=λ2(Vk+1i+2,j+Vk+1i+1,j−1)+λ3(Vk+1i+2,j+1+Vk+1i+2,j−1+Vk+1i,j−1)+λ4Vki+1,j+λ5(Vki+2,j+Vki,j+Vki+1,j+1+Vki+1,j−1)+λ6(Vki+2,j+1+Vki,j+1+Vki+2,j−1+Vki,j−1)+2518h20fk+12i+1,j+536h20(fk+12i+2,j+fk+12i,j+fk+12i+1,j+1+fk+12i+1,j−1)+h2072(fk+12i+2,j+1+fk+12i,j+1+fk+12i+2,j−1+fk+12i,j−1)−Fi+1,j, |
rhsi+1,j+1=λ2(Vk+1i+2,j+1+Vk+1i+1,j+2)+λ3(Vk+1i+2,j+2+Vk+1i,j+2+Vk+1i+2,j)+λ4Vki+1,j+1+λ5(Vki+2,j+1+Vki,j+1+Vki+1,j+2+Vki+1,j)+λ6(Vki+2,j+2+Vki,j+2+Vki+2,j+Vki,j)+2518h20fk+12i+1,j+1+536h20(fk+12i+2,j+1+fk+12i,j+1+fk+12i+1,j+2+fk+12i+1,j)+h2072(fk+12i+2,j+2+fk+12i,j+2+fk+12i+2,j+fk+12i,j)−Fi+1,j+1, |
rhsi,j+1=λ2(Vk+1i−1,j+1+Vk+1i,j+2)+λ3(Vk+1i+1,j+2+Vk+1i−1,j+2+Vk+1i−1,j)+λ4Vki,j+1+λ5(Vki+1,j+1+Vki−1,j+1+Vki,j+2+Vki,j)+λ6(Vki+1,j+2+Vki−1,j+2+Vki+1,j+Vki−1,j)+2518h20fk+12i,j+1+536h20(fk+12i+1,j+1+fk+12i−1,j+1+fk+12i,j+2+fk+12i,j)+h2072(fk+12i+1,j+2+fk+12i−1,j+2+fk+12i+1,j+fk+12i−1,j)−Fi,j+1, |
and
Fi,j=k−1∑s=1(ak−s+1−ak−s)(2518h20Vsi,j+536h20(Vsi+1,j+Vsi−1,j+Vsi,j+1+Vsi,j−1)+h2072(Vsi+1,j+1+Vsi−1,j+1+Vsi+1,j−1+Vsi−1,j−1)). |
Similarly, the inverted matrix equation (2.11) will give explicit group equation
[Vk+1i,jVk+1i+1,jVk+1i+1,j+1Vk+1i,j+1]=1d[ϕ1ϕ2ϕ3ϕ2ϕ2ϕ1ϕ2ϕ3ϕ3ϕ2ϕ1ϕ2ϕ2ϕ3ϕ2ϕ1][rhsi,jrhsi+1,jrhsi+1,j+1rhsi,j+1], | (2.12) |
where
ϕ1=λ31−2λ1λ22−2λ22λ3−λ1λ23,ϕ2=λ21λ2+2λ1λ2λ3+λ2λ23,ϕ3=2λ1λ22+λ21λ3+2λ22λ3−λ33,d=(−4λ22+(λ1−λ3)2)(λ1+λ3)2. |
In the proposed method, firstly, group of four points are computed for the different iterations using Eq (2.12) till the required convergence is attained. After the required convergence, the SP compact scheme Eq (2.10) is used directly once for computing of reaming points. Figures 1 and 2 show the grid points on the x-y plane for FDM and FEGM at various time levels when m=9 respectively.
In this section, the stability of the proposed method is discussed.
The Eq (2.12) can also be written as
AV1=BV0+f12,k=0,AVk+1=BVk−h20k−1∑s=1(ak−s+1−ak−s)CVs+h20fk+12,k>0, | (3.1) |
where
A=[R1R30R2R1R3R2R1⋱R30R2R1],B=[P1P30P2P1P3P2P1⋱P30P2P1], |
R1=[G1G3G2G1G3G2G1⋱G3G2G1],C=[Q1Q30Q2Q1Q3Q2Q1⋱Q30Q2Q1],f=[K1K1⋮K1K1], |
R2=[G6G4G8G6G4G8G6⋱G4G8G6],R3=[G7G9G5G7G9G5G7⋱G9G5G7], |
P1=[H1H3H2H1H3H2H1⋱H3H2H1],P2=[H6H4H8H6H4H8H6⋱H4H8H6], |
P3=[H7H9H5H7H9H5H7⋱H9H5H7],Q1=[L1L3L2L1L3L2L1⋱L3L2L1], |
Q2=[L6L4L8L6L4L8L6⋱L4L8L6],Q3=[L7L9L5L7L9L5L7⋱L9L5L7],K1=[W1W1⋮W1W1], |
G1=[λ1−λ2−λ3−λ2−λ2λ1−λ2−λ3−λ3−λ2λ1−λ2−λ2−λ3−λ2λ1],G2=[00−λ3−λ200−λ2−λ300000000],G3=[00000000−λ3−λ200−λ2−λ300], |
G4=[0000000000000−λ300],G5=[0000000−λ300000000],G6=[0−λ2−λ30000000000−λ3−λ20], |
G7=[0000−λ200−λ3−λ300−λ20000],G8=[00−λ30000000000000],G9=[00000000−λ30000000], |
H1=[λ4λ5λ6λ5λ5λ4λ5λ6λ6λ5λ4λ5λ5λ6λ5λ4],H2=[00λ6λ500λ5λ600000000],H3=[00000000λ6λ500λ5λ600], |
H4=[0000000000000λ600],H5=[0000000λ600000000],H6=[0λ5λ60000000000λ6λ50], |
H7=[0000λ500λ6λ600λ50000],H8=[00λ60000000000000],H9=[00000000λ60000000], |
L1=118[25521454522552141452255252145225],L2=118[00145400521400000000],L3=118[00000000145400541400], |
L4=[000000000000017200],L5=[000000017200000000],L6=118[05214000000000014520], |
L7=118[00005200141400520000],L8=[001720000000000000],L9=[000000001720000000],W1=[fi,jfi+1,jfi+1,j+1fi,j+1]. |
It can observe that Eq (3.1) form the particular structure as
[A(N−2)2×(N−2)2]Vk+1=[B(N−2)2×(N−2)2]Vk−h20k−1∑s=1(ak−s+1−ak−s)[C(N−2)2×(N−2)2]Vs+h20fk+12. |
Proposition 3.1. The proposed scheme Eq (2.12) is unconditionally stable.
Proof. Let Vki,j represents approximate and vki,j represents exact solutions for the time FSDE respectively, then the error is defined as ϵki,j=vki,j−Vki,j. So, from Eq (3.1),
AE1=BE0,k=0,AEk+1=BEk−h20k−1∑s=1(ak−s+1−ak−s)CEs,k>0, | (3.2) |
where
Ek+1=[Ek+11Ek+12⋮Ek+1m−2Ek+1m−1],Ek+1i=[ϵk+11ϵk+12⋮ϵk+1m−2ϵk+1m−1],ϵk+1i=[ϵk+1i,jϵk+1i+1,jϵk+1i+1,j+1ϵk+1i,j+1],i,j=1,2,...,m−1. |
From Eq (3.1) we know
A=G1I+(G2+G3)Q+G6I+(G4+G8)Q+G7I+(G5+G9)Q, | (3.3) |
B=H1I+(H2+H3)Q+H6I+(H4+H8)Q+H7I+(H5+H9)Q, | (3.4) |
C=L1I+(L2+L3)Q+L6I+(L4+L8)Q+L7I+(L5+L9)Q, | (3.5) |
where I and Q are two matrices, I represents identity matrix and Q represents unity values having each diagonal forthwith above and below the main diagonal, and elsewhere zero.
Suppose maximum eigenvalues are represented with ψ, χ and η for the matrices A, B and C respectively, then using Mathematica software, we get
ψ=98(g1h20+4),χ=(296+7972h20g0),η=12172. | (3.6) |
From Eq (3.2), when k = 0,
E1=A−1BE0,‖E1‖≤‖A−1B‖‖E0‖=132+121h2g1348+79h2g0‖E0‖. |
But since a1=σ((32)1−α−(12)1−α)=g0(31−α−1) and g1=a1−g0=g0(31−α−2). Also we know that 31−α<3, so,
31−α−2<1,g0(31−α−2)<g0,∵g0>0,g1<g0. |
Hence,
‖E1‖≤‖E0‖,∵g0>g1. |
Suppose
‖Er‖≤‖E0‖,r=2,3,...,k, | (3.7) |
and for r=k+1,
‖Ek+1‖=‖A−1(BEk−h20k−1∑s=1(ak−s+1−ak−s)CEs))‖≤‖A−1B‖‖Ek‖+h20k−1∑s=1(ak−s+1−ak−s)‖A−1C‖‖Es‖≤(‖A−1B‖+h20k−1∑s=1(ak−s+1−ak−s)‖A−1C‖)‖E0‖by using Eq. (3.7)=(324+81h20g1348+h20g0+121h20(ak−a1)72(348+79h20g0))‖E0‖=(324+81h20g1+1.68h20(ak−a1)348+79h20g0)‖E0‖=(132+(72.02)h2(g1+(ak−a1))348+79h20g0)‖E0‖. |
‖Ek+1‖≤‖E0‖,∵(ak−a1)<0. |
So, by mathematical induction, we prove that FEGM is unconditionally stable.
Suppose ek+12i,j,ek+12i+1,j,ek+12i+1,j+1 and ek+12i,j+1 represent different truncation errors, then,
Rk+12={Rk+121,1,Rk+121,2,...,Rk+121,M2−14,Rk+122,1,Rk+122,2,...,Rk+12M1−14,M2−14}, |
where
Rk+12i,j={ek+12i,j,ek+12i+1,j,ek+12i+1,j+1,ek+12i,j+1},i,j={1,2,...,M−14}, |
so from Eq (2.10) we have
‖Rk+12‖≤φ0(ν2−γ+h40), | (4.1) |
where φ0 is a constant.
Proposition 4.1. The FEGS equation (2.12) is unconditionally convergent with the order of convergence O(ν2−α+h4).
Proof. Since from Eq (4.1),
‖R(k−1)+12‖≤φ0(ν2−α+h40), | (4.2) |
then,
‖R(k−1)+12‖−‖Rk+12‖≤0,‖R(k−1)+12‖≤‖Rk+12‖. | (4.3) |
Since E0=0, then from Eq (2.10), we have
AE1=R12,k=0,AEk+1=BEk−h20k−1∑s=1(ak−s+1−ak−s)CEs+Rk+12,k>0. | (4.4) |
When k=0,
AE1=R12,‖E1‖≤‖A−1‖‖R12‖=1λ1+2λ2+λ3‖R12‖=1348+79h20g0‖R12‖,‖E1‖≤μ0‖R12‖,whereμ0=1348+79h20g0andμ0∈(0,1),‖E1‖≤‖R12‖. |
Assume that
‖Es‖≤‖R(s−1)+12‖,s=2,3,...,k, | (4.5) |
and now from Eq (4.4),
AEk+1=BEk−h20k−1∑s=1(ak−s+1−ak−s)CEs+Rk+12. | (4.6) |
By taking norm function on both sides of Eq (4.6),
‖Ek+1‖≤‖A−1B‖‖Ek‖−h20k−1∑s=1(ak−s+1−ak−s)‖A−1C‖‖Es‖+‖A−1‖‖Rk+12‖≤(324+81h20g1348+79h20g0+1.68h20(ak−a1)348+79h20g0+1348+79h20g0)‖Rk+12‖(by using Eqs (4.3) and (4.5))=(325+81h20g1+1.68h20(ak−a1)348+79h20g0)‖Rk+12‖=γ‖Rk+12‖, |
where γ=133+121h2g1+1.68h20(ak−a1)348+79h20g0, but since h∈(0,1), g0>g1, and (a1−ak)<0, then γ∈(0,1), therefore,
‖Ek+1‖≤‖Rk+12‖≤φ0(ν2−α+h40). |
Therefore, we get
‖Ek+1‖≤φ0(ν2−α+h40),∀k=0,1,2...,N−1. |
Thus the proposed scheme is conditionally stable.
The proposed scheme can be written in matrix form:
G1V1=G2V0+G3Υ12,k=0,G1Vk+1=G2Vk+G3Υk+12−k−1∑s=1(ak−s+1−ak−s)G3Vs,k≥1,V0i,j=b0(xi,yj),1≤i≤M,1≤j≤M,Vk0,j=b1(0,yj),1≤j≤M,0≤k≤N,VkL,j=b2(L,yj),1≤j≤M,0≤k≤N,Vki,0=b3(xi,0),1≤i≤M,0≤k≤N,Vki,L=b4(xi,L),1≤i≤M,0≤k≤N, | (5.1) |
where
G1=[λ1−λ2⋯−λ2−λ3⋯0−λ2λ1−λ2−λ3−λ2−λ3⋮−λ2λ1−λ2⋯−λ3−λ2−λ3⋮⋯−λ2λ1−λ2−λ3−λ2−λ3−λ2λ1−λ2−λ3−λ2⋮⋯−λ2λ1−λ2−λ2λ1−λ2−λ2λ1−λ20⋯⋯−λ2λ1], |
G2=[λ4λ5⋯λ5λ6⋯0λ5λ4λ5λ6λ5λ6⋮λ5λ4λ5⋯λ6λ5λ6⋮⋯λ5λ4λ5λ6λ5λ6λ5λ4λ5λ6λ5⋮⋯λ5λ4λ5λ5λ4λ5λ5λ4λ50⋯⋯λ5λ4], |
G3=[ρ1ρ2⋯ρ2ρ3⋯0ρ2ρ1ρ2ρ3ρ2ρ3⋮ρ2ρ1ρ2⋯ρ3ρ2ρ3⋮⋯ρ2ρ1ρ2ρ3ρ2ρ3ρ2ρ1ρ2ρ3ρ2⋮⋯ρ2ρ1ρ2ρ2ρ1ρ2ρ2ρ1ρ20⋯⋯ρ2ρ1], |
Υk=[Υk0,Υk1,Υk2,...,Υkn]T,Υk+12=f(xi,yj,tk+12),ρ1=25h2018,ρ2=5h2036andρ3=h2072. |
Proposition 5.1. The difference equation (2.12) is uniquely solvable.
Proof. Since λ1=172(240+100h20g0), λ2=172(48−10h20g0), λ3=172(12−h20g0), and h0,g0>0, then,
|λ1|=103+25h20g018 |
and
3|λ2|+2|λ3|≤73+49h20g0<103+2518h20g0=|λ1|. |
Hence, |λ1|>3|λ2|+2|λ1|, which shows that matrix G1 is strictly diagonally dominant and G1 is non-singular. This completes the proof.
The proposed method is simulated using the Intel Core i-7, 2.40GHz GHz, 6GB of RAM with Windows 8 using Mathematica software, and the experiments were done using the proposed method with SOR iterative technique as an acceleration factor (ω=1.8) with different mesh sizes (n=10,14,18,22,30) and different time steps. Furthermore, throughout the experiments, the L∞-norm convergence criteria ζ=10−5 is used. Also, the C2-order and C1-order of convergence are used for the computational order of spatial and temporal convergence using [25]
C2−order=log2(‖L∞(16ν,2h0)‖‖L∞(ν,h0)‖), | (6.1) |
C1−order=log2(∥L∞(2ν,h0)∥∥L∞(ν,h0)∥), | (6.2) |
where L∞ is the maximum error.
Some examples are presented below to show the efficiency of FEGM.
Problem 1. [26]
∂αV∂tα=∂2V∂x2+∂2V∂y2+(2Γ(3−α)t2−α+2t2)sin(x)sin(y),0<x, y<1,0<t≤1, |
with initial and Dirichlet boundary conditions.
The analytic solution for Problem 1 is
V(x,y,t)=t2sin(x)sin(y). |
Problem 2. [12]
∂2V∂t2=∂2V∂x1+∂2V∂y2+(Γ(2+α)−2t1+α)ex+y,0<x, y<1,0<t≤1, |
with initial and Dirichlet boundary conditions.
The analytic solution for Problem 2 is
V(x,y,t)=ex+yt1+α. |
The number of iterations, error analysis, and execution times are shown for the comparison between FEGM and SP methods from Tables 1–5. The execution times in FEGM are decreased by (4.7−28.49)%, (2−23)%, (9.16−28.13)%, (8.9−25.39)% and (6.98−27.79)% compared to SP method in Tables 1–5 respectively. Similarly, in Table 6, the comparison between the proposed method and high-order standard point method[27] is presented, which shows the proposed method gives better results. Figures 3 and 4 represent the exact and approximate solution for Problem 1, respectively, which depicts the effectiveness of the FEGM. Likewise, In Figure 5, the compression of execution times between the SP and proposed methods are shown, which shows the proposed method is efficient in terms of execution timings. Table 7 shows the computational complexity per iteration, while the computational effort is shown in Tables 8 and 9, which depict that the FEGM requires less number of operations during computations as compared to the standard SP method. Tables 10 and 11 represent the spatial convergence order for Problems 1 and 2, respectively. Similarly, Tables 12 and 13 represent the temporal convergence order for the first and second Problem respectively, which depict the experimental and theoretical convergence orders in agreement.
h0/ν | No. of iteration | Execution time | Maximum-error | Average-error | ||||
FEGM | SP | FEGM | SP | FEGM | SP | FEGM | SP | |
h0=ν=110 | 43 | 49 | 7.39 | 7.76 | 1.6404×10−4 | 1.653 ×10−4 | 7.9408 ×10−5 | 7.3891 ×10−5 |
h0=ν=118 | 41 | 48 | 66.19 | 79.82 | 6.6571 ×10−5 | 6.5544×10−5 | 2.8320 ×10−5 | 2.9081 ×10−5 |
h0=ν=122 | 42 | 54 | 148.1 | 207.1 | 4.5739 ×10−5 | 4.6717 ×10−5 | 1.9997 ×10−5 | 2.085 ×10−5 |
h0=ν=130 | 43 | 56 | 530.45 | 658.68 | 3.0559 ×10−5 | 3.2268 ×10−5 | 1.1887 ×10−5 | 1.2180 ×10−5 |
h0/ν | No. of iteration | Execution time | Maximum-error | Average-error | ||||
FEGM | SP | FEGM | SP | FEGM | SP | FEGM | SP | |
h0=ν=110 | 44 | 50 | 7.75 | 7.94 | 2.1534 ×10−4 | 2.2065 ×10−4 | 1.0469 ×10−4 | 1.0634 ×10−4 |
h0=ν=114 | 44 | 43 | 27.36 | 27.28 | 1.2736 ×10−4 | 1.2786×10−4 | 5.9175 ×10−5 | 6.0624 ×10−5 |
h0=ν=118 | 42 | 48 | 68.29 | 80.18 | 8.0887 ×10−5 | 8.5709 ×10−5 | 3.0710 ×10−5 | 3.9889×10−5 |
h0=ν=122 | 41 | 55 | 150.40 | 197.87 | 6.1207 ×10−5 | 6.2773 ×10−5 | 2.8554 ×10−5 | 2.8809 ×10−5 |
h0=ν=130 | 42 | 56 | 523.25 | 648.55 | 3.7373×10−5 | 4.4510 ×10−5 | 1.6460 ×10−5 | 1.7874 ×10−5 |
h0/ν | No. of iteration | Execution time | Maximum-error | Average-error | ||||
FEGM | SP | FEGM | SP | FEGM | SP | FEGM | SP | |
h0=ν=110 | 44 | 50 | 7.75 | 7.94 | 2.1534 ×10−4 | 2.2065 ×10−4 | 1.0469 ×10−4 | 1.0634 ×10−4 |
h0=ν=114 | 44 | 43 | 27.36 | 27.28 | 1.2736 ×10−4 | 1.2786×10−4 | 5.9175 ×10−5 | 6.0624 ×10−5 |
h0=ν=118 | 42 | 48 | 68.29 | 80.18 | 8.0887 ×10−5 | 8.5709 ×10−5 | 3.0710 ×10−5 | 3.9889×10−5 |
h0=ν=122 | 41 | 55 | 150.40 | 197.87 | 6.1207 ×10−5 | 6.2773 ×10−5 | 2.8554 ×10−5 | 2.8809 ×10−5 |
h0=ν=130 | 42 | 56 | 523.25 | 648.55 | 3.7373×10−5 | 4.4510 ×10−5 | 1.6460 ×10−5 | 1.7874 ×10−5 |
ν/h0 | No. of iteration | Execution time | Maximum-error | Average-error | ||||
FEGM | SP | FEGM | SP | FEGM | SP | FEGM | SP | |
h0=ν=110 | 46 | 52 | 7.65 | 8.72 | 2.9442 ×10−4 | 2.8915 ×10−4 | 1.3091 ×10−4 | 1.2849 ×10−4 |
h0=ν=114 | 48 | 47 | 30.93 | 29.53 | 1.3893 ×10−4 | 1.4377 ×10−4 | 3.7322 ×10−5 | 3.9455 ×10−5 |
h0=ν=118 | 48 | 52 | 77.2 | 84.75 | 8.5499 ×10−5 | 8.5075 ×10−5 | 2.4058 ×10−5 | 2.4340 ×10−5 |
h0=ν=122 | 47 | 57 | 172.53 | 203.76 | 5.5120 ×10−5 | 5.5018 ×10−5 | 1.9837 ×10−5 | 1.7813 ×10−5 |
h0=ν=130 | 47 | 65 | 588.74 | 789.14 | 2.9259 ×10−5 | 2.9669 ×10−5 | 1.2478 ×10−5 | 1.2549 ×10−5 |
h0/ν | No. of iteration | Execution time | Maximum-error | Average-error | ||||
FEGM | SP | FEGM | SP | FEGM | SP | FEGM | SP | |
h0=ν=110 | 47 | 53 | 7.92 | 8.82 | 2.2868 ×10−4 | 2.2917 ×10−4 | 1.0194 ×10−4 | 1.0189 ×10−4 |
h0=ν=114 | 48 | 48 | 31.23 | 30.46 | 2.3640 ×10−4 | 2.3848 ×10−4 | 1.1941 ×10−4 | 1.2003 ×10−4 |
h0=ν=118 | 49 | 52 | 78.34 | 85.3 | 2.1361 ×10−4 | 2.1290 ×10−4 | 1.0621 ×10−4 | 1.0863 ×10−4 |
h0=ν=122 | 48 | 57 | 169.15 | 207.6 | 1.9686 ×10−4 | 1.8908 ×10−4 | 9.5981 ×10−5 | 9.5644 ×10−5 |
h0=ν=130 | 48 | 65 | 599.97 | 775.49 | 1.5244 ×10−4 | 1.4938 ×10−4 | 7.3782 ×10−5 | 7.3922 ×10−5 |
h0/ν | No. of iteration | Maximum-error | Average-error | |||
FEGM | [27] | FEGM | [27] | FEGM | [27] | |
h0=ν=110 | 43 | 53 | 1.6404 ×10−4 | 1.2428 ×10−2 | 7.9408 ×10−5 | 8.8490 ×10−3 |
h0=ν=118 | 41 | 52 | 6.6571 ×10−5 | 7.1213 ×10−3 | 2.8320 ×10−5 | 3.6917 ×10−3 |
h0=ν=122 | 42 | 55 | 4.7739 ×10−5 | 2.6959 ×10−3 | 1.9997 ×10−5 | 1.3580 ×10−3 |
h0=165,ν=135 | 41 | 58 | 2.5368 ×10−5 | 2.0605 ×10−3 | 5.0090 ×10−6 | 1.0285 ×10−3 |
Technique | operations per iteration | |
+/- | */ ÷ | |
SP | (26+8(k−1))m2 | (8+4(k−1))m2 |
FEGM | (28+8(k−1))(m−1)2 + (26+8(k−1))(2m−1) | (12+4(k−1))(m−1)2 + (8+4(k−1))(2m−1) |
h0/ν | SP method | FEGM | ||
Number of iteration | Total operations | Number of iteration | Total operations | |
h0=ν=110 | 49 | 695800 | 43 | 631498 |
h0=ν=118 | 48 | 3701376 | 41 | 3232686 |
h0=ν=122 | 54 | 7474896 | 42 | 5924940 |
h0=ν=130 | 56 | 19252800 | 43 | 15000378 |
k/m | SP method | FEGM | ||
Number of iteration | Total operations | Number of iteration | Total operations | |
h0=ν=110 | 53 | 752600 | 47 | 690242 |
h0=ν=118 | 52 | 4009824 | 49 | 3863454 |
h0=ν=122 | 57 | 7890168 | 48 | 6771360 |
h0=ν=130 | 65 | 22347000 | 48 | 16744608 |
α=0.4 | α=0.5 | ||||
ν/h0 | Maximum error | C2-order | ν/h0 | Maximum error | C2-order |
ν=h0=0.5 | 1.4394 ×10−4 | — | ν=h0=0.5 | 3.1088 ×10−4 | — |
ν=0.031, h0=0.25 | 1.0910 ×10−5 | 3.72 | ν=0.031, h0=0.25 | 2.2509 ×10−5 | 3.78 |
ν=h0=0.25 | 2.1929 ×10−4 | — | ν=h0=0.25 | 3.546 ×10−4 | — |
ν=0.016, h0=0.12 | 9.1371 ×10−6 | 4.58 | ν=0.016, h0=0.12 | 2.1414 ×10−5 | 4.04 |
α=0.6 | α=0.8 | ||||
ν/h0 | Maximum error | C2-order | ν/h0 | Maximum error | C2-order |
ν=h0=0.5 | 5.2435 ×10−4 | — | ν=h0=0.5 | 9.9004 ×10−4 | — |
ν=0.031, h0=0.25 | 2.7191 ×10−5 | 4.26 | ν=0.031, h0=0.25 | 9.9004 ×10−5 | 4.13 |
ν=h0=0.25 | 5.0421 ×10−4 | — | ν=h0=0.25 | 7.2673 ×10−4 | — |
ν=0.016, h0=0.12 | 3.1980 ×10−5 | 3.97 | ν=0.016, h0=0.12 | 3.7243 ×10−5 | 4.28 |
α=0.7 | α=0.8 | ||||
ν/h0 | Maximum error | C2-order | ν/h0 | Maximum error | C2-order |
ν=h0=0.5 | 1.6072 ×10−3 | — | ν=h0=0.5 | 3.4202 ×10−3 | — |
ν=0.031, h0=0.25 | 1.0543 ×10−4 | 3.93 | ν=0.031, h0=0.25 | 1.7636 ×10−4 | 4.27 |
ν=h0=0.25 | 1.3545 ×10−3 | — | ν=h0=0.25 | 1.6955 ×10−3 | — |
ν=0.016, h0=0.12 | 5.9784 ×10−5 | 4.50 | ν=0.016, h0=0.12 | 8.2806 ×10−5 | 4.35 |
α=0.3 | α=0.5 | ||||
ν/h0 | Maximum error | C2-order | ν/h0 | Maximum error | C2-order |
ν=h0=0.5 | 3.832 ×10−3 | — | ν=h0=0.5 | 8.6771 ×10−4 | — |
ν=0.031, h0=0.25 | 3.2105 ×10−4 | 3.57 | ν=0.031, h0=0.25 | 4.8772 ×10−5 | 4.15 |
ν=h0=0.25 | 4.5916 ×10−4 | — | ν=h0=0.25 | 4.1734 ×10−4 | — |
ν=0.016, h0=0.12 | 2.5978 ×10−5 | 4.14 | ν=0.016, h0=0.12 | 2.3009 ×10−5 | 4.18 |
ν | α=0.3 | α=0.8 | ||
L∞ | C1− Order | L∞ | C1− Order | |
ν=110 | 7.8675×10−4 | — | 2.1609×10−4 | — |
ν=120 | 3.2208 ×10−5 | 1.28 | 7.1112 ×10−5 | 1.60 |
ν= 140 | 1.7253 ×10−5 | 1.29 | 2.6203 ×10−5 | 1.44 |
ν=180 | 6.2057×10−6 | 1.72 | 8.3392×10−6 | 1.65 |
ν | α=0.1 | α=0.8 | ||
L∞ | C1− Order | L∞ | C1− Order | |
ν=110 | 4.7691×10−4 | — | 2.1429×10−4 | — |
ν=120 | 1.6086 ×10−4 | 1.56 | 8.2679 ×10−5 | 1.37 |
ν= 140 | 5.1170 ×10−5 | 1.65 | 3.2827 ×10−5 | 1.33 |
ν=180 | 1.7848×10−5 | 1.51 | 1.3010×10−5 | 1.51 |
In this article, the 2-D fractional sub-diffusion equation is solved using the fractional explicit group method with weak singularity at initial time t=0, where the standard point finite difference scheme is used for the development of the fourth-order grouping scheme. The fractional explicit group method reduces the computational complexity and execution time by comparing it with the standard point fourth-order method without deteriorating the accuracy of the solutions. Furthermore, the unconditional stability and convergence of the proposed scheme are proved using the matrix analysis via mathematical induction, which confirms the feasibility and reliability of the new formulation.
This work was supported by the Ministry of Higher Education under Fundamental Research Grant Scheme (FRGS/1/2019/STG06/UTM/02/13, FRGS/1/2020/STG05/UTM/02/12).
We declare no conflicts of interest in this paper.
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1. | Rasel Islam, M. Abul Kawser, M. Sohel Rana, M. Nurul Islam, Mathematical analysis of soliton solutions in space-time fractional Klein-Gordon model with generalized exponential rational function method, 2024, 12, 26668181, 100942, 10.1016/j.padiff.2024.100942 | |
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3. | Jalu A. Prakosa, Norma Alias, Chifayah Astuti, 2023, Performance comparison of applying integer and fractional order calculus to DC motor speed control experiments, 979-8-3503-4389-2, 79, 10.1109/ICRAMET60171.2023.10366595 | |
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5. | Waleed Hamali, Abdulah A. Alghamdi, Exact solutions to the fractional nonlinear phenomena in fluid dynamics via the Riccati-Bernoulli sub-ODE method, 2024, 9, 2473-6988, 31142, 10.3934/math.20241501 |
h0/ν | No. of iteration | Execution time | Maximum-error | Average-error | ||||
FEGM | SP | FEGM | SP | FEGM | SP | FEGM | SP | |
h0=ν=110 | 43 | 49 | 7.39 | 7.76 | 1.6404×10−4 | 1.653 ×10−4 | 7.9408 ×10−5 | 7.3891 ×10−5 |
h0=ν=118 | 41 | 48 | 66.19 | 79.82 | 6.6571 ×10−5 | 6.5544×10−5 | 2.8320 ×10−5 | 2.9081 ×10−5 |
h0=ν=122 | 42 | 54 | 148.1 | 207.1 | 4.5739 ×10−5 | 4.6717 ×10−5 | 1.9997 ×10−5 | 2.085 ×10−5 |
h0=ν=130 | 43 | 56 | 530.45 | 658.68 | 3.0559 ×10−5 | 3.2268 ×10−5 | 1.1887 ×10−5 | 1.2180 ×10−5 |
h0/ν | No. of iteration | Execution time | Maximum-error | Average-error | ||||
FEGM | SP | FEGM | SP | FEGM | SP | FEGM | SP | |
h0=ν=110 | 44 | 50 | 7.75 | 7.94 | 2.1534 ×10−4 | 2.2065 ×10−4 | 1.0469 ×10−4 | 1.0634 ×10−4 |
h0=ν=114 | 44 | 43 | 27.36 | 27.28 | 1.2736 ×10−4 | 1.2786×10−4 | 5.9175 ×10−5 | 6.0624 ×10−5 |
h0=ν=118 | 42 | 48 | 68.29 | 80.18 | 8.0887 ×10−5 | 8.5709 ×10−5 | 3.0710 ×10−5 | 3.9889×10−5 |
h0=ν=122 | 41 | 55 | 150.40 | 197.87 | 6.1207 ×10−5 | 6.2773 ×10−5 | 2.8554 ×10−5 | 2.8809 ×10−5 |
h0=ν=130 | 42 | 56 | 523.25 | 648.55 | 3.7373×10−5 | 4.4510 ×10−5 | 1.6460 ×10−5 | 1.7874 ×10−5 |
h0/ν | No. of iteration | Execution time | Maximum-error | Average-error | ||||
FEGM | SP | FEGM | SP | FEGM | SP | FEGM | SP | |
h0=ν=110 | 44 | 50 | 7.75 | 7.94 | 2.1534 ×10−4 | 2.2065 ×10−4 | 1.0469 ×10−4 | 1.0634 ×10−4 |
h0=ν=114 | 44 | 43 | 27.36 | 27.28 | 1.2736 ×10−4 | 1.2786×10−4 | 5.9175 ×10−5 | 6.0624 ×10−5 |
h0=ν=118 | 42 | 48 | 68.29 | 80.18 | 8.0887 ×10−5 | 8.5709 ×10−5 | 3.0710 ×10−5 | 3.9889×10−5 |
h0=ν=122 | 41 | 55 | 150.40 | 197.87 | 6.1207 ×10−5 | 6.2773 ×10−5 | 2.8554 ×10−5 | 2.8809 ×10−5 |
h0=ν=130 | 42 | 56 | 523.25 | 648.55 | 3.7373×10−5 | 4.4510 ×10−5 | 1.6460 ×10−5 | 1.7874 ×10−5 |
ν/h0 | No. of iteration | Execution time | Maximum-error | Average-error | ||||
FEGM | SP | FEGM | SP | FEGM | SP | FEGM | SP | |
h0=ν=110 | 46 | 52 | 7.65 | 8.72 | 2.9442 ×10−4 | 2.8915 ×10−4 | 1.3091 ×10−4 | 1.2849 ×10−4 |
h0=ν=114 | 48 | 47 | 30.93 | 29.53 | 1.3893 ×10−4 | 1.4377 ×10−4 | 3.7322 ×10−5 | 3.9455 ×10−5 |
h0=ν=118 | 48 | 52 | 77.2 | 84.75 | 8.5499 ×10−5 | 8.5075 ×10−5 | 2.4058 ×10−5 | 2.4340 ×10−5 |
h0=ν=122 | 47 | 57 | 172.53 | 203.76 | 5.5120 ×10−5 | 5.5018 ×10−5 | 1.9837 ×10−5 | 1.7813 ×10−5 |
h0=ν=130 | 47 | 65 | 588.74 | 789.14 | 2.9259 ×10−5 | 2.9669 ×10−5 | 1.2478 ×10−5 | 1.2549 ×10−5 |
h0/ν | No. of iteration | Execution time | Maximum-error | Average-error | ||||
FEGM | SP | FEGM | SP | FEGM | SP | FEGM | SP | |
h0=ν=110 | 47 | 53 | 7.92 | 8.82 | 2.2868 ×10−4 | 2.2917 ×10−4 | 1.0194 ×10−4 | 1.0189 ×10−4 |
h0=ν=114 | 48 | 48 | 31.23 | 30.46 | 2.3640 ×10−4 | 2.3848 ×10−4 | 1.1941 ×10−4 | 1.2003 ×10−4 |
h0=ν=118 | 49 | 52 | 78.34 | 85.3 | 2.1361 ×10−4 | 2.1290 ×10−4 | 1.0621 ×10−4 | 1.0863 ×10−4 |
h0=ν=122 | 48 | 57 | 169.15 | 207.6 | 1.9686 ×10−4 | 1.8908 ×10−4 | 9.5981 ×10−5 | 9.5644 ×10−5 |
h0=ν=130 | 48 | 65 | 599.97 | 775.49 | 1.5244 ×10−4 | 1.4938 ×10−4 | 7.3782 ×10−5 | 7.3922 ×10−5 |
h0/ν | No. of iteration | Maximum-error | Average-error | |||
FEGM | [27] | FEGM | [27] | FEGM | [27] | |
h0=ν=110 | 43 | 53 | 1.6404 ×10−4 | 1.2428 ×10−2 | 7.9408 ×10−5 | 8.8490 ×10−3 |
h0=ν=118 | 41 | 52 | 6.6571 ×10−5 | 7.1213 ×10−3 | 2.8320 ×10−5 | 3.6917 ×10−3 |
h0=ν=122 | 42 | 55 | 4.7739 ×10−5 | 2.6959 ×10−3 | 1.9997 ×10−5 | 1.3580 ×10−3 |
h0=165,ν=135 | 41 | 58 | 2.5368 ×10−5 | 2.0605 ×10−3 | 5.0090 ×10−6 | 1.0285 ×10−3 |
Technique | operations per iteration | |
+/- | */ ÷ | |
SP | (26+8(k−1))m2 | (8+4(k−1))m2 |
FEGM | (28+8(k−1))(m−1)2 + (26+8(k−1))(2m−1) | (12+4(k−1))(m−1)2 + (8+4(k−1))(2m−1) |
h0/ν | SP method | FEGM | ||
Number of iteration | Total operations | Number of iteration | Total operations | |
h0=ν=110 | 49 | 695800 | 43 | 631498 |
h0=ν=118 | 48 | 3701376 | 41 | 3232686 |
h0=ν=122 | 54 | 7474896 | 42 | 5924940 |
h0=ν=130 | 56 | 19252800 | 43 | 15000378 |
k/m | SP method | FEGM | ||
Number of iteration | Total operations | Number of iteration | Total operations | |
h0=ν=110 | 53 | 752600 | 47 | 690242 |
h0=ν=118 | 52 | 4009824 | 49 | 3863454 |
h0=ν=122 | 57 | 7890168 | 48 | 6771360 |
h0=ν=130 | 65 | 22347000 | 48 | 16744608 |
α=0.4 | α=0.5 | ||||
ν/h0 | Maximum error | C2-order | ν/h0 | Maximum error | C2-order |
ν=h0=0.5 | 1.4394 ×10−4 | — | ν=h0=0.5 | 3.1088 ×10−4 | — |
ν=0.031, h0=0.25 | 1.0910 ×10−5 | 3.72 | ν=0.031, h0=0.25 | 2.2509 ×10−5 | 3.78 |
ν=h0=0.25 | 2.1929 ×10−4 | — | ν=h0=0.25 | 3.546 ×10−4 | — |
ν=0.016, h0=0.12 | 9.1371 ×10−6 | 4.58 | ν=0.016, h0=0.12 | 2.1414 ×10−5 | 4.04 |
α=0.6 | α=0.8 | ||||
ν/h0 | Maximum error | C2-order | ν/h0 | Maximum error | C2-order |
ν=h0=0.5 | 5.2435 ×10−4 | — | ν=h0=0.5 | 9.9004 ×10−4 | — |
ν=0.031, h0=0.25 | 2.7191 ×10−5 | 4.26 | ν=0.031, h0=0.25 | 9.9004 ×10−5 | 4.13 |
ν=h0=0.25 | 5.0421 ×10−4 | — | ν=h0=0.25 | 7.2673 ×10−4 | — |
ν=0.016, h0=0.12 | 3.1980 ×10−5 | 3.97 | ν=0.016, h0=0.12 | 3.7243 ×10−5 | 4.28 |
α=0.7 | α=0.8 | ||||
ν/h0 | Maximum error | C2-order | ν/h0 | Maximum error | C2-order |
ν=h0=0.5 | 1.6072 ×10−3 | — | ν=h0=0.5 | 3.4202 ×10−3 | — |
ν=0.031, h0=0.25 | 1.0543 ×10−4 | 3.93 | ν=0.031, h0=0.25 | 1.7636 ×10−4 | 4.27 |
ν=h0=0.25 | 1.3545 ×10−3 | — | ν=h0=0.25 | 1.6955 ×10−3 | — |
ν=0.016, h0=0.12 | 5.9784 ×10−5 | 4.50 | ν=0.016, h0=0.12 | 8.2806 ×10−5 | 4.35 |
α=0.3 | α=0.5 | ||||
ν/h0 | Maximum error | C2-order | ν/h0 | Maximum error | C2-order |
ν=h0=0.5 | 3.832 ×10−3 | — | ν=h0=0.5 | 8.6771 ×10−4 | — |
ν=0.031, h0=0.25 | 3.2105 ×10−4 | 3.57 | ν=0.031, h0=0.25 | 4.8772 ×10−5 | 4.15 |
ν=h0=0.25 | 4.5916 ×10−4 | — | ν=h0=0.25 | 4.1734 ×10−4 | — |
ν=0.016, h0=0.12 | 2.5978 ×10−5 | 4.14 | ν=0.016, h0=0.12 | 2.3009 ×10−5 | 4.18 |
ν | α=0.3 | α=0.8 | ||
L∞ | C1− Order | L∞ | C1− Order | |
ν=110 | 7.8675×10−4 | — | 2.1609×10−4 | — |
ν=120 | 3.2208 ×10−5 | 1.28 | 7.1112 ×10−5 | 1.60 |
ν= 140 | 1.7253 ×10−5 | 1.29 | 2.6203 ×10−5 | 1.44 |
ν=180 | 6.2057×10−6 | 1.72 | 8.3392×10−6 | 1.65 |
ν | α=0.1 | α=0.8 | ||
L∞ | C1− Order | L∞ | C1− Order | |
ν=110 | 4.7691×10−4 | — | 2.1429×10−4 | — |
ν=120 | 1.6086 ×10−4 | 1.56 | 8.2679 ×10−5 | 1.37 |
ν= 140 | 5.1170 ×10−5 | 1.65 | 3.2827 ×10−5 | 1.33 |
ν=180 | 1.7848×10−5 | 1.51 | 1.3010×10−5 | 1.51 |
h0/ν | No. of iteration | Execution time | Maximum-error | Average-error | ||||
FEGM | SP | FEGM | SP | FEGM | SP | FEGM | SP | |
h0=ν=110 | 43 | 49 | 7.39 | 7.76 | 1.6404×10−4 | 1.653 ×10−4 | 7.9408 ×10−5 | 7.3891 ×10−5 |
h0=ν=118 | 41 | 48 | 66.19 | 79.82 | 6.6571 ×10−5 | 6.5544×10−5 | 2.8320 ×10−5 | 2.9081 ×10−5 |
h0=ν=122 | 42 | 54 | 148.1 | 207.1 | 4.5739 ×10−5 | 4.6717 ×10−5 | 1.9997 ×10−5 | 2.085 ×10−5 |
h0=ν=130 | 43 | 56 | 530.45 | 658.68 | 3.0559 ×10−5 | 3.2268 ×10−5 | 1.1887 ×10−5 | 1.2180 ×10−5 |
h0/ν | No. of iteration | Execution time | Maximum-error | Average-error | ||||
FEGM | SP | FEGM | SP | FEGM | SP | FEGM | SP | |
h0=ν=110 | 44 | 50 | 7.75 | 7.94 | 2.1534 ×10−4 | 2.2065 ×10−4 | 1.0469 ×10−4 | 1.0634 ×10−4 |
h0=ν=114 | 44 | 43 | 27.36 | 27.28 | 1.2736 ×10−4 | 1.2786×10−4 | 5.9175 ×10−5 | 6.0624 ×10−5 |
h0=ν=118 | 42 | 48 | 68.29 | 80.18 | 8.0887 ×10−5 | 8.5709 ×10−5 | 3.0710 ×10−5 | 3.9889×10−5 |
h0=ν=122 | 41 | 55 | 150.40 | 197.87 | 6.1207 ×10−5 | 6.2773 ×10−5 | 2.8554 ×10−5 | 2.8809 ×10−5 |
h0=ν=130 | 42 | 56 | 523.25 | 648.55 | 3.7373×10−5 | 4.4510 ×10−5 | 1.6460 ×10−5 | 1.7874 ×10−5 |
h0/ν | No. of iteration | Execution time | Maximum-error | Average-error | ||||
FEGM | SP | FEGM | SP | FEGM | SP | FEGM | SP | |
h0=ν=110 | 44 | 50 | 7.75 | 7.94 | 2.1534 ×10−4 | 2.2065 ×10−4 | 1.0469 ×10−4 | 1.0634 ×10−4 |
h0=ν=114 | 44 | 43 | 27.36 | 27.28 | 1.2736 ×10−4 | 1.2786×10−4 | 5.9175 ×10−5 | 6.0624 ×10−5 |
h0=ν=118 | 42 | 48 | 68.29 | 80.18 | 8.0887 ×10−5 | 8.5709 ×10−5 | 3.0710 ×10−5 | 3.9889×10−5 |
h0=ν=122 | 41 | 55 | 150.40 | 197.87 | 6.1207 ×10−5 | 6.2773 ×10−5 | 2.8554 ×10−5 | 2.8809 ×10−5 |
h0=ν=130 | 42 | 56 | 523.25 | 648.55 | 3.7373×10−5 | 4.4510 ×10−5 | 1.6460 ×10−5 | 1.7874 ×10−5 |
ν/h0 | No. of iteration | Execution time | Maximum-error | Average-error | ||||
FEGM | SP | FEGM | SP | FEGM | SP | FEGM | SP | |
h0=ν=110 | 46 | 52 | 7.65 | 8.72 | 2.9442 ×10−4 | 2.8915 ×10−4 | 1.3091 ×10−4 | 1.2849 ×10−4 |
h0=ν=114 | 48 | 47 | 30.93 | 29.53 | 1.3893 ×10−4 | 1.4377 ×10−4 | 3.7322 ×10−5 | 3.9455 ×10−5 |
h0=ν=118 | 48 | 52 | 77.2 | 84.75 | 8.5499 ×10−5 | 8.5075 ×10−5 | 2.4058 ×10−5 | 2.4340 ×10−5 |
h0=ν=122 | 47 | 57 | 172.53 | 203.76 | 5.5120 ×10−5 | 5.5018 ×10−5 | 1.9837 ×10−5 | 1.7813 ×10−5 |
h0=ν=130 | 47 | 65 | 588.74 | 789.14 | 2.9259 ×10−5 | 2.9669 ×10−5 | 1.2478 ×10−5 | 1.2549 ×10−5 |
h0/ν | No. of iteration | Execution time | Maximum-error | Average-error | ||||
FEGM | SP | FEGM | SP | FEGM | SP | FEGM | SP | |
h0=ν=110 | 47 | 53 | 7.92 | 8.82 | 2.2868 ×10−4 | 2.2917 ×10−4 | 1.0194 ×10−4 | 1.0189 ×10−4 |
h0=ν=114 | 48 | 48 | 31.23 | 30.46 | 2.3640 ×10−4 | 2.3848 ×10−4 | 1.1941 ×10−4 | 1.2003 ×10−4 |
h0=ν=118 | 49 | 52 | 78.34 | 85.3 | 2.1361 ×10−4 | 2.1290 ×10−4 | 1.0621 ×10−4 | 1.0863 ×10−4 |
h0=ν=122 | 48 | 57 | 169.15 | 207.6 | 1.9686 ×10−4 | 1.8908 ×10−4 | 9.5981 ×10−5 | 9.5644 ×10−5 |
h0=ν=130 | 48 | 65 | 599.97 | 775.49 | 1.5244 ×10−4 | 1.4938 ×10−4 | 7.3782 ×10−5 | 7.3922 ×10−5 |
h0/ν | No. of iteration | Maximum-error | Average-error | |||
FEGM | [27] | FEGM | [27] | FEGM | [27] | |
h0=ν=110 | 43 | 53 | 1.6404 ×10−4 | 1.2428 ×10−2 | 7.9408 ×10−5 | 8.8490 ×10−3 |
h0=ν=118 | 41 | 52 | 6.6571 ×10−5 | 7.1213 ×10−3 | 2.8320 ×10−5 | 3.6917 ×10−3 |
h0=ν=122 | 42 | 55 | 4.7739 ×10−5 | 2.6959 ×10−3 | 1.9997 ×10−5 | 1.3580 ×10−3 |
h0=165,ν=135 | 41 | 58 | 2.5368 ×10−5 | 2.0605 ×10−3 | 5.0090 ×10−6 | 1.0285 ×10−3 |
Technique | operations per iteration | |
+/- | */ ÷ | |
SP | (26+8(k−1))m2 | (8+4(k−1))m2 |
FEGM | (28+8(k−1))(m−1)2 + (26+8(k−1))(2m−1) | (12+4(k−1))(m−1)2 + (8+4(k−1))(2m−1) |
h0/ν | SP method | FEGM | ||
Number of iteration | Total operations | Number of iteration | Total operations | |
h0=ν=110 | 49 | 695800 | 43 | 631498 |
h0=ν=118 | 48 | 3701376 | 41 | 3232686 |
h0=ν=122 | 54 | 7474896 | 42 | 5924940 |
h0=ν=130 | 56 | 19252800 | 43 | 15000378 |
k/m | SP method | FEGM | ||
Number of iteration | Total operations | Number of iteration | Total operations | |
h0=ν=110 | 53 | 752600 | 47 | 690242 |
h0=ν=118 | 52 | 4009824 | 49 | 3863454 |
h0=ν=122 | 57 | 7890168 | 48 | 6771360 |
h0=ν=130 | 65 | 22347000 | 48 | 16744608 |
α=0.4 | α=0.5 | ||||
ν/h0 | Maximum error | C2-order | ν/h0 | Maximum error | C2-order |
ν=h0=0.5 | 1.4394 ×10−4 | — | ν=h0=0.5 | 3.1088 ×10−4 | — |
ν=0.031, h0=0.25 | 1.0910 ×10−5 | 3.72 | ν=0.031, h0=0.25 | 2.2509 ×10−5 | 3.78 |
ν=h0=0.25 | 2.1929 ×10−4 | — | ν=h0=0.25 | 3.546 ×10−4 | — |
ν=0.016, h0=0.12 | 9.1371 ×10−6 | 4.58 | ν=0.016, h0=0.12 | 2.1414 ×10−5 | 4.04 |
α=0.6 | α=0.8 | ||||
ν/h0 | Maximum error | C2-order | ν/h0 | Maximum error | C2-order |
ν=h0=0.5 | 5.2435 ×10−4 | — | ν=h0=0.5 | 9.9004 ×10−4 | — |
ν=0.031, h0=0.25 | 2.7191 ×10−5 | 4.26 | ν=0.031, h0=0.25 | 9.9004 ×10−5 | 4.13 |
ν=h0=0.25 | 5.0421 ×10−4 | — | ν=h0=0.25 | 7.2673 ×10−4 | — |
ν=0.016, h0=0.12 | 3.1980 ×10−5 | 3.97 | ν=0.016, h0=0.12 | 3.7243 ×10−5 | 4.28 |
α=0.7 | α=0.8 | ||||
ν/h0 | Maximum error | C2-order | ν/h0 | Maximum error | C2-order |
ν=h0=0.5 | 1.6072 ×10−3 | — | ν=h0=0.5 | 3.4202 ×10−3 | — |
ν=0.031, h0=0.25 | 1.0543 ×10−4 | 3.93 | ν=0.031, h0=0.25 | 1.7636 ×10−4 | 4.27 |
ν=h0=0.25 | 1.3545 ×10−3 | — | ν=h0=0.25 | 1.6955 ×10−3 | — |
ν=0.016, h0=0.12 | 5.9784 ×10−5 | 4.50 | ν=0.016, h0=0.12 | 8.2806 ×10−5 | 4.35 |
α=0.3 | α=0.5 | ||||
ν/h0 | Maximum error | C2-order | ν/h0 | Maximum error | C2-order |
ν=h0=0.5 | 3.832 ×10−3 | — | ν=h0=0.5 | 8.6771 ×10−4 | — |
ν=0.031, h0=0.25 | 3.2105 ×10−4 | 3.57 | ν=0.031, h0=0.25 | 4.8772 ×10−5 | 4.15 |
ν=h0=0.25 | 4.5916 ×10−4 | — | ν=h0=0.25 | 4.1734 ×10−4 | — |
ν=0.016, h0=0.12 | 2.5978 ×10−5 | 4.14 | ν=0.016, h0=0.12 | 2.3009 ×10−5 | 4.18 |
ν | α=0.3 | α=0.8 | ||
L∞ | C1− Order | L∞ | C1− Order | |
ν=110 | 7.8675×10−4 | — | 2.1609×10−4 | — |
ν=120 | 3.2208 ×10−5 | 1.28 | 7.1112 ×10−5 | 1.60 |
ν= 140 | 1.7253 ×10−5 | 1.29 | 2.6203 ×10−5 | 1.44 |
ν=180 | 6.2057×10−6 | 1.72 | 8.3392×10−6 | 1.65 |
ν | α=0.1 | α=0.8 | ||
L∞ | C1− Order | L∞ | C1− Order | |
ν=110 | 4.7691×10−4 | — | 2.1429×10−4 | — |
ν=120 | 1.6086 ×10−4 | 1.56 | 8.2679 ×10−5 | 1.37 |
ν= 140 | 5.1170 ×10−5 | 1.65 | 3.2827 ×10−5 | 1.33 |
ν=180 | 1.7848×10−5 | 1.51 | 1.3010×10−5 | 1.51 |