Processing math: 100%
Research article

A new fourth-order grouping iterative method for the time fractional sub-diffusion equation having a weak singularity at initial time

  • Received: 09 October 2022 Revised: 25 January 2023 Accepted: 07 February 2023 Published: 10 April 2023
  • MSC : 35R11, 65N06

  • 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

    Related Papers:

    [1] Emadidin Gahalla Mohmed Elmahdi, Jianfei Huang . Two linearized finite difference schemes for time fractional nonlinear diffusion-wave equations with fourth order derivative. AIMS Mathematics, 2021, 6(6): 6356-6376. doi: 10.3934/math.2021373
    [2] Fouad Mohammad Salama, Faisal Fairag . On numerical solution of two-dimensional variable-order fractional diffusion equation arising in transport phenomena. AIMS Mathematics, 2024, 9(1): 340-370. doi: 10.3934/math.2024020
    [3] Ajmal Ali, Tayyaba Akram, Azhar Iqbal, Poom Kumam, Thana Sutthibutpong . A numerical approach for 2D time-fractional diffusion damped wave model. AIMS Mathematics, 2023, 8(4): 8249-8273. doi: 10.3934/math.2023416
    [4] Junying Cao, Zhongqing Wang, Ziqiang Wang . Stability and convergence analysis for a uniform temporal high accuracy of the time-fractional diffusion equation with 1D and 2D spatial compact finite difference method. AIMS Mathematics, 2024, 9(6): 14697-14730. doi: 10.3934/math.2024715
    [5] Abdul-Majeed Ayebire, Saroj Sahani, Priyanka, Shelly Arora . Numerical study of soliton behavior of generalised Kuramoto-Sivashinsky type equations with Hermite splines. AIMS Mathematics, 2025, 10(2): 2098-2130. doi: 10.3934/math.2025099
    [6] Yanjie Zhou, Xianxiang Leng, Yuejie Li, Qiuxiang Deng, Zhendong Luo . A novel two-grid Crank-Nicolson mixed finite element method for nonlinear fourth-order sin-Gordon equation. AIMS Mathematics, 2024, 9(11): 31470-31494. doi: 10.3934/math.20241515
    [7] Fouad Mohammad Salama, Nur Nadiah Abd Hamid, Norhashidah Hj. Mohd Ali, Umair Ali . An efficient modified hybrid explicit group iterative method for the time-fractional diffusion equation in two space dimensions. AIMS Mathematics, 2022, 7(2): 2370-2392. doi: 10.3934/math.2022134
    [8] Ailing Zhu, Yixin Wang, Qiang Xu . A weak Galerkin finite element approximation of two-dimensional sub-diffusion equation with time-fractional derivative. AIMS Mathematics, 2020, 5(5): 4297-4310. doi: 10.3934/math.2020274
    [9] Fouad Mohammad Salama, Nur Nadiah Abd Hamid, Umair Ali, Norhashidah Hj. Mohd Ali . Fast hybrid explicit group methods for solving 2D fractional advection-diffusion equation. AIMS Mathematics, 2022, 7(9): 15854-15880. doi: 10.3934/math.2022868
    [10] Din Prathumwan, Thipsuda Khonwai, Narisara Phoochalong, Inthira Chaiya, Kamonchat Trachoo . An improved approximate method for solving two-dimensional time-fractional-order Black-Scholes model: a finite difference approach. AIMS Mathematics, 2024, 9(7): 17205-17233. doi: 10.3934/math.2024836
  • 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:

    αVtα=2Vx2+2Vy2+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),0x,yM,0tN,

    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(1α)t0f(x)(tx)αdx. (1.2)

    The Caputo derivative approximated using the L1 gives the accuracy 2α [9,10], but the presence of kernel (tx)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[2α2,2α]+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,j2Vki,j+Vki1,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)
    Vki1,j=Vki,jh01!Vx|ki,j+h202!Vxx|ki,jh303!Vxxx|ki,j+... (2.2)

    By adding Eqs (2.1) and (2.2) and after rearranging, we get

    2Vki,jx2=Vki+1,j2uki,j+Vki1,jh20+2h204!4Vki,jx4+...=Vki+1,j2Vki,j+Vki1,jh20+O(h20). (2.3)

    Therefore, the Taylor series expansions at points uki+1,j and uki,j+1 are

    δ2xh20Vki,j=2Vx2|ki,jh20124Vx4|ki,jh403606Vx6|ki,j+O(h60), (2.4)
    δ2yh20Vki,j=2Vy2|ki,jh20124Vy4|ki,j+h403606Vy6|ki,j+O(h60). (2.5)

    The difference operator δ2x, which maintain the three-point stencil is given by []

    δ2xh20(1+112δ2x)Vki,j=2Vx2|ki,jh402404Vx4|ki,j+O(h60), (2.6)

    and

    δ2yh20(1+112δ2y)Vki,j=2Vy2|ki,jh402404Vy4|ki,j+O(h60). (2.7)

    The fractional derivative is approximated using the Central difference formula as [24]

    αtαV(x,y,t)|k+12i,j=1Γ(1α)tk+120Vt(x,y,ε)(tk+12ε)αε=1Γ(1α)[tk+120Vt(x,y,ε)((k+12)νε)αε+ttk+12tk(Vk+1i,jVki,jν+O(ν))((k+12)νε)αε]=1Γ(1α)ks0=1[sν(s01)νVs0i,jVs01i,jν+(εts012)Vtt(xi,yj,cs0)×((k+12νε)αε)]+(k+12)νkν(Vk+1i,jVki,jν+O(ν))((k+12)νε)αε=1Γ(1α)ks0=1Vs0i,jVs01i,jνsν(s01)ν((k+12)νε)αε+1Γ(1α)ks0=1sν(s1)ν(εts012)Vtt(xi,yj,cs0)((k+12)νε)αε+1Γ(1α)[Vk+1i,jVki,jν+O(ν)](k+12)νkν[((k+12)νε)αε]ε=1να(1α)Γ(1α)ks0=1[Vs0i,jVs01i,j][(ks0+32)1α(ks0+12)1α]+1να(1α)Γ(1α)(Vk+1i,jVki,j)121α+1Γ(1α)ks0=1sν(s1)ν(εts012)Vtt(xi,yj,cs0)((k+12)νε)αε+1Γ(1α)(1α)21αO(ν)2α.

    Therefore, after some simplifications, the Crank-Nicolson (C-N) Caputo fractional derivative

    αtαV(x,y,t)|k+12i,j=a1V|ki,j+k1s=1(aks+1aks)V|si,jakV|0i,j+σVk+1i,j+Vki,j21α+O(ν2α),σ=1ναΓ(2α),as=σ((s+12)1α(s12)1α),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+k1s=1(aks+1aks)Vsi,jakV0i,j+σVk+1i,j+Vki,j21α=(1+112δ2x)1×δ2xh20Vk+12i,j+(1+112δ2y)1δ2yh20Vk+12i,j+fk+12i,j+O(ν2α+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+1i1,j+Vk+1i,j+1+Vk+1i,j1)+λ3(Vk+1i+1,j+1+Vk+1i1,j+1+Vk+1i+1,j1+Vk+1i1,j1)+λ4Vki,j+λ5(Vki+1,j+Vki1,j+Vki,j+1+Vki,j1)+λ6(Vki+1,j+1+Vki1,j+1+Vki+1,j1+Vki1,j1)+2518h20fk+12i,j+536h20(fk+1ti+1,j+fk+1ti1,j+fk+1ti,j+1+fk+1ti,j1)+h2072(fk+12i+1,j+1+fk+12i1,j+1+fk+12i+1,j1+fk+12i1,j1)k1s=1(aks+1aks)(2518h20Vsi,j+536h20(Vsi+1,j+Vsi1,j+Vsi,j+1+Vsi,j1)+h2072(Vsi+1,j+1+Vsi1,j+1+Vsi+1,j1+Vsi1,j1))+O(ν2α+h40), (2.10)

    where

    g0=σ21α,g1=a1g0,λ1=172(240+100h20g0),λ2=172(4810h20g0),λ3=172(12h20g0),λ4=172(240+100h20g1),λ5=172(4810h20g1),λ6=172(12h20g1).

    Now, using Eq (2.10) will give the following system for the group of four points:

    [λ1λ2λ3λ2λ2λ1λ2λ3λ3λ2λ1λ2λ2λ3λ2λ1][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+1i1,j+Vk+1i,j1)+λ3(Vk+1i1,j+1+Vk+1i+1,j1+Vk+1i1,j1)+λ4Vki,j+λ5(Vki+1,j+Vki1,j+Vki,j+1+Vki,j1)+λ6(Vki+1,j+1+Vki1,j+1+Vki+1,j1+Vki1,j1)+2518h20fk+12i,j+536h20(fk+12i+1,j+fk+12i1,j+fk+12i,j+1+fk+12i,j1)+h2072(fk+12i+1,j+1+fk+12i1,j+1+fk+12i+1,j1+fk+12i1,j1)Fi,j,
    rhsi+1,j=λ2(Vk+1i+2,j+Vk+1i+1,j1)+λ3(Vk+1i+2,j+1+Vk+1i+2,j1+Vk+1i,j1)+λ4Vki+1,j+λ5(Vki+2,j+Vki,j+Vki+1,j+1+Vki+1,j1)+λ6(Vki+2,j+1+Vki,j+1+Vki+2,j1+Vki,j1)+2518h20fk+12i+1,j+536h20(fk+12i+2,j+fk+12i,j+fk+12i+1,j+1+fk+12i+1,j1)+h2072(fk+12i+2,j+1+fk+12i,j+1+fk+12i+2,j1+fk+12i,j1)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+1i1,j+1+Vk+1i,j+2)+λ3(Vk+1i+1,j+2+Vk+1i1,j+2+Vk+1i1,j)+λ4Vki,j+1+λ5(Vki+1,j+1+Vki1,j+1+Vki,j+2+Vki,j)+λ6(Vki+1,j+2+Vki1,j+2+Vki+1,j+Vki1,j)+2518h20fk+12i,j+1+536h20(fk+12i+1,j+1+fk+12i1,j+1+fk+12i,j+2+fk+12i,j)+h2072(fk+12i+1,j+2+fk+12i1,j+2+fk+12i+1,j+fk+12i1,j)Fi,j+1,

    and

    Fi,j=k1s=1(aks+1aks)(2518h20Vsi,j+536h20(Vsi+1,j+Vsi1,j+Vsi,j+1+Vsi,j1)+h2072(Vsi+1,j+1+Vsi1,j+1+Vsi+1,j1+Vsi1,j1)).

    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=λ312λ1λ222λ22λ3λ1λ23,ϕ2=λ21λ2+2λ1λ2λ3+λ2λ23,ϕ3=2λ1λ22+λ21λ3+2λ22λ3λ33,d=(4λ22+(λ1λ3)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.

    Figure 1.  Four points in computation of grouping method.
    Figure 2.  Different points in the FEGM at time levels k+1,k,k1,...,1 with mesh size m=9.

    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=BVkh20k1s=1(aks+1aks)CVs+h20fk+12,k>0, (3.1)

    where

    A=[R1R30R2R1R3R2R1R30R2R1],B=[P1P30P2P1P3P2P1P30P2P1],
    R1=[G1G3G2G1G3G2G1G3G2G1],C=[Q1Q30Q2Q1Q3Q2Q1Q30Q2Q1],f=[K1K1K1K1],
    R2=[G6G4G8G6G4G8G6G4G8G6],R3=[G7G9G5G7G9G5G7G9G5G7],
    P1=[H1H3H2H1H3H2H1H3H2H1],P2=[H6H4H8H6H4H8H6H4H8H6],
    P3=[H7H9H5H7H9H5H7H9H5H7],Q1=[L1L3L2L1L3L2L1L3L2L1],
    Q2=[L6L4L8L6L4L8L6L4L8L6],Q3=[L7L9L5L7L9L5L7L9L5L7],K1=[W1W1W1W1],
    G1=[λ1λ2λ3λ2λ2λ1λ2λ3λ3λ2λ1λ2λ2λ3λ2λ1],G2=[00λ3λ200λ2λ300000000],G3=[00000000λ3λ200λ2λ300],
    G4=[0000000000000λ300],G5=[0000000λ300000000],G6=[0λ2λ30000000000λ3λ20],
    G7=[0000λ200λ3λ300λ20000],G8=[00λ30000000000000],G9=[00000000λ30000000],
    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(N2)2×(N2)2]Vk+1=[B(N2)2×(N2)2]Vkh20k1s=1(aks+1aks)[C(N2)2×(N2)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,jVki,j. So, from Eq (3.1),

    AE1=BE0,k=0,AEk+1=BEkh20k1s=1(aks+1aks)CEs,k>0, (3.2)

    where

    Ek+1=[Ek+11Ek+12Ek+1m2Ek+1m1],Ek+1i=[ϵk+11ϵk+12ϵk+1m2ϵk+1m1],ϵk+1i=[ϵk+1i,jϵk+1i+1,jϵk+1i+1,j+1ϵk+1i,j+1],i,j=1,2,...,m1.

    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=A1BE0,E1A1BE0=132+121h2g1348+79h2g0E0.

    But since a1=σ((32)1α(12)1α)=g0(31α1) and g1=a1g0=g0(31α2). Also we know that 31α<3, so,

    31α2<1,g0(31α2)<g0,g0>0,g1<g0.

    Hence,

    E1E0,g0>g1.

    Suppose

    ErE0,r=2,3,...,k, (3.7)

    and for r=k+1,

    Ek+1=A1(BEkh20k1s=1(aks+1aks)CEs))A1BEk+h20k1s=1(aks+1aks)A1CEs(A1B+h20k1s=1(aks+1aks)A1C)E0by using Eq. (3.7)=(324+81h20g1348+h20g0+121h20(aka1)72(348+79h20g0))E0=(324+81h20g1+1.68h20(aka1)348+79h20g0)E0=(132+(72.02)h2(g1+(aka1))348+79h20g0)E0.
    Ek+1E0,(aka1)<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,M214,Rk+122,1,Rk+122,2,...,Rk+12M114,M214},

    where

    Rk+12i,j={ek+12i,j,ek+12i+1,j,ek+12i+1,j+1,ek+12i,j+1},i,j={1,2,...,M14},

    so from Eq (2.10) we have

    Rk+12φ0(ν2γ+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(ν2α+h4).

    Proof. Since from Eq (4.1),

    R(k1)+12φ0(ν2α+h40), (4.2)

    then,

    R(k1)+12Rk+120,R(k1)+12Rk+12. (4.3)

    Since E0=0, then from Eq (2.10), we have

    AE1=R12,k=0,AEk+1=BEkh20k1s=1(aks+1aks)CEs+Rk+12,k>0. (4.4)

    When k=0,

    AE1=R12,E1A1R12=1λ1+2λ2+λ3R12=1348+79h20g0R12,E1μ0R12,whereμ0=1348+79h20g0andμ0(0,1),E1R12.

    Assume that

    EsR(s1)+12,s=2,3,...,k, (4.5)

    and now from Eq (4.4),

    AEk+1=BEkh20k1s=1(aks+1aks)CEs+Rk+12. (4.6)

    By taking norm function on both sides of Eq (4.6),

    Ek+1A1BEkh20k1s=1(aks+1aks)A1CEs+A1Rk+12(324+81h20g1348+79h20g0+1.68h20(aka1)348+79h20g0+1348+79h20g0)Rk+12(by using Eqs (4.3) and (4.5))=(325+81h20g1+1.68h20(aka1)348+79h20g0)Rk+12=γRk+12,

    where γ=133+121h2g1+1.68h20(aka1)348+79h20g0, but since h(0,1), g0>g1, and (a1ak)<0, then γ(0,1), therefore,

    Ek+1Rk+12φ0(ν2α+h40).

    Therefore, we get

    Ek+1φ0(ν2α+h40),k=0,1,2...,N1.

    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+12k1s=1(aks+1aks)G3Vs,k1,V0i,j=b0(xi,yj),1iM,1jM,Vk0,j=b1(0,yj),1jM,0kN,VkL,j=b2(L,yj),1jM,0kN,Vki,0=b3(xi,0),1iM,0kN,Vki,L=b4(xi,L),1iM,0kN, (5.1)

    where

    G1=[λ1λ2λ2λ30λ2λ1λ2λ3λ2λ3λ2λ1λ2λ3λ2λ3λ2λ1λ2λ3λ2λ3λ2λ1λ2λ3λ2λ2λ1λ2λ2λ1λ2λ2λ1λ20λ2λ1],
    G2=[λ4λ5λ5λ60λ5λ4λ5λ6λ5λ6λ5λ4λ5λ6λ5λ6λ5λ4λ5λ6λ5λ6λ5λ4λ5λ6λ5λ5λ4λ5λ5λ4λ5λ5λ4λ50λ5λ4],
    G3=[ρ1ρ2ρ2ρ30ρ2ρ1ρ2ρ3ρ2ρ3ρ2ρ1ρ2ρ3ρ2ρ3ρ2ρ1ρ2ρ3ρ2ρ3ρ2ρ1ρ2ρ3ρ2ρ2ρ1ρ2ρ2ρ1ρ2ρ2ρ1ρ20ρ2ρ1],
    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(4810h20g0), λ3=172(12h20g0), 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=105 is used. Also, the C2-order and C1-order of convergence are used for the computational order of spatial and temporal convergence using [25]

    C2order=log2(L(16ν,2h0)L(ν,h0)), (6.1)
    C1order=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]

    αVtα=2Vx2+2Vy2+(2Γ(3α)t2α+2t2)sin(x)sin(y),0<x, y<1,0<t1,

    with initial and Dirichlet boundary conditions.

    The analytic solution for Problem 1 is

    V(x,y,t)=t2sin(x)sin(y).

    Problem 2. [12]

    2Vt2=2Vx1+2Vy2+(Γ(2+α)2t1+α)ex+y,0<x, y<1,0<t1,

    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 15. The execution times in FEGM are decreased by (4.728.49)%, (223)%, (9.1628.13)%, (8.925.39)% and (6.9827.79)% compared to SP method in Tables 15 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.

    Table 1.  Numerical results for the Problem 1, when α=0.5.
    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 1.6404×104 1.653 ×104 7.9408 ×105 7.3891 ×105
    h0=ν=118 41 48 66.19 79.82 6.6571 ×105 6.5544×105 2.8320 ×105 2.9081 ×105
    h0=ν=122 42 54 148.1 207.1 4.5739 ×105 4.6717 ×105 1.9997 ×105 2.085 ×105
    h0=ν=130 43 56 530.45 658.68 3.0559 ×105 3.2268 ×105 1.1887 ×105 1.2180 ×105

     | Show Table
    DownLoad: CSV
    Table 2.  Numerical results for the Problem 1, when α=0.75.
    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 ×104 2.2065 ×104 1.0469 ×104 1.0634 ×104
    h0=ν=114 44 43 27.36 27.28 1.2736 ×104 1.2786×104 5.9175 ×105 6.0624 ×105
    h0=ν=118 42 48 68.29 80.18 8.0887 ×105 8.5709 ×105 3.0710 ×105 3.9889×105
    h0=ν=122 41 55 150.40 197.87 6.1207 ×105 6.2773 ×105 2.8554 ×105 2.8809 ×105
    h0=ν=130 42 56 523.25 648.55 3.7373×105 4.4510 ×105 1.6460 ×105 1.7874 ×105

     | Show Table
    DownLoad: CSV
    Table 3.  Numerical results for the Problem 2, where α=0.1.
    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 ×104 2.2065 ×104 1.0469 ×104 1.0634 ×104
    h0=ν=114 44 43 27.36 27.28 1.2736 ×104 1.2786×104 5.9175 ×105 6.0624 ×105
    h0=ν=118 42 48 68.29 80.18 8.0887 ×105 8.5709 ×105 3.0710 ×105 3.9889×105
    h0=ν=122 41 55 150.40 197.87 6.1207 ×105 6.2773 ×105 2.8554 ×105 2.8809 ×105
    h0=ν=130 42 56 523.25 648.55 3.7373×105 4.4510 ×105 1.6460 ×105 1.7874 ×105

     | Show Table
    DownLoad: CSV
    Table 4.  Numerical results for the Problem 2, where α=0.5.
    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 ×104 2.8915 ×104 1.3091 ×104 1.2849 ×104
    h0=ν=114 48 47 30.93 29.53 1.3893 ×104 1.4377 ×104 3.7322 ×105 3.9455 ×105
    h0=ν=118 48 52 77.2 84.75 8.5499 ×105 8.5075 ×105 2.4058 ×105 2.4340 ×105
    h0=ν=122 47 57 172.53 203.76 5.5120 ×105 5.5018 ×105 1.9837 ×105 1.7813 ×105
    h0=ν=130 47 65 588.74 789.14 2.9259 ×105 2.9669 ×105 1.2478 ×105 1.2549 ×105

     | Show Table
    DownLoad: CSV
    Table 5.  Numerical results for the Problem 2, where α=0.75.
    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 ×104 2.2917 ×104 1.0194 ×104 1.0189 ×104
    h0=ν=114 48 48 31.23 30.46 2.3640 ×104 2.3848 ×104 1.1941 ×104 1.2003 ×104
    h0=ν=118 49 52 78.34 85.3 2.1361 ×104 2.1290 ×104 1.0621 ×104 1.0863 ×104
    h0=ν=122 48 57 169.15 207.6 1.9686 ×104 1.8908 ×104 9.5981 ×105 9.5644 ×105
    h0=ν=130 48 65 599.97 775.49 1.5244 ×104 1.4938 ×104 7.3782 ×105 7.3922 ×105

     | Show Table
    DownLoad: CSV
    Table 6.  Comparison of the Proposed method with standard point method[27] for Example 2 when α=0.5.
    h0/ν No. of iteration Maximum-error Average-error
    FEGM [27] FEGM [27] FEGM [27]
    h0=ν=110 43 53 1.6404 ×104 1.2428 ×102 7.9408 ×105 8.8490 ×103
    h0=ν=118 41 52 6.6571 ×105 7.1213 ×103 2.8320 ×105 3.6917 ×103
    h0=ν=122 42 55 4.7739 ×105 2.6959 ×103 1.9997 ×105 1.3580 ×103
    h0=165,ν=135 41 58 2.5368 ×105 2.0605 ×103 5.0090 ×106 1.0285 ×103

     | Show Table
    DownLoad: CSV
    Figure 3.  Approximate solution for the Problem 1, where h0=ν=135.
    Figure 4.  Exact solution for the Problem 1, where h0=ν=135.
    Figure 5.  Execution time (in sec) for different mesh sizes for the Problems 1 and 2.
    Table 7.  The number of computing operations required for the FEGM and SP Technique.
    Technique operations per iteration
    +/- */ ÷
    SP (26+8(k1))m2 (8+4(k1))m2
    FEGM (28+8(k1))(m1)2 + (26+8(k1))(2m1) (12+4(k1))(m1)2 + (8+4(k1))(2m1)

     | Show Table
    DownLoad: CSV
    Table 8.  The total computation effort for the Problem 1, where α=12.
    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

     | Show Table
    DownLoad: CSV
    Table 9.  The total computation effort for the Problem 2, where α=34.
    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

     | Show Table
    DownLoad: CSV
    Table 10.  The spatial convergence order for the Problem 1.
    α=0.4 α=0.5
    ν/h0 Maximum error C2-order ν/h0 Maximum error C2-order
    ν=h0=0.5 1.4394 ×104 ν=h0=0.5 3.1088 ×104
    ν=0.031, h0=0.25 1.0910 ×105 3.72 ν=0.031, h0=0.25 2.2509 ×105 3.78
    ν=h0=0.25 2.1929 ×104 ν=h0=0.25 3.546 ×104
    ν=0.016, h0=0.12 9.1371 ×106 4.58 ν=0.016, h0=0.12 2.1414 ×105 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 ×104 ν=h0=0.5 9.9004 ×104
    ν=0.031, h0=0.25 2.7191 ×105 4.26 ν=0.031, h0=0.25 9.9004 ×105 4.13
    ν=h0=0.25 5.0421 ×104 ν=h0=0.25 7.2673 ×104
    ν=0.016, h0=0.12 3.1980 ×105 3.97 ν=0.016, h0=0.12 3.7243 ×105 4.28

     | Show Table
    DownLoad: CSV
    Table 11.  The spatial convergence order for the Problem 2.
    α=0.7 α=0.8
    ν/h0 Maximum error C2-order ν/h0 Maximum error C2-order
    ν=h0=0.5 1.6072 ×103 ν=h0=0.5 3.4202 ×103
    ν=0.031, h0=0.25 1.0543 ×104 3.93 ν=0.031, h0=0.25 1.7636 ×104 4.27
    ν=h0=0.25 1.3545 ×103 ν=h0=0.25 1.6955 ×103
    ν=0.016, h0=0.12 5.9784 ×105 4.50 ν=0.016, h0=0.12 8.2806 ×105 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 ×103 ν=h0=0.5 8.6771 ×104
    ν=0.031, h0=0.25 3.2105 ×104 3.57 ν=0.031, h0=0.25 4.8772 ×105 4.15
    ν=h0=0.25 4.5916 ×104 ν=h0=0.25 4.1734 ×104
    ν=0.016, h0=0.12 2.5978 ×105 4.14 ν=0.016, h0=0.12 2.3009 ×105 4.18

     | Show Table
    DownLoad: CSV
    Table 12.  Temporal convergence order for the Problem 1, when h0=18.
    ν α=0.3 α=0.8
    L C1 Order L C1 Order
    ν=110 7.8675×104 2.1609×104
    ν=120 3.2208 ×105 1.28 7.1112 ×105 1.60
    ν= 140 1.7253 ×105 1.29 2.6203 ×105 1.44
    ν=180 6.2057×106 1.72 8.3392×106 1.65

     | Show Table
    DownLoad: CSV
    Table 13.  Temporal convergence order for the Problem 2, when h0=18.
    ν α=0.1 α=0.8
    L C1 Order L C1 Order
    ν=110 4.7691×104 2.1429×104
    ν=120 1.6086 ×104 1.56 8.2679 ×105 1.37
    ν= 140 5.1170 ×105 1.65 3.2827 ×105 1.33
    ν=180 1.7848×105 1.51 1.3010×105 1.51

     | Show Table
    DownLoad: CSV

    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.



    [1] I. Podlubny, Fractional differential equations, Academic Press, 1999.
    [2] A. H. Bhrawy, E. H. Doha, D. Baleanu, S. S. Ezz-Eldien, A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations, J. Comput. Phys., 293 (2015), 142–156. https://doi.org/10.1016/j.jcp.2014.03.039 doi: 10.1016/j.jcp.2014.03.039
    [3] S. S. Ezz-Eldien, E. H. Doha, Fast and precise spectral method for solving pantograph type Volterra integro-differential equations, Numer. Algorithms, 81 (2019), 57–77. https://doi.org/10.1007/s11075-018-0535-x doi: 10.1007/s11075-018-0535-x
    [4] S. S. Ezz-Eldien, Y. Wang, M. A. Abdelkawy, M. A. Zaky, A. A. Aldraiweesh, J. T. Machado, Chebyshev spectral methods for multi-order fractional neutral pantograph equations, Nonlinear Dyn., 100 (2020), 3785–3797. https://doi.org/10.1007/s11071-020-05728-x doi: 10.1007/s11071-020-05728-x
    [5] M. M. Alsuyuti, E. H. Doha, S. S. Ezz-Eldien, I. K. Youssef, Spectral Galerkin schemes for a class of multi-order fractional pantograph equations, J. Comput. Appl. Math., 384 (2021), 113157. https://doi.org/10.1016/j.cam.2020.113157 doi: 10.1016/j.cam.2020.113157
    [6] R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1–77. https://doi.org/10.1016/S0370-1573(00)00070-3 doi: 10.1016/S0370-1573(00)00070-3
    [7] O. P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dyn., 29 (2002), 145–155. https://doi.org/10.1023/A:1016539022492 doi: 10.1023/A:1016539022492
    [8] R. Gorenflo, F. Mainardi, D. Moretti, P. Paradisi, Time fractional diffusion: a discrete random walk approach, Nonlinear Dyn., 29 (2002), 129–143. https://doi.org/10.1023/A:1016547232119 doi: 10.1023/A:1016547232119
    [9] X. J. Yang, The fractional calculus, New York: Academic Press, 1974.
    [10] Z. Z. Sun, X. N. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), 193–209. https://doi.org/10.1016/j.apnum.2005.03.003 doi: 10.1016/j.apnum.2005.03.003
    [11] J. Y. Shen, Z. Z. Sun, R. Du, Fast finite difference schemes for time-fractional diffusion equations with a weak singularity at initial time, East Asian J. Appl. Math., 8 (2018), 834–858. https://doi.org/10.4208/eajam.010418.020718 doi: 10.4208/eajam.010418.020718
    [12] M. R. Cui, Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation, Numer. Algorithms, 62 (2013), 383–409. https://doi.org/10.1007/s11075-012-9589-3 doi: 10.1007/s11075-012-9589-3
    [13] P. Zhuang, F. Liu, Finite difference approximation for two-dimensional time fractional diffusion equation, J. Algorithms Comput. Technol., 1 (2007), 1–16. https://doi.org/10.1260/174830107780122667 doi: 10.1260/174830107780122667
    [14] Y. N. Zhang, Z. Z. Sun, Error analysis of a compact ADI scheme for the 2D fractional subdiffusion equation, J. Sci. Comput., 59 (2014), 104–128. https://doi.org/10.1007/s10915-013-9756-2 doi: 10.1007/s10915-013-9756-2
    [15] C. C. Ji, Z. Z Sun, The high-order compact numerical algorithms for the two-dimensional fractional sub-diffusion equation, Appl. Math. Comput., 269 (2015), 775–791. https://doi.org/10.1016/j.amc.2015.07.088 doi: 10.1016/j.amc.2015.07.088
    [16] T. Wang, Y. M. Wang, A modified compact ADI method and its extrapolation for two-dimensional fractional subdiffusion equations, J. Appl. Mathe. Comput., 52 (2016), 439–476. https://doi.org/10.1007/s12190-015-0949-8 doi: 10.1007/s12190-015-0949-8
    [17] S. Y. Zhai, X. L. Feng, Investigations on several compact ADI methods for the 2D time fractional diffusion equation, Numer. Heat Transf. Part B Fund., 69 (2016), 364–376. https://doi.org/10.1080/10407790.2015.1097231 doi: 10.1080/10407790.2015.1097231
    [18] K. L. Ming, N. H. M. Ali, New explicit group iterative methods in the solution of three dimensional hyperbolic telegraph equations, J. Comput. Phys., 294 (2015), 382–404. https://doi.org/10.1016/j.jcp.2015.03.052 doi: 10.1016/j.jcp.2015.03.052
    [19] M. A. Khan, N. Alias, I. Khan, F. M. Salama, S. M. Eldin, A new implicit high-order iterative scheme for the numerical simulation of the two-dimensional time fractional Cable equation, Sci. Rep., 13 (2023), 1549. https://doi.org/10.1038/s41598-023-28741-7 doi: 10.1038/s41598-023-28741-7
    [20] A. T. Balasim, N. H. M. Ali, Group iterative methods for the solution of two-dimensional time-fractional diffusion equation, AIP Conf. Proc., 1750 (2016), 030003. https://doi.org/10.1063/1.4954539 doi: 10.1063/1.4954539
    [21] A. Ajmal, N. H. M. Ali, On skewed grid point iterative method for solving 2D hyperbolic telegraph fractional differential equation, Adv. Differ. Equ., 2019 (2019), 303. https://doi.org/10.1186/s13662-019-2238-6 doi: 10.1186/s13662-019-2238-6
    [22] M. A. Khan, N. H. M. Ali, High-order compact scheme for the two-dimensional fractional Rayleigh-Stokes problem for a heated generalized second-grade fluid, Adv. Differ. Equ., 2020 (2020), 1–21. https://doi.org/10.1186/s13662-020-02689-8 doi: 10.1186/s13662-020-02689-8
    [23] M. R. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228 (2009), 7792–7804. https://doi.org/10.1016/j.jcp.2009.07.021 doi: 10.1016/j.jcp.2009.07.021
    [24] I. Karatay, N. Kale, S. R. Bayramoglu, A new difference scheme for time fractional heat equations based on the Crank-Nicholson method, Fract. Calc. Appl. Anal., 16 (2013), 892–910. https://doi.org/10.2478/s13540-013-0055-2 doi: 10.2478/s13540-013-0055-2
    [25] M. Abbaszadeh, A. Mohebbi, A fourth-order compact solution of the two-dimensional modified anomalous fractional sub-diffusion equation with a nonlinear source term, Comput. Math. Appl., 66 (2013), 1345–1359. https://doi.org/10.1016/j.camwa.2013.08.010 doi: 10.1016/j.camwa.2013.08.010
    [26] A. T. Balasim, N. H. M. Ali, A rotated Crank-Nicolson iterative method for the solution of two-dimensional time-fractional diffusion equation, Indian J. Sci. Technol., 8 (2015), 1–8. https://doi.org/10.17485/ijst/2015/v8i32/92045 doi: 10.17485/ijst/2015/v8i32/92045
    [27] M. A. Khan, N. H. M Ali, Fourth-order compact iterative scheme for the two-dimensional time fractional sub-diffusion equations, Math. Stat., 8 (2020), 52–57. https://doi.org/10.13189/ms.2020.081309 doi: 10.13189/ms.2020.081309
  • This article has been cited by:

    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
    2. Fouad Mohammad Salama, Faisal Fairag, On numerical solution of two-dimensional variable-order fractional diffusion equation arising in transport phenomena, 2024, 9, 2473-6988, 340, 10.3934/math.2024020
    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
    4. Nasir Ali, Muhammad Waseem, Maimoona Safdar, Ali Akgül, Fikadu Tesgera Tolasa, Iterative solutions for nonlinear equations via fractional derivatives: adaptations and advances, 2024, 32, 2769-0911, 10.1080/27690911.2024.2333816
    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
  • Reader Comments
  • © 2023 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(1413) PDF downloads(43) Cited by(5)

Figures and Tables

Figures(5)  /  Tables(13)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog