Research article

Approximation of Caputo time-fractional diffusion equation using redefined cubic exponential B-spline collocation technique

  • Received: 04 November 2020 Accepted: 10 January 2021 Published: 28 January 2021
  • MSC : 35XX, 65N12

  • The purpose of this work is to find the numerical solution of the Caputo time-fractional diffusion equation using the modified cubic exponential B-spline (CExpB-spline) collocation technique. First, the CExpB-spline functions are modified and then used to discretize the space derivatives. Three numerical examples are considered for checking the efficiency and accuracy of the method. The obtained results are compared with those reported earlier showing that the present technique gives highly accurate results. Von Neumann stability is carried out which gives the guarantee that the technique is unconditionally stable. The rate of convergence is also obtained. Furthermore, this technique is efficient and requires less storage.

    Citation: Mohammad Tamsir, Neeraj Dhiman, Deependra Nigam, Anand Chauhan. Approximation of Caputo time-fractional diffusion equation using redefined cubic exponential B-spline collocation technique[J]. AIMS Mathematics, 2021, 6(4): 3805-3820. doi: 10.3934/math.2021226

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  • The purpose of this work is to find the numerical solution of the Caputo time-fractional diffusion equation using the modified cubic exponential B-spline (CExpB-spline) collocation technique. First, the CExpB-spline functions are modified and then used to discretize the space derivatives. Three numerical examples are considered for checking the efficiency and accuracy of the method. The obtained results are compared with those reported earlier showing that the present technique gives highly accurate results. Von Neumann stability is carried out which gives the guarantee that the technique is unconditionally stable. The rate of convergence is also obtained. Furthermore, this technique is efficient and requires less storage.



    Fractional differential equations have been applied in modelling of many fields of physics and processes such as earthquakes, optics, finance, hydrology, traffic flow, fluid mechanics, fractional kinetics, mathematical biology, measurement of visco-elastic material properties, electrical network, electro-chemistry electro-magnetic, signal processing, control theory, acoustics, material sciences [2,5,8,9,17,22,39,41,42,43,44,45,46]. The nonlocal property is the most important advantage of these equations showing state of a complex system does not depend only on its current state but also depends on its all-previous states. Due to this reason, the fractional calculus has becoming more and more popular.

    The fractional diffusion equations (FDEs) are generalized forms of classical diffusion equations. These equations have been solved by many researchers. For example, authors of [6,11,24,26,32] have been applied finite element method, compact FDM, Crank-Nicolson FDM, B-spline based method, implicit difference approximation, respectively, to solve FDEs. Khader [14] considered an efficient method based upon Chebyshev approximations. Murio [20] developed an implicit unconditionally stable method. Sun et al. [25] applied a semi-analytical FEM to solve a class of these equations. Tadjeran et al. [27] applied aforesaid method with spatial extrapolation to solve a class of variable coefficients FDEs. Lin and Xu [15] constructed a method based on FDM and Legendre spectral method while Ghanbari and Atangana [7] presented a method based on the product-integration rule for solving aforesaid equations. Çelik and Duman [3] examined a Crank-Nicolson method with the Riesz fractional derivative. Zhai and Feng [35] introduced a block-centered FDM for solving these equations.

    Additionally, Zhai et al. [34,36] constructed unconditionally stable FDMs to solve the time-space fractional diffusion and three-dimensional time-fractional subdiffusion equations, respectively. Wu and Zhai [33] proposed a high order FDM to solve the 2D time-fractional convection-dominated diffusion equation while Zhai et al. [37,38] proposed a high-order compact FDM and ADI method to solve the 3D fractional convection-diffusion equation. Furthermore, Hanert [10] proposed a flexible numerical method to discretize the space-time FDEs, where pseudo-spectral expansion has been used for discretization of the time derivative and either high order pseudo-spectral or low order FE and FD have been used for space derivative. Murillo and Yuste [19] considered explicit difference scheme, where three-point centered formula has been used to approximate the spatial derivative. Verma et al. [31] studied nonlinear diffusion equations analytically and numerically using the classical Lie symmetry method.

    In this paper, we consider a Caputo time-fractional diffusion equation

    cDαtu(x,tn)=2u(x,t)x2+f(x,t),0x1, 0tT, (1.1)

    subject to the initial condition

    u(x,0)=φ(x),0x1, (1.2)

    the Dirichlet boundary conditions

    u(0,t)=0,  u(1,t)=0,0tT, (1.3)

    where cDαtu(x,tn) denotes the Caputo time-fractional derivative [40] as

    cDαtu(x,tn)=1Γ(1α)t0(tζ)αu(x,ζ)ζdζ. (1.4)

    First, the Caputo time-fractional derivative is discretized. For this, we partition the time domain [0,T] uniformly as 0=t0<t1,...,<tN=T, where interval Δt=tntn1=TN for n=1,2,,N. Here N is the number of time mesh. By the definition of Caputo derivative at time t=tn, the time-fractional derivative is approximated as

    cDαtu(x,tn)=1Γ(1α)tn0(tnζ)αu(x,ζ)ζdζ=ΔtαΓ(2α)n1l=0(u(x,tnl)u(x,tnl1))ˉbl+TrnΔt, (2.1)

    where ˉbl=(l+1)1αl1α  0lN and TrnΔt gives the truncation error ˉcuΔt2α, where the constant ˉcu is only related to u(x,t).

    Lemma 3.1 The below relations should be hold for the coefficients ˉbl

    (1)ˉb0=1,

    (2)ˉbl>0,  0lN,

    (3)ˉbl1>ˉbl,  1lN.

    Proof. For the proof of Lemma 3.1, see [16].

    Next, we use CExpB-spline collocation technique to discretize the space derivatives. First, we partition the space domain [a,b] uniformly as a=x0<x1,...,xM=b with space size Δx=h=xm+1xm=baM where m=0,1,,M. For discrete form, we denote unm=u(xm,tn) for m=0,1,...,M and n=0,1,...,N. The CExpB-spline functions Epm(x) for m=1, 0, 1,,M+1 are defined as [28,29,30]:

    Epm(x)=1h3{ˆγ5((xm2x)1ˆκ(sinh(ˆκ(xm2x)))), x[xm2,xm1)ˆγ1+ˆγ2(xmx)+ˆγ3eˆκ(xmx)+ˆγ4eˆκ(xmx), x[xm1,xm)ˆγ1+ˆγ2(xxm)+ˆγ3eˆκ(xmx)+ˆγ4eˆκ(xmx), x[xm,xm+1)ˆγ5((xxm+2)1ˆκ(sinh(ˆκ(xxm+2)))),x[xm+1,xm+2)0,otherwise (2.2)

    where

    ˆγ1=ˆκhcˆκhcs, ˆγ2=ˆκ2(c(c1)+s2(ˆκhcs)(1c)), ˆγ3=14((1c)eˆκh+s(eˆκh1)(ˆκhcs)(1c)),ˆγ4=14(eˆκh(c1)+s(eˆκh1)(ˆκhcs)(1c)), ˆγ5=ˆκ2(ˆκhcs), s=sinh(ˆκh), c=cosh(ˆκh).

    In Eq (2.2), the free parameter ˆκ have been used for obtaining the different forms of CExpB-spline functions. The set of Epm(x)  m=1, 0, 1,,M+1 forms a basis over the problem domain. We assume that the approximation uM to the exact u(x,t) at the point (xm,tn) is expressed in terms of linear combinations of the CExpB-spline functions and unknown time-dependent quantities as follows:

    uM(xm,tn)=M+1i=1Ci(t)Epi(x), (2.3)

    where Ci(t) are the unknown quantities which we have to evaluate for the approximated solution uM(x,t) at (xi,tj). Since each CExpB-spline covers four elements, so each element is covered by four CExpB-splines. So the variation of the uM(x,t), over the element, can be written as:

    uM(x,t)=m+1i=m1Ci(t)Epi(x). (2.4)

    Using Eq (2.4), the u(x,t) and its first two derivatives at the knots in terms of Cnm are given as:

    unm=sˆκh2(ˆκhcs)Cnm1+Cnm+sˆκh2(ˆκhcs)Cnm+1, (2.5)
    (ux)nm=ˆκ(c1)2(ˆκhcs)(Cnm1Cnm+1), (2.6)
    (uxx)nm=ˆκ2s2(ˆκhcs)(Cnm12Cnm+Cnm+1), (2.7)

    Now, we modify the CExpB-spline basis functions which generate a new set of CExpB-spline basis functions. The procedure, for modifying the basis functions, is given as [1,12,13]:

    {^Ep0(x)=4Ep1(x)+Ep0(x),^Ep1(x)=Ep1(x)Ep1(x),^Epm(x)=Epm(x), m=2,3,,M2,^EpM1(x)=EpM1(x)EpM+1(x),^EpM(x)=EpM(x)+4EpM+1(x), (2.8)

    where {^Ep0(x),^Ep1(x)^EpM(x)} forms a basis over the problem domain. Next, the modified form of the approximated solutions, as a linear combination of modified CExpB-spline functions, is given by

    {uM(x0,tn)=0, for m=0,uM(xm,tn)=Mj=0^Epj(x)Cj(t), for m=1,2,,M1,uM(xM,tn)=0, for m=M. (2.9)

    Next, the initial vector (C0m, m=0, 1,, M) can be obtained by initial condition and boundary values of its derivatives as:

    {uM(x0,0)=0, for m=0,uM(xm,0)=φm, for m=1,2,,M1,uM(xM,0)=0, for m=M. (2.10)

    Eq (2.10) yields a (M+1)×(M+1) system as:

    [ˆκh(c1)(ˆκhcs) 0sˆκh2(ˆκhcs) 1  sˆκh2(ˆκhcs)0 sˆκh2(ˆκhcs) 1  sˆκh2(ˆκhcs)sˆκh2(ˆκhcs) 1 sˆκh2(ˆκhcs) 0 ˆκh(c1)(ˆκhcs)][C01C02C03C0M1C0M]=[0φ1φ2φM10]. (2.11)

    Now, discretizing the Eq (1.1) by using (2.1) and Crank-Nicolson method, we have

    ΔtαΓ(2α)n1l=0(unlmunl1m)(ˉbl)=12{(uxx)nm+(uxx)n1m}+12(fnm+fn1m),m=0,1,...,M, n=1,2,...,N. (2.12)

    Using Eqs (2.4) and (2.6) in Eq (2.12) and simplifying the terms, we have

    (ΔtαΓ(2α)k0ω10.5ω2)Cnm1+(ΔtαΓ(2α)k0+ω2)Cnm+(ΔtαΓ(2α)k0ω10.5ω2)Cnm+1=ΔtαΓ(2α)[n1l=1(ω1Cl1m1+Cl1m+ω1Cl1m+1)(ˉbl1ˉbl)+ˉbn1(ω1C0m1+C0m+ω1C0m+1)]+0.5ω2(Cn1m12Cn1m+Cn1m+1)+12(fnm+fn1m), m=1,2,...,M, n=1,2,...,N. (2.13)

    where

    ω1=sˆκh2(ˆκhcs), and ω2=ˆκ2s2(ˆκhcs).

    For m=0, from Eq (2.13), we get

    (ΔtαΓ(2α)(1+2ω1)ˉb0)Cn0=(1+2ω1)ΔtαΓ(2α)[n1l=1((ˉbl1ˉbl)Cl10+ˉbn1C00)]+12(fn0+fn10),  n=1,2,...,N. (2.14)

    For m = M, from Eq (2.13), we get

    (ΔtαΓ(2α)(1+2ω1)ˉb0)CnM=(1+2ω1)ΔtαΓ(2α)[n1l=1((ˉbl1ˉbl)Cl1M+ˉbn1C0M)]+12(fnM+fn1M),  n=1,2,...,N. (2.15)

    At time step tn, n=1,2,...,N, the Eqs (2.13)–(2.15) can be formulated into a (M+1)×(M+1) linear system as follows:

    (ˆA00ˆB1ˆB2ˆB1ˆB1ˆB2ˆB1ˆB1ˆB2ˆB100ˆA)(Cn0Cn1Cn2CnM1CnM)=(R0R1R2RM1RM), (2.16)

    where

    ˆA=ΔtαΓ(2α)(1+2ω1)ˉb0, ˆB1=ΔtαΓ(2α)k0ω10.5ω2, ˆB2=ΔtαΓ(2α)k0+ω2,R0=ΔtαΓ(2α)[n1l=1((ˉbl1ˉbl)(1+2ω1)Cl10+ˉbn1(1+2ω1)C00)]+12(fn0+fn10),n=1,2,...,N,Rm=ΔtαΓ(2α)[n1l=1((ω1Cl1m1+Cl1m+ω1Cl1m+1)(ˉbl1ˉbl)+ˉbn1(ω1C0m1+C0m+ω1C0m+1))]+0.5(ω2Cn1m12ω2Cn1m+ω2Cn1m+1)+12(fnm+fn1m),m=1,2,...,M1,n=1,2,...,N,RM=ΔtαΓ(2α)[n1l=1((ˉbl1ˉbl)(1+2ω1)Cl1M+ˉbn1(1+2ω1)C0M)]+12(fnM+fn1M),n=1,2,...,N.

    The von-Neumann stability analysis [4,18,21,23] have been done in this section. Taking f(x,t)=0 in Eq (1.1) and discretizing, we get

    (ΔtαΓ(2α)k0ω10.5ω2)Cnm1+(ΔtαΓ(2α)k0+ω2)Cnm+(ΔtαΓ(2α)k0ω10.5ω2)Cnm+1=ΔtαΓ(2α)[n1l=1(ω1Cl1m1+Cl1m+ω1Cl1m+1)(ˉbl1ˉbl)+ˉbn1(ω1C0m1+C0m+ω1C0m+1)]+0.5ω2(Cn1m12Cn1m+Cn1m+1). (3.1)

    Simplifying and rearranging the terms, we get

    (ΔtαΓ(2α)ˉb0ω10.5ω2)Cnm1+(ΔtαΓ(2α)ˉb0+ω2)Cnm+(ΔtαΓ(2α)ˉb0ω10.5ω2)Cnm+1=0.5ω2(Cn1m12Cn1m+Cnlm+1)+ΔtαΓ(2α)n1l=1(ω1Cnlm1+Cnlm+ω1Cnlm+1)(ˉbl1ˉbl)+ΔtαΓ(2α)ˉbn1(ω1C0m1+C0m+ω1C0m+1), (3.2)

    which can be written as

    ACnm1+BCnm+ACnm+1=DCn1m1+ECn1m+DCn1m+1+F, (3.3)

    where

    A=(ΔtαΓ(2α)ˉb0ω10.5ω2), B=(ΔtαΓ(2α)ˉb0+ω2), D=0.5ω2, S=ω2

    and

    F=ΔtαΓ(2α)n1l=1(ω1Cnlm1+Cnlm+ω1Cnlm+1)(ˉbl1ˉbl)+ΔtαΓ(2α)ˉbn1(ω1C0m1+C0m+ω1C0m+1).

    Now we consider one Fourier mode out of the full solution Cnm=δnekiϕ as trial solutions at a given point xi, where ϕ=θh. The θ and h are the mode number and element size respectively, and k=1. Substituting the trial solution in above equation and simplifying the terms, we get

    δn(2Acosϕ+B)=δn1(2Dcosϕ+D)+ΔtαΓ(2α)(1+2ω1cosϕ)ˉF,

    where ˉF=n1l=1(ˉbl1ˉbl)δnl+ˉbn1δ0.

    Now we define

    δn1max=max0jn1|δj| (3.4)

    Using (3.4), we get

    ˉF=n1l=1(ˉbl1ˉbl)δnl+ˉbn1δ0n1l=1(ˉbl1ˉbl+ˉbn1)δn1max=ˉb0δn1max. (3.5)

    Now, using above equation and the values of A, B, D and E in Eq (3.3), we have

    |δ||ΔtαΓ(2α)(1+2ω1cosϕ)ˉb0ω2(1+2sin2(ϕ))ΔtαΓ(2α)(1+2ω1cosϕ)ˉb0+ω2(1+2sin2(ϕ))|. (3.6)

    From (3.6), we get |δ|1 , and hence the method is unconditionally stable for the discretized system of the Caputo time-fractional diffusion equation.

    In this section, we consider numerical examples of the Caputo time-fractional diffusion equation in order to check the accuracy and efficiency of the method. We use the following error norms :

    L2=hMj=0|uexactjunumericalj|2,  Lmax=max0jM|uexactjunumericalj|. (4.1)

    The ROC is calculated by

    ROC=log(Eh1/Eh2)log(h1/h2) and ROC=log(Ek1/Ek2)log(k1/k2), (4.2)

    where, Eh1 and Eh2 represent the errors at space mesh sizes h1 and h2, respectively and Ek1 and Ek2 represent the errors at time mesh sizes k1 and k2, respectively.

    First, we consider the Caputo time-fractional diffusion equation

    cDαtu(x,tn)=2u(x,t)t2+f(x,t),      x[0,1],  t0,  0<α<1, (4.3)

    with the exact solution

    u(x,t)=t2sin(2πx),

    where

    f(x,t)=(2Γ(3α)t2α+4π2t2)sin(2πx).

    We solve the Caputo time-fractional diffusion Eq (4.3) for different values of M and N. Table 1 shows the comparison of the present method with the finite element method [6] and cubic B-spline collocation method [24] for different values of M, fixed Δt=0.001 at the time-fractional order α=0.5  in terms of L2 and Lmax error norms. This table shows that the present method gives better results than the results obtained by those available in [6,24]. Moreover, one can notice from Table 1 that the error norms are decreasing as we increase the space as well as time mesh sizes. The analytical and approximate solutions together with the absolute errors are presented in Figure 1 for ˆκ=11, Δx=0.001, and α=0.5 at Δt=0.2, Δt=0.1, and Δt=0.05, respectively. Figure 2 shows the comparison between analytical and approximate solutions for M=64, Δt=0.001, and α=0.5 at different t. From these figures, we notice that there is an excellent agreement between analytical and approximate solutions. Also, the absolute errors are decreasing on increasing the time interval.

    Table 1.  Comparison of present results and the results available in [6,24], for Example 1.
    M Present Method Lmax Present Method L2 Cubic B-spline collocation method [24] Lmax Finite element method [6] L2
    4 1.28e-03 9.06e-03 1.72e-01 1.09e-01
    8 2.73e-04 1.93e-04 4.80e-02 3.20e-02
    16 1.09e-04 7.67e-05 1.23e-02 8.42e-03
    32 8.65e-05 6.12e-05 3.08e-03 2.15e-03
    64 8.23e-05 5.82e-05 7.70e-04 5.42e-04

     | Show Table
    DownLoad: CSV
    Figure 1.  Analytical and approximate solutions together with absolute errors for ˆκ=11, Δx=0.001, and α=0.5 at (a) Δt=0.2, (b) Δt=0.1, and (c) Δt=0.05 (Example 1).
    Figure 2.  Analytical and approximate u(x,t) for M=64, Δt=0.001 and α=0.5 at different t (Example 1).

    Next, we consider the Caputo time-fractional diffusion equation

    cDαtu(x,tn)=2u(x,t)x2+f(x,t),  x[0,1],  t0,  0<α<1, (4.4)

    with the exact solution

    u(x,t)=x2(x1)2ext(2+α),

    where

    f(x,t)=0.5t2exx2(x1)2Γ(α+3)t(2+α)ex(x4+6x3+x28x+2).

    We solve the Caputo time-fractional diffusion Eq (4.4) for various values of M. Table 2 shows the maximum absolute error norms Lmax for different values of time-fractional orders i. e. α=0.2, 0.5  and 0.8 at fixed time mesh size of N=50 for different space mesh sizes. One can see from Tables 2 and 3 that the error norms are decreasing as we increase the mesh sizes. As we can notice from this table that the present method is of order O((Δx)2,(Δt)1+α). The approximate solutions for time levels t= 0.1, 0.4, 0.6 and 0.8 are plotted in Figure 3 for M=32, N=50 and time-fractional order α=0.5. One can notice that the amplitude of the approximate solutions is increasing on incresing the time t.

    Table 2.  The Lmax error norms and convergence order for different M and α at fixed N=50 (Example 2).
    M α=0.2 ROC  α=0.5 ROC α=0.8 ROC
    2 6.67e-01 - 1.94e-01 - 1.287e-02 -
    4 1.715e-01 1.96 5.01e-02 1.95 2.987e-03 2.10
    8 4.205-02 2.03 1.23e-02 2.02 6.958e-04 2.10
    16 1.040e-02 2.02 3.99e-03 1.99 1.710e-04 2.02
    32 2.051e-03 2.05 7.81e-04 1.98 4.297e-05 1.99

     | Show Table
    DownLoad: CSV
    Table 3.  The Lmax error norms and convergence order for different N and α at fixed M=50 (Example 2).
    N α=0.2 ROC  α=0.5 ROC α=0.8 ROC
    2 9.270e-04 - 5.478e-04 - 3.733e-04 -
    4 3.927e-04 1.23 1.987e-04 1.46 1.032e-04 1.85
    8 1.618e-04 1.27 7.182e-05 1.47 2.733e-05 1.91
    16 6.901e-05 1.22 2.609e-05 1.46 7.793e-05 1.81
    32 2.950e-05 1.22 9.472e-06 1.46 2.139e-06 1.86

     | Show Table
    DownLoad: CSV
    Figure 3.  Numerical solutions for M=32, N=50 and α=0.5 at different t (Example 2).

    Now, we consider the Caputo time-fractional diffusion equation

    cDαtu(x,tn)=2u(x,t)x2+f(x,t),  0<x,α<1, t>0, (4.5)

    where

    f(x,t)=tβex(Γ(β+1)Γ(β+1α)tα1),

    subject to the initial and boundary conditions

     u(x,0)=0,   0<x<1,
    u(0,t)=tβ,  u(1,t)=etβ   t0 ,

    and with the exact solution

    u(x,t)=extβ.

    Now, we solve the Caputo time-fractional diffusion Eq (4.5) for different values of M and N. Table 4 shows the comparison in term of Lmax error norms obtained by the present method and the method based on the product-integration (PI) rule presented in [7] for Δx=1101, β=6, different values of N and α at t=1. As we can see that the present method gives more accurate results than the method presented in [7]. Also, we notice that the error norms are decreasing as we increase the time mesh sizes. Figure 4 shows the analytical and approximate solutions together with absolute error norms for Δx=1101, α=0.85, β=5, Δt=0.01 at T=0.35 showing that errors are decreasing on increasing the grid sizes.

    Table 4.  The Lmax error norms for different time-fractional orders with Δx=1101, β=6 at T=1 (Example 3).
    N Present Method for α=0.5 Method based on PI rule [7] for α=0.5 Present method for α=0.7 Method based on PI rule [7] for α=0.7 Present method for α=0.9 Method based on PI rule [7] for α=0.9
    10 2.34e-03 7.63e-03 8.41e-03 1.07e-02 7.59e-03 1.14e-02
    20 6.23e-04 2.03e-03 1.25e-03 2.76e-04 2.71e-03 3.53e-03
    40 2.54e-04 5.27e-04 6.18e-04 7.03e-04 6.78e-04 8.89e-04
    80 7.22e-05 1.36e-04 1.57e-04 1.78e-04 1.54e-04 2.23e-04
    160 3.31e-05 3.55e-05 4.49e-05 4.56e-05 4.35e-05 5.67e-05

     | Show Table
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    Figure 4.  Analytical and approximate solutions together with absolute errors for p=0.01, Δx=0.001, and α=0.5 at (a) Δt=0.08, (b) Δt=0.04, and (c) Δt=0.02 (Example 3).

    A modified CExpB-spline collocation technique has been presented for solving the Caputo time-fractional diffusion equation. The modified CExpB-spline collocation technique is used to discretize the space derivatives. The three examples of the Caputo time-fractional diffusion equation have been considered. The obtained results show that the present method gives more accurate results than the results obtained in [6,7,24]. It is observed numerically that the method is second-order accurate in space and (1+α) order in time. The stability analysis shows that the method is unconditionally stable. Moreover, the implementation of the present method is easy and needs low memory storage which is the advantage. The present method can easily be extended for solving higher dimensional fractional PDEs.

    The authors have no conflict of interest.



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