Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation

Xiaoyong Xu
  • Corresponding author
  • School of Science, East China University of Technology, Nanchang, Jiangxi 330013, China
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar
and Fengying Zhou

Abstract

In this paper, a discrete orthogonal spline collocation method combining with a second-order Crank-Nicolson weighted and shifted Grünwald integral (WSGI) operator is proposed for solving time-fractional wave equations based on its equivalent partial integro-differential equations. The stability and convergence of the schemes have been strictly proved. Several numerical examples in one variable and in two space variables are given to demonstrate the theoretical analysis.

1 Introduction

Recently, fractional partial differential equations (FPDEs) have attracted more and more attention, which can be used to describe some physical and chemical phenomenon more accurately than the classical integer-order differential equations. For example, when studying universal electromagnetic responses involving the unification of diffusion and wave propagation phenomena, there are processes that are modeled by equations with time fractional derivatives of order γ ∈ (1, 2) [1]. Generally, the analytical solutions of fractional partial differential equations are difficult to obtain, so many authors have resorted to numerical solution techniques based on convergence and stability. Various kinds of numerical methods for solving FPDEs have been proposed by researchers, such as finite element method [2, 3], finite difference method [4, 5, 6], meshless method [7, 8], wavelets method [9], spline collocation method [10, 11, 12] and so forth.

In this study, we consider the following two-dimensional time-fractional diffusion-wave equation

0CDtγu(x,y,t)=Δu(x,y,t)u(x,y,t)+f(x,y,t),(x,y,t)Ω×(0,T]

subject to the initial condition

u(x,y,0)=ϕ(x,y),u(x,y,0)t=φ(x,y),(x,y)Ω,

and the boundary condition

u(x,y,t)=0,(x,y,t)Ω×(0,T],

where Δ is Laplace operator, Ω = [0, 1] × [0, 1] with boundary Ω, ϕ(x, y), φ(x, y) and f(x, y, t) are given sufficiently smooth functions in their respective domains and 0CDtγ denotes the Caputo derivative of order γ (1 < γ < 2), which reads as follows:

0CDtγu(x,y,t)=1Γ(2γ)0t2u(x,y,s)s2(ts)1γds,

in which Γ(⋅) is the Gamma function. Without loss of generality, we assume that ϕ(x, y) ≡ 0 in(1.2), since we can solve the equation for v(x, y, t) = u(x, y, t) − ϕ(x, y) in general.

Most of the numerical algorithms in [1, 2, 3, 4, 5, 6, 7, 8] employed the L1 scheme to approximate fractional derivatives. Recently, Tian et al. [13] proposed second-and third-order approximations for Riemann-Liouville fractional derivative via the weighted and shifted Grünwald difference (WSGD) operators. Thereafter, some related research work covering the WSGD idea were done by many scholars. In [14], Liu et al developed a high-order local discontinuous Galerkin method combined with WSGD approximation for a Caputo time-fractional sub-diffusion equation. In [15], Chen considered the numerical solutions of the multi-term time fractional diffusion and diffusion-wave equations with variable coefficients, which the time fractional derivative was approximated by WSGD operator. In [16], Yang proposed a new numerical approximation, using WSGD operator with second order in time direction and orthogonal spline collocation method in spatial direction, for the two-dimensional distributed-order time fractional reaction-diffusion equation. Following the idea of WSGD operator, Wang and Vong [17] used compact finite difference WSGI scheme for the temporal Caputo fractional diffusion-wave equation. However, the numerical methods with WSGI approximation have been rarely studied. Cao et al.[18] applied the idea of WSGI approximation combining with finite element method to solve the time fractional wave equation.

Orthogonal spline collocation (OSC) method has evolved as a valuable technique for solving different types of partial differential equations [19, 20, 21, 22, 23]. The popularity of OSC is due to its conceptual simplicity, wide applicability and easy implementation. Comparing with finite difference method and the Galerkin finite element method, OSC method has the following advantages: the calculation of the coefficients in the equation determining the approximate solution is fast since there is no need to calculate the integrals; and it provides approximations to the solution and spatial derivatives. Moreover, OSC scheme always leads to the almost block diagonal linear system, which can be solved by the software packages efficiently [24]. Another feature of OSC method lies in its super-convergence [25].

Motivated and inspired by the work mentioned above, the main goal of this paper is to propose a high-order OSC approximation method combined with second order WSGI operator for solving two-dimensional time-fractional wave equation, which is abbreviated as WSGI-OSC in forthcoming sections. The remainder of the paper is organized as follows. In Section 2, some notations and preliminaries are presented. In Section 3, the fully discrete scheme combining WSGI operator with second order and orthogonal spline collocation scheme is formulated. Stability and convergence analysis of WSGI-OSC scheme are presented in Section 4. Section 5 provides detailed description of the WSGI-OSC scheme. In Section 6, several numerical experiments are carried out to confirm the convergence analysis. Finally, the conclusion is drawn in Section 7.

2 Discrete-time OSC scheme

2.1 Preliminaries

In this section, we will introduce some notations and basic lemmas. For some positive integers Nx and Ny, δx and δy are two uniform partitions of I = [0, 1] which are defined as follows:

δx:0=x0<x1<<xNx=1,δy:0=y0<y1<<yNy=1,

and hix=xixi1,Iix=(xi1,xi),1iNx,andhjy=yjyj1,Ijy=(yj1,yj),1jNy, h=max(max1iNxhix,max1jNyhjy). Let Mr(δx) and Mr(δy) be the space of piecewise polynomial of degree at most r ≥ 3, defined by

Mr(δx)={vC1[0,1]:v|IixPr,1iNx,v(0)=v(1)=0},Mr(δy)={vC1[0,1]:v|IjyPr,1jNy,v(0)=v(1)=0},

where Pr denotes the set of polynomial of degree at most r. It is easy to know that the dimension of the spaces Mx(δx) and My(δy) are (r − 1)Nx := Mx and (r − 1)Ny := My, respectively.

Let δ = δxδy be a quasi-uniform partition of Ω, and Mr(δ) = Mr(δx) ⊗ Mr(δy) with the dimension of M>x × My. Let {λj}j=1r1 denotes the nodes for the {r − 1}-point Gaussian quadrature rule on the interval I with corresponding weights {ωj}j=1r1. Denote by

Gx={ξi,lx}i,l=1Nx,r1andGy={ξj,my}j,m=1Ny,r1

as the sets of Gauss points in x and y direction, respectively, where

ξi,lx=xi1+hixλl,ξj,my=yj1+hjyλm,1l,mr1.

Let 𝓖 = {ξ = (ξx, ξy) : ξx ∈ 𝓖x, ξy ∈ 𝓖y}. For the functions u and v defined on 𝓖, the inner product 〈u, v〉 and norm ∥vMr are respectively defined by

u,v=i=1Nxj=1Nyhixhjyl=1r1m=1r1ωlωm(uv)(ξi,l,ξj,m),vMr2=v,v.

For m a nonnegative integer, let Hm(Ω) denotes the usual Sobolev space with norm

vHm=(l=0mi+j=li+jvxiyj2)12,

where the norm ∥⋅∥ denotes the usual L2 norm, sometimes it is written as ∥⋅∥H0 for convenience. The following important lemmas are required in our forthcoming analysis. First, we introduce the differentiable (resp. twice differentiable) map W : [0, T] → Mr(δ) by

Δ(uW)=0onG×[0,T],

where u is the solution of the Eqs.(1.1)-(1.3) . Then we have the following estimates for uW and its time derivatives.

Lemma 2.1

[26] Ifl u/∂ tlHr+3−j, for allt ∈ [0, T], l = 0, 1, 2, j = 0, 1, 2, andWis defined by (2.1), then there exists a constantCsuch that

l(uW)tlHjChr+1jlutlHr+3j.

Lemma 2.2

[26] Ifi u/∂ tiHr+3, fort ∈ [0, T], i = 0, 1, then

l+i(uW)xl1yl2tiMrChr+1liutiHr+3,

where 0 ≤ l = l1 + l2 ≤ 4.

Lemma 2.3

[27] If u, vMr(δ), then

Δu,v=u,Δv,

and there exists a positive constant C such that

Δu,uCu20.

Lemma 2.4

[28] The norms ∥⋅∥Mrand ∥⋅∥ are equivalent onMr(δ).

Throughout the paper, we denote C > 0 a constant which is independent of mesh sizes h and τ. The following Young's inequality will also be used repeatedly,

XYεX2+14εY2,X,YR,ε>0.

2.2 Construction of the fully discrete orthogonal spline collocation scheme

In this subsection, we consider discrete-time OSC schemes for solving the Eqs.(1.1)-(1.3). Our main idea of the proposed method is to transform the time fractional diffusion-wave equation into its equivalent partial integro-differential equation. To construct the continuous-time OSC scheme to the solution u of (1.1), we introduce the Riemann-Liouville fractional integral which is defined by

0Itαu(x,y,t)=1Γ(α)0tu(x,y,s)(ts)1αds,

where 0 < α = γ − 1 < 1.

We integrate the equation(1.1) using Riemann-Liouville fractional integral operator 0Itα defined in (2.7), then the problem is transformed into its equivalent partial integro-differential equation as follows

ut(x,y,t)0ItαΔu(x,y,t)+0Itαu(x,y,t)=0Itαf(x,y,t)+φ(x,y).

Let tk = , k = 0, 1, ⋯, N, where τ = T/N is the time step size. For the convenience of description, we define Dtun+1=un+1unτ,andun+12=un+1+un2, where unu(x, y, tn). Based on the idea of weighted and shifted Grünwald difference operator, Wang and Vong ([17]) established the second order accuracy approximation formula of the Riemann-Liouville fractional integral operator 0Itαun+1, which is called as WSGI approximation,

0Itαun+1=ταk=0nλk(α)un+1k+E~0Itαun+1+E~,

where = O(τ2) and

λ0(α)=(1α2)ω0(α),λk(α)=(1α2)ωk(α)+α2ωk1(α),k1,

here

ωk(α)=(1)kαk,ω0(α)=1,ωk(α)=(1+α1k)ωk1(α),k1.

By using the Crank-Nicolson difference scheme and WSGI approximation formula to discretize the equation (2.8), we obtain the semi-discrete scheme in time direction

Dtun+10Itαun+12+0Itαun+12=gn+12+En+12,

where gn+12=0Itαfn+12+φ(x,y),En+12=E~+Ecn+12=O(τ2),Ecn+12=Dtun+12ut(tn+12)=O(τ2). Then by using (2.9), (2.12), the fully discrete WSGI-OSC scheme for Eqs(1.1) consists in finding {uhn}n=0N1Mr(δ) such that

uhn+1uhnτταk=0nλk(α)uhn+12k+ταk=0nλk(α)uhn+12k=gn+12.

For the needs of analysis, we give the following equivalent Galerkin weak formulation of the equation(2.12) by multiplying the equation with vH01 and integrating with respect to spatial domain Ω

(Dtun+1,v)+(0Itαun+12,v)+(0Itαun+12,v)=(gn+12,v)+(En+12,v).

We take the space Mr(δ) ⊂ H01 and obtain the fully discrete scheme as follows:

(uhn+1uhnτ,vh)+ταk=0nλk(α)(uhn+12k,vh)+ταk=0nλk(α)(uhn+12k,vh)=(gn+12,vh),vhMr(δ)

3 Stability and convergence analysis of WSGI-OSC scheme

In this section, we will give the stability and convergence analysis for fully-discrete WSGI-OSC scheme (2.13). To this end, we further need the following lemmas.

Lemma 3.1

[17] Let{λk(α)}k=0defined in (2.10), then for any positive integerkand real vector (v1, v2, ⋯, vk)T ∈ 𝓡k, it holds that

n=0k1(p=0nλp(α)vn+1p)vn+10.

Lemma 3.2

(Gronwall’s ineqality) [29] Assume thatknandpnare nonnegative sequence, and the sequenceϕnsatisfies

ϕ0g0,ϕnϕ0+l=0n1pl+l=0n1klpl,n1,

where, g0 ≥ 0. Then the sequenceϕn satisfies

ϕn(g0+l=0n1pl)exp(l=0n1kl),n1.

Theorem 3.1

The fully-discrete WSGI-OSC scheme (2.15) is unconditionally stable for sufficiently smallτ > 0, it holds

||uhL+1||2C(||uh0||2+max0nN1||gn+12||2),1LN1.

Proof

Taking vh=uhn+12=un+1+un2 in (2.15) and applying the Cauchy-Schwarz inequality and Young inequality, it gives that

12τ(||uhn+1||2||uhn||2)+ταk=0nλk(α)[(uhn+12k,vh)+(uhn+12k,vh)]12(||gn+12||2+||uhn+12||2).

Summing (3.2) for n from 0 to L(0 ≤ nN − 1), we obtain

12τn=0L(||uhn+1||2||uhn||2)+ταn=0Lk=0nλk(α)[(uhn+12k,vh)+(uhn+12k,vh)]12n=0L(||gn+12||2+||uhn+12||2).

Multiplying the above equation by 2τ, also using Lemma 1, then dropping the nonnegative terms

2τα+1n=0Lk=0nλk(α)[(uhn+12k,vh)+(uhn+12k,vh)],

we have

||uhL+1||2||uh0||2+τn=0L(||gn+12||2+||uhn+12||2)||uh0||2+Tmax0nN1||gn+12||2+τn=0L||uhn+12||2||uh0||2+Tmax0nN1||gn+12||2+τn=0L12(||uhn+1||2+||uhn||2).

Then, it gives that,

(112τ)||uhL+1||2(1+12τ)||uh0||2+Tmax0nN1||gn+12||2+τn=1L||uhn||2.

Provided the time step τ is sufficiently small, there exists a positive constant C such that

||uhL+1||2C(||uh0||2+Tmax0nN1||gn+12||2+τn=1L||uhn||2).

Using Gronwall’s Lemma 3.2, we get

||uhL+1||2C(||uh0||2+max0nN1||gn+12||2).

The proof is complete.

Theorem 3.2

Supposeuis the exact solution of (1.1)-(1.3), anduhn(0 ≤ nN − 1) is the solution of the problem (2.13) withuh0 = W0, then there exists a positive constantC, independent ofhandτsuch that

u(tn)uhn2C(τ2+hr+1).

Proof

With W defined in (2.1), we set

ηn=Wnun,ζn=uhnWn,0nN,

thus we have

unuhn=ηn+ζn.

Because the estimate of ηn are provided by Lemma 2.2, it is sufficient to bound ζn, then use the triangle inequality to bound unuhn. Firstly, from(1.1), (2.1), (2.13), and(2.15), then for vhMr(δ), we obtain

(ηn+1ηnτ,vh)+ταk=0nλk(α)(ηn+12k,vh)+ταk=0nλk(α)(ηn+12k,vh)=ταk=0nλk(α)(ζn+12k,vh)(ζn+1ζnτ,vh)+(En+12,vh),

where En+12 is defined in (2.12). Taking vh=ηn+12 in (3.11), we have

(ηn+1ηnτ,ηn+12)+ταk=0nλk(α)(ηn+12k,ηn+12)+ταk=0nλk(α)(ηn+12k,ηn+12)=ταk=0nλk(α)(ζn+12k,ηn+12)(ζn+1ζnτ,ηn+12)+(En+12,ηn+12).

Multiplying (3.12) by 2τ, and summing from n = 0 to n = L − 1 (1 ≤ nN + 1), it follows that

n=0L1(||ηn+1||2||ηn||2)+2τα+1n=0L1k=0nλk(α)[(ηn+12k,ηn+12)+(ηn+12k,ηn+12)]=2τα+1n=0L1k=0nλk(α)(ζn+12k,ηn+12)2τn=0L1(ζn+1ζnτ,ηn+12)+2τn=0L1(En+12,ηn+12)=I1+I2+I3.

Next, we will give the estimate of I1, I2 and I3, respectively.

I1=2τα+1n=0L1k=0nλk(α)(ζn+12k,ηn+12)=2τα+1n=0L1(0Itn+1αζ+0Itnαζ2E~,ηn+12)=2τα+1n=0L1(1Γ(α)0tn+1ζ(x,y,s)(tn+1s)1αds+1Γ(α)0tnζ(x,y,s)(tns)1αds2E~,ηn+12)τn=0L1(1Γ(α)α[(tn+1s)α|0tn+1+(tns)α|0tn]max0stn+1||ζ(x,y,s)||+||2E~||)||ηn+12||τΓ(α+1)n=0L1(2Tαmax0tT||ζ(x,y,t)||+||E~||)||ηn+12||Cτn=0L1(2Tαmax0tT||ζ(x,y,t)||+||E~||)||ηn+12||Cτn=0L1(τ4+max0tT||ζ(x,y,t)||2+||ηn+12||2).

Taking advantages of mean value theorem and Cauchy-Schwarz inequality as well as Young inequality, we have tntn+θtn+1

I2+I3=2τn=0L1(ζn+1ζnτ,ηn+12)+2τn=0L1(En+12,ηn+12)=τn=0L1(||ζt(x,y,tn+θ||2+||En+12||2+2||ηn+12||2).

Using Lemma 1, we obtain

2τα+1n=0Lk=0nλk(α)[(ηn+12k,η)+(ηn+12k,η)]0.

Substituting (3.14), (3.15), (3.16) in (3.13) and removing the nonnegative terms, we attain

||ηL||2||η0||2+Cτn=0L1(τ4+max0tT||ζ(x,y,t)||2+||ηn+12||2)+τn=0L1(||ζt(x,y,tn+θ||2+||En+12||2+2||ηn+12||2),

that is

(1Cτ)||ηL||2Cτn=0L1||ηn||2+Cτn=0L1(τ4+max0tT||ζ(x,y,t)||2+||ζt(x,y,tn+θ||2).

Using the Gronwall’s inequality, Lemma 2.2 and triangle inequality, in the case that the time step τ is sufficiently small, there exists a positive constant C such that

||ηL||2exp(CT).Cτn=0L1(τ4+Ch2r+2||u||Hr+32+Ch2r+2||ut||Hr+32)C(τ4+h2r+2)

and

||u(tL)uhL||2(||ηL||+||ζL||)2C(τ4+h2r+2)

which completes the proof.

4 Description of the WSGI-OSC scheme

It can be observed from the fully discrete scheme (2.13) that we need to handle a two-dimensional partial differential equation for each time level, that is

(1+12τα+1λ0(α))uhn+112τα+1λ0(α)Δuhn+1=12τα+1k=1n+1λk(α)(Δuhn+1k+uhn+1k)12τα+1k=0nλk(α)(Δuhnk+uhnk)+τgn+1+gn2+uhn

We denote α0=12τα+1λ0(α),β0=12τα+1, then the above equation can be rewritten as

(1+α0)uhn+1α0Δuhn+1=β0k=1n+1λk(α)(Δuhn+1kuhn+1k)+β0k=0nλk(α)(Δuhnkuhnk)+τgn+1+gn2+uhn,n=0,,N1.

For applying the numerical schemes, firstly, we usually represent uhn by the base functions of Mr(δ), then solve the coefficients of the representation formula. Letting

Mr(δx)=span{Φ1,Φ2,,ΦMx1,ΦMx},Mr(δy)=span{Ψ1,Ψ2,,ΨMy1,ΨMy},

then

uhn(x,y)=j=1Myi=1Mxu^i,jnΦi(x)Ψj(y),

where {u^i,jn}i,j=1Mx,My are unknown coefficients to be determined. Setting

u^=[u^1,1n,u^1,2n,,u^1,Myn,u^2,1n,u^2,2n,,u^Mx,Myn]T,

then the equation (4.2) can be written in the following form by Kronecker product

{(1+α0)(BxBy)+α0(AxBy+BxAy)}u^n+1=β0{AxBy+BxAy+BxBy}(k=1n+1λk(α)u^n+1k+k=0nλk(α)u^nk)+(BxBy)u^n+12τ(G1n+1+G2n),n=0,,N1,

where

Ax=(ai,jx)i,j=1Mx,ai,jx=Φj(ξix),Bx=(bi,jx)i,j=1Mx,bi,jx=Φj(ξix),Ay=(ai,jy)i,j=1My,ai,jy=Ψj(ξiy),By=(bi,jy)i,j=1My,bi,jy=Ψj(ξiy),

and

G1n+1=[gn+1(ξ1x,ξ1y),gn+1(ξ1x,ξ2y),,gn+1(ξ1x,ξMyy),gn+1(ξ2x,ξ1y),,gn+1(ξMxx,ξMyy)]T,

G2n=[gn(ξ1x,ξ1y),gn(ξ1x,ξ2y),,gn(ξ1x,ξMyy),gn(ξ2x,ξ1y),gn(ξ2x,ξ2y),,gn(ξMxx,ξMyy)]T.

The matrices Ax, Bx, Ay and By are Mx × My having the following structure,

××××××××××××××××××××.

We carry out the WSGI-OSC scheme in piecewise Hermite cubic spline space M3(δ), which satisfies zero boundary condition. Detailedly, we choose the basis of cubic Hermite polynomials [30], namly, for 1 ≤ iK − 1, it follows that

ϕi(x)=2(xxi1)3h3+3(xxi1)2h2,xi1xxi,2(xxi1)3h3+3(xxi+1)2h2,xixxi+1,0,x<xi1orx>xi+1,

and

ψi(x)=(xxi1)2(xxi)h2,xi1xxi,(xxi)(xxi+1)2h2,xixxi+1,0,x<xi1orx>xi+1.

Note that functions ϕi(x), ψi(x) satisfy zero boundary conditions ϕi(0) = ϕi(1) = ψi(0) = ψi(1) = 0. Renumber the basis functions and let

{ψ0,ϕ1,ψ1,ϕ2,,ϕK1,ψK1,ψK}={Φ1,Φ2,Φ3,,Φ2K},

then

M3(δx)=span{Φ1,Φ2,Φ3,,Φ2K},M3(δy)=span{Φ1,Φ2,Φ3,,Φ2K}.

In order to recover the coefficient matrix of the equations (4.3), we need to calculate the values of the basis functions at the Gauss point and their second-order derivatives. They are defined as follows:

H1(uj)=(1+2uj)(1uj)2,H2(uj)=uj(1uj)2hk,H3(uj)=uj2(32uj),H4(uj)=uj2(uj1)hk,I1(uj)=(12uj6)/hk2,I2(uj)=(6uj4)/hk,I3(uj)=(612uj)/hk2,I4(uj)=(6uj2)/hk,

where u1=(33)/6,u2=(3+3)/6,Hi and Ii denotes the formulas of Hermite polynomials and their second-order derivatives at Gauss points, respectively. Based on the above descriptions of basis functions, we give an example of matrix Ax and Bx in the case of Nx = Ny = 5 and hk = 1/Nx. We have

Ax=I2(u1)I3(u1)I4(u1)0000000I2(u2)I3(u2)I4(u2)00000000I1(u1)I2(u1)I3(u1)I4(u1)000000I1(u2)I2(u2)I3(u2)I4(u2)00000000I1(u1)I2(u1)I3(u1)I4(u1)000000I1(u2)I2(u2)I3(u2)I4(u2)00000000I1(u1)I2(u1)I3(u1)I4(u1)000000I1(u2)I2(u2)I3(u2)I4(u2)00000000I1(u1)I2(u1)I4(u1)0000000I1(u2)I2(u2)I4(u2).

Bx=H2(u1)H3(u1)H4(u1)0000000H2(u2)H3(u2)H4(u2)00000000H1(u1)H2(u1)H3(u1)H4(u1)000000H1(u2)H2(u2)H3(u2)H4(u2)00000000H1(u1)H2(u1)H3(u1)H4(u1)000000H1(u2)H2(u2)H3(u2)H4(u2)00000000H1(u1)H2(u1)H3(u1)H4(u1)000000H1(u2)H2(u2)H3(u2)H4(u2)00000000H1(u1)H2(u1)H4(u1)0000000H1(u2)H2(u2)H4(u2).

It can be seen from the tensor product calculation that the WSGI-OSC scheme requires the solution of an almost block diagonal linear system at each time level, which can be solved efficiently by the software package COLROW [24].

5 Numerical experiments

In this section, four examples are given to demonstrate our theoretical analysis. In our implementations, we adopt the space of piecewise Hermite bicubics(r = 3) on uniform partitions of I in both x and y directions with Nx = Ny = K. The forcing term f(x, y, t) is approximated by the piecewise Hermite interpolant projection in the Guass points. To check the accuracy of WSGI-OSC scheme, we present L and L2 errors at T = 1 and the corresponding convergence order defined by

Convergence orderlog(em/em+1)log(hm/hm+1),

where hm = 1/K is the time step size and em is the norm of the corresponding error.

Example 1

We consider the following one-dimensional time-fractional diffusion-wave equation

0cDtγu(x,t)=2u(x,t)x2u(x,t)+f(x,t),0<x<1,0<t1,u(x,0)=0,u(x,0)t=0,0x1,u(0,t)=u(1,t)=0,0t1,

where f(x,t)=Γ(4)Γ(4α)t3γx2(1x)2ex2t3ex(14x+4x3). The analytical solution of this equation is u(x, t) = t3x2(1 − x)2ex.

From the theoretical analysis, the numerical convergence order of WSGI-OSC (4.2) is expected to be O(τ2 + h4) when r = 3. In order to check the second order accuracy in time direction, we select τ = h so that the error caused by the spatial approximation can be negligible. Table 1 lists L and L2 errors and the corresponding convergence orders of WSGI-OSC scheme for γ ∈ (1, 2). We observe that our scheme generates the temporal accuracy with the order 2. To test the spatial approximation accuracy, Table 2 shows that our scheme has the accuracy of 4 in spatial direction, where the temporal step size τ = h2 is fixed. Numerical solution and global error for γ = 1.3, h = 1/32, τ = 1/32 are shown in Figure 1.

Figure 1
Figure 1

Numerical solution (a) and global error (b) for Example 1 with γ = 1.3, h = 1/32, τ = 1/32.

Citation: Open Mathematics 18, 1; 10.1515/math-2020-0007

Table 1

The L, L2 errors and temporal convergence orders with τ = h for Example 1.

γτL errorConvergence orderL2 errorConvergence order
1.11107.0727×10−54.4681×10−5
1201.7932×10−51.97981.1012×10−52.0206
1404.5623×10−61.97472.7487×10−62.0022
1801.1483×10−61.99036.8758×10−71.9992
1.31102.6081×10−41.7238×10−4
1206.6648×10−51.96844.2518×10−52.0194
1401.6825×10−61.98601.0577×10−52.0072
1804.2263×10−71.99312.6387×10−62.0030
1.51104.1657×10−42.7911×10−4
1201.0633×10−41.97016.8593×10−52.0247
1402.6736×10−51.99161.7020×10−52.0108
1806.7115×10−61.99414.2405×10−62.0050
1.71105.3422×10−43.6265×10−4
1201.3701×10−41.96328.9343×10−52.0212
1403.4419×10−51.99302.2160×10−52.0114
1808.6292×10−61.99595.5175×10−62.0059
1.91105.7600×10−43.9339×10−4
1201.4884×10−41.95239.7244×10−52.0163
1403.7391×10−51.99302.4112×10−52.0119
1809.3633×10−61.99765.9996×10−62.0068
1.951105.6941×10−43.8862×10−4
1201.4696×10−41.95409.6061×10−52.0163
1403.6917×10−51.99312.3812×10−52.0123
1809.2425×10−61.99795.9232×10−62.0072

Table 2

The L, L2 errors and spatial convergence orders with τ = h2 for Example 1.

γhL errorConvergence orderL2 errorConvergence order
1.11102.4371×10−61.7740×10−6
1201.5377×10−73.98631.0837×10−74.0329
1409.6290×10−93.99726.6928×10−94.0172
1806.0225×10−103.99894.1576×10−104.0088
1.31103.8377×10−62.6750×10−6
1202.4364×10−73.97741.6332×10−74.0338
1401.5241×10−83.99871.0085×10−84.0174
1809.5308×10−103.99926.2644×10−104.0089
1.51104.7527×10−63.2535×10−6
1203.0159×10−73.97811.9851×10−74.0347
1401.8863×10−83.99901.2256×10−84.0177
1801.1798×10−93.99907.6129×10−104.0089
1.71105.1530×10−63.4857×10−6
1203.2579×10−73.98342.1258×10−74.0354
1402.0382×10−83.99861.3123×10−84.0178
1801.2754×10−93.99828.1509×10−104.0090
1.91104.6730×10−63.0735×10−6
1202.9311×10−73.99481.8735×10−74.0361
1401.8412×10−83.99271.1563×10−84.0181
1801.1509×10−93.99997.1819×10−104.0090
1.951104.3316×10−62.8280×10−6
1202.7151×10−73.99581.7235×10−74.0364
1401.7062×10−83.99221.0637×10−84.0182
1801.0665×10−93.99996.6066×10−104.0091

Example 2

Consider the following one-dimensional fractional diffusion-wave equation

0cDtγu(x,t)=2u(x,t)x2u(x,t)+f(x,t),0<x<1,0<t1,u(x,0)=0,u(x,0)t=sinπx,0x1,u(0,t)=u(1,t)=0,0t1,

where f(x,t)=[2Γ(3γ)t2γ+(t2t)π2+(t2t)]sinπx. The analytical solution of this equation is u(x, t) = (t2t)sinπx.

In order to test the temporal accuracy of the proposed method, we choose τ = h to avoid contamination of the spatial error. The maximum L, L2 errors and temporal convergence orders are shown in Table 3. To check the convergence order in space, the time step τ and space step h are chosen such that τ = h2, and γ = 1.1, 1.3, 1.5, 1.7, 1.9, 1.95. Table 4 presents the maximum L, L2 errors and spatial convergence orders. The results in Tables 3 and 4 demonstrate the expected convergence rates of 2 order in time and 4 order in space simultaneously. Numerical solution and global error at T = 1 with γ = 1.5, h = 1/32, τ = 1/32 are shown in Figure 2.

Figure 2
Figure 2

Numerical solution (a) and global error (b) for Example 2 with γ = 1.5 at T = 1 (h = 1/32, τ = 1/32).

Citation: Open Mathematics 18, 1; 10.1515/math-2020-0007

Table 3

The L, L2 errors and temporal convergence orders with τ = h for Example 2.

γτL errorConvergence orderL2 errorConvergence order
1.11102.7779×10−51.8686×10−5
1206.9405×10−62.00094.5452×10−62.0395
1401.7225×10−62.01051.1135×10−62.0292
1804.2704×10−72.01212.7427×10−72.0215
1.31106.8399×10−54.6042×10−5
1201.6912×10−52.01591.1079×10−52.0551
1404.1818×10−62.01582.7032×10−62.0352
1801.0358×10−62.01346.6503×10−72.0232
1.51101.0251×10−46.9519×10−5
1202.5114×10−52.02921.6555×10−52.0701
1406.2025×10−62.01764.0327×10−62.0375
1801.5384×10−62.01149.9335×10−72.0214
1.71101.4424×10−49.9816×10−5
1203.5642×10−52.01682.4001×10−52.0562
1408.8717×10−62.00635.8868×10−62.0275
1802.2116×10−62.00411.4572×10−62.0143
1.91101.9061×10−41.2932×10−4
1204.7290×10−52.01103.1561×10−52.0347
1401.1810×10−52.00157.7937×10−62.0178
1802.9519×10−62.00031.9367×10−62.0087
1.951102.0110×10−41.3455×10−4
1204.9930×10−52.00993.2930×10−52.0306
1401.2482×10−52.00018.1369×10−62.0169
1803.1218×10−61.99942.0222×10−62.0085

Table 4

The L, L2 errors and spatial convergence orders with τ = h2 for Example 2.

γhL errorConvergence orderL2 errorConvergence order
1.11109.8873×10−87.9806×10−8
1206.3013×10−93.97195.0124×10−93.9929
1404.0052×10−103.97573.1726×10−103.9818
1802.5425×10−113.97752.0139×10−113.9776
1.31102.0590×10−71.1973×10−7
1201.2348×10−84.05967.0248×10−94.0912
1407.5090×10−104.03954.2273×10−104.0547
1804.6084×10−114.02632.5826×10−114.0328
1.51103.1378×10−71.8224×10−7
1201.9140×10−84.03511.0838×10−84.0716
1401.1827×10−94.01656.6114×10−104.0350
1807.3513×10−114.00794.0836×10−114.0170
1.71104.2637×10−72.5157×10−7
1202.6414×10−84.01271.5170×10−84.0516
1401.6453×10−94.00499.3246×10−104.0241
1801.0270×10−104.00195.7825×10−114.0113
1.91106.2873×10−73.5996×10−7
1203.9276×10−84.00072.1898×10−84.0389
1402.4548×10−94.00001.3510×10−94.0188
1801.5342×10−104.00008.3898×10−114.0092
1.951106.7414×10−73.9216×10−7
1204.2262×10−83.99562.3894×10−84.0367
1402.6423×10−93.99951.4745×10−94.0184
1801.6515×10−103.99999.1572×10−114.0091

Example 3

Consider the following two-dimensional fractional diffusion-wave equation

0cDtγu(x,y,t)Δu(x,y,t)+u(x,y,t)=f(x,y,t),u(x,y,0)=0,u(x,y,0)t=0,(x,y)Ω,u(x,y,t)=0,(x,y,t)Ω×(0,T],

where Ω=[0,1]×[0,1],T=1,f(x,y,t)=[Γ(4)Γ(4γ)t3γxy(1x)(1y)+t3xy(73y3xxy)]ex+y. The exact solution of the equation is u(x, y, t) = t3xy(1 − x)(1 − y)ex+y.

Similar to the selection of parameters in Examples 1 and 2, Tables 5 and 6 list the maximum L, L2 errors and convergence orders, respectively. The similar convergence rates in time and space are also obtained. As we hope, the convergence order of all numerical results match that of the theoretical analysis. Figure 3 plots the numerical solution and global error at T = 1 with γ = 1.7, h = 1/32, τ = 1/32.

Figure 3
Figure 3

Numerical solution (a) and global error (b) for Example 3 with γ = 1.7 at T = 1 (h = 1/32, τ = 1/32).

Citation: Open Mathematics 18, 1; 10.1515/math-2020-0007

Table 5

The L, L2 errors and temporal convergence orders for Example 3.

γNL errorConvergence orderL2 errorConvergence order
1.1101.6611×10−48.3486×10−5
157.5461×10−51.94613.7876×10−51.9493
204.2909×10−51.96242.1509×10−51.9669
252.7589×10−51.97921.3841×10−51.9754
1.3105.1729×10−42.6164×10−4
152.3249×10−41.97241.1769×10−41.9704
201.3148×10−41.98136.6585×10−51.9799
258.4565×10−51.97794.2760×10−51.9847
1.5107.7899×10−43.9475×10−4
153.4829×10−41.98531.7651×10−41.9850
201.9648×10−41.98999.9627×10−51.9882
251.2607×10−41.98866.3896×10−51.9905
1.7109.8958×10−45.0659×10−4
154.3990×10−41.99952.2433×10−42.0090
202.4748×10−41.99951.2609×10−42.0028
251.5841×10−41.99948.0685×10−52.0006
1.9101.1985×10−36.2808×10−4
155.3173×10−42.00442.7856×10−42.0052
202.9891×10−42.00221.5657×10−42.0026
251.9123×10−42.00151.0018×10−42.0014

Table 6

The L, L2 errors and spatial convergence orders for Example 3.

γNL errorConvergence orderL2 errorConvergence order
1.1101.7277×10−65.8164×10−7
153.7129×10−73.79211.1583×10−73.9799
201.2237×10−73.85823.6753×10−83.9902
255.1343×10−83.89221.5074×10−83.9942
1.3104.5383×10−62.0806×10−6
158.9315×10−74.00914.1188×10−73.9946
202.8185×10−74.00921.3042×10−73.9974
251.1523×10−74.00835.3439×10−83.9984
1.5107.1532×10−63.3974×10−6
151.4118×10−64.00216.7176×10−73.9976
204.4624×10−74.00362.1262×10−73.9988
251.8263×10−74.00378.7104×10−83.9993
1.7109.2188×10−64.4527×10−6
151.8187×10−64.00318.7956×10−74.0000
205.7483×10−74.00382.7830×10−73.9999
252.3526×10−74.00361.1399×10−73.9999
1.9101.1444×10−55.7230×10−6
152.2505×10−64.01101.1299×10−64.0011
207.1020×10−74.00913.5746×10−74.0005
252.9046×10−74.00681.4641×10−74.0003

Example 4

Consider the following two-dimensional fractional diffusion-wave equation

0cDtγu(x,y,t)Δu(x,y,t)+u(x,y,t)=f(x,y,t),u(x,y,0)=0,u(x,y,0)t=0,(x,y)Ω,u(x,y,t)=0,(x,y,t)Ω×(0,T]

where Ω=[0,1]×[0,1],T=1,f(x,y,t)=[Γ(3+γ)2+(2π2+1)tγ]t2sinπxsinπy. The exact solution of the equation is u(x, y, t) = t2+γ sinπx sin πy.

Tables 7 and 8 display L and L2 errors and the corresponding convergence orders in time and space for some γ ∈ (1, 2). Once again, the expected convergence rates with second-order accuracy in time direction and fourth-order accuracy in spatial direction can be observed from two tables. Numerical solution and global error at T = 1 with γ = 1.9, h = 1/32, τ = 1/32 are displayed in Figure 4.

Figure 4
Figure 4

Numerical solution (a) and global error (b) for Example 4 with γ = 1.9 at T = 1 (h = 1/32, τ = 1/32).

Citation: Open Mathematics 18, 1; 10.1515/math-2020-0007

Table 7

The L, L2 errors and temporal convergence orders for Example 4.

γNL errorConvergence orderL2 errorConvergence order
1.1108.8381×10−44.4449×10−4
153.9978×10−41.95662.0073×10−41.9607
202.2738×10−41.96151.1385×10−41.9711
251.4625×10−41.97757.3238×10−51.9771
1.3103.1514×10−31.5847×10−3
151.4225×10−31.96177.1426×10−41.9654
208.0814×10−41.96564.0464×10−41.9752
255.1939×10−41.98112.6009×10−41.9807
1.5105.3058×10−32.6680×10−3
152.3861×10−31.97091.1981×10−31.9745
201.3534×10−31.97116.7766×10−41.9808
258.6906×10−41.98514.3519×10−41.9847
1.7107.2062×10−33.6236×10−3
153.2347×10−31.97551.6242×10−31.9792
201.8321×10−31.97609.1737×10−41.9857
251.1754×10−31.98935.8858×10−41.9889
1.9108.0346×10−34.0402×10−3
153.6198×10−31.96651.8175×10−31.9701
202.0516×10−31.97361.0273×10−31.9833
251.3162×10−31.98936.5910×10−41.9888

Table 8

The L, L2 errors and spatial convergence orders for Example 4.

γNL errorConvergence orderL2 errorConvergence order
1.1101.2725×10−56.5169×10−5
152.5700×10−63.94531.2858×10−54.0029
208.0847×10−74.02024.0665×10−54.0014
253.3300×10−73.97511.6653×10−54.0009
1.3103.6079×10−51.8230×10−5
157.1968×10−63.97583.6064×10−63.9964
202.2773×10−63.99971.1417×10−63.9982
259.3472×10−73.99074.6773×10−73.9989
1.5105.7773×10−52.9133×10−5
151.1491×10−53.98295.7620×10−63.9968
203.6402×10−63.99591.8240×10−63.9984
251.4930×10−63.99427.4725×10−73.9991
1.7107.6488×10−53.8541×10−5
151.5190×10−53.98687.6189×10−63.9981
204.8133×10−63.99482.4113×10−63.9991
251.9734×10−63.99599.8780×10−73.9994
1.9108.4694×10−54.2666×10−5
151.6803×10−53.98928.4290×10−63.9997
205.3240×10−63.99512.6670×10−63.9999
252.1823×10−63.99681.0924×10−64.0000

6 Conclusion

In this paper, we have constructed a Crank-Nicolson WSGI-OSC method for the two-dimensional time-fractional diffusion-wave equation. The original fractional diffusion-wave equation is transformed into its equivalent partial integro-differential equations, then Crank-Nicolson orthogonal spline collocation method with WSGI approximation is developed. The proposed method holds a higher convergence order than the convergence order O(τ3−γ) of general L1 approximation. The stability and convergence analysis are derived. Some numerical examples are also given to confirm our theoretical analysis.

Acknowledgement

The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper. This work was supported by the National Natural Science Foundation of China (Grant No.11601076) and the Ph.D. Research Start-up Fund Project of East China University of Technology (Grant No.DHBK2019213).

References

  • [1]

    A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

  • [2]

    Y.D. Zhang, Y.M. Zhao, F.L. Wang, and Y.F. Tang, High-accuracy finite element method for 2D time fractional diffusion-wave equation on anisotropic meshes, Int. J. Comput. Math. 95 (2018), no. 1, 218–230.

    • Crossref
    • Export Citation
  • [3]

    L.M. Li, D. Xu, and M. Luo, Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation, J. Comput. Phys. 255 (2013), 471–485,

    • Crossref
    • Export Citation
  • [4]

    Z.Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56 (2006), no. 2, 193–209,

    • Crossref
    • Export Citation
  • [5]

    J.F. Huang, Y.F. Tang, L. Vázquez, and J.Y. Yang, Two finite difference schemes for time fractional diffusion-wave equation, Numer. Algorithms 64 (2013), no. 4, 707–720,

    • Crossref
    • Export Citation
  • [6]

    Y.N. Zhang, Z.Z. Sun, and X. Zhao, Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation, SIAM J. Numer. Anal. 50 (2012), no. 3, 1535–1555,

    • Crossref
    • Export Citation
  • [7]

    M. Aslefallah and E. Shivanian, An efficient meshless method based on RBFs for the time fractional diffusion-wave equation, Afr. Mat. 29 (2018), no. 7-8, 1203–1214,

    • Crossref
    • Export Citation
  • [8]

    M. Dehghan, M. Abbaszadeh, and A. Mohebbi, Analysis of a meshless method for the time fractional diffusion-wave equation, Numer. Algorithms 73 (2016), no. 2, 445–476,

    • Crossref
    • Export Citation
  • [9]

    M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini, and C. Cattanic, Wavelets method for the time fractional diffusion-wave equation, Phys. Lett. A 379 (2015), no. 3, 71–76,

    • Crossref
    • Export Citation
  • [10]

    G. Fairweather, X.H. Yang, D. Xu, and H.X. Zhang, An ADI Crank-Nicolson orthogonal spline collocation method for the two-dimensional fractional diffusion-wave equation, J. Sci. Comput. 65 (2015), no. 3, 1217–1239,

    • Crossref
    • Export Citation
  • [11]

    M. Yaseen, M. Abbas, T. Nazir, and D. Baleanu, A finite difference scheme based on cubic trigonometric B-splines for a time fractional diffusion-wave equation, Adv. Differ. Equ. 2017 (2017), 274,

    • Crossref
    • Export Citation
  • [12]

    A. Esen, O. Tasbozan, Y. Ucar, and N.M. Yagmurlu, A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations, Tbilisi Math. J. 8 (2015), no. 2, 181–193,

    • Crossref
    • Export Citation
  • [13]

    W.Y. Tian, H. Zhou, and W.H. Deng, A class of second order difference approximations for solving space fractional diffusion equations, Math. Comput. 84 (2015), no. 294, 1298–1314,

    • Crossref
    • Export Citation
  • [14]

    Y. Liu, M. Zhang, H. Li, and J.C. Li, High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional subdiffusion equation, Comput. Math. Appl. 73 (2017), no. 6, 1298–1314,

    • Crossref
    • Export Citation
  • [15]

    H. Chen, S. Lü, and W. Chen, A unified numerical scheme for the multi-term time fractional diffusion and diffusion-wave equations with variable coefficients, J. Comput. Appl. Math. 330 (2018), 380–397,

    • Crossref
    • Export Citation
  • [16]

    X. Yang, H. Zhang, and D. Xu, WSGD-OSC Scheme for two-dimensional distributed order fractional reaction-diffusion equation, J. Sci. Comput. 76 (2018), no. 3, 1502–1520,

    • Crossref
    • Export Citation
  • [17]

    Z. Wang and S. Vong, Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation, J. Comput. Phys. 277 (2014), 1–15,

    • Crossref
    • Export Citation
  • [18]

    Y. Cao, B.L. Yin, Y. Liu, and H. Li, Crank-Nicolson WSGI difference scheme with finite element method for multi-dimensional time-fractional wave problem, Comput. Appl. Math. 37 (2018), no. 4, 5126–5145,

    • Crossref
    • Export Citation
  • [19]

    B. Bialecki and G. Fairweather, Orthogonal spline collocation methods for partial differential equations, J. Comput. Appl. Math. 128 (2001), no. 1-2, 55–82,

    • Crossref
    • Export Citation
  • [20]

    C.E. Greenwell-Yanik and G. Fairweather, Analyses of spline collocation methods for parabolic and hyperbolic problems in two space variables, SIAM J. Numer. Anal. 23 (1986), no. 2, 282–296,

    • Crossref
    • Export Citation
  • [21]

    C. Li, T.G. Zhao, W.H. Deng, and Y.J. Wu, Orthogonal spline collocation methods for the subdiffusion equation, J. Comput. Appl. Math. 255 (2014), 517–528,

    • Crossref
    • Export Citation
  • [22]

    L. Qiao and D. Xu, Orthogonal spline collocation scheme for the multi-term time-fractional diffusion equation, Int. J. Comput. Math. 95 (2018), no. 8, 1478–1493,

    • Crossref
    • Export Citation
  • [23]

    H.X. Zhang, X.H. Yang, and D. Xu, A high-order numerical method for solving the 2D fourth-order reaction-diffusion equation, Numer. Algorithms 80 (2019), no. 3, 849–877,

    • Crossref
    • Export Citation
  • [24]

    G. Fairweather and I. Gladwell, Algorithms for almost block diagonal linear systems, SIAM Rev. 46 (2004), no. 1, 49–58,

    • Crossref
    • Export Citation
  • [25]

    A.K. Pani, G. Fairweather, and R.I. Fernandes, ADI orthogonal spline collocation methods for parabolic partial integro-differential equations, IMA J. Numer. Anal. 30 (2010), no. 1, 248–276,

    • Crossref
    • Export Citation
  • [26]

    C.E. Greenwell-Yanik and G. Fairweather, Analyses of spline collocation methods for parabolic and hyperbolic problems in two space variables, SIAM J. Numer. Anal. 23 (1996), no. 2, 282–296,

    • Crossref
    • Export Citation
  • [27]

    M.P. Robinson and G. Fairweather, Orthogonal spline collocation methods for Schrödinger-type equations in one space variable, Numer. Math. 68 (1994), no. 3, 355–376,

    • Crossref
    • Export Citation
  • [28]

    B. Li, G. Fairweather, and B. Bialecki, Discrete-time orthogonal spline collocation methods for Schrödinger equations in two space variables, SIAM J. Numer. Anal. 35 (1998), no. 2, 453–477,

    • Crossref
    • Export Citation
  • [29]

    A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer, Berlin, 1997.

  • [30]

    S. Arora, I. Kaur, H. Kumar, and V.K. Kukreja, A robust technique of cubic hermite collocation for solution of two phase nonlinear model, Journal of King Saud University – Engineering Sciences 29 (2017), no. 2, 159–165,

    • Crossref
    • Export Citation

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

  • [2]

    Y.D. Zhang, Y.M. Zhao, F.L. Wang, and Y.F. Tang, High-accuracy finite element method for 2D time fractional diffusion-wave equation on anisotropic meshes, Int. J. Comput. Math. 95 (2018), no. 1, 218–230.

    • Crossref
    • Export Citation
  • [3]

    L.M. Li, D. Xu, and M. Luo, Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation, J. Comput. Phys. 255 (2013), 471–485,

    • Crossref
    • Export Citation
  • [4]

    Z.Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56 (2006), no. 2, 193–209,

    • Crossref
    • Export Citation
  • [5]

    J.F. Huang, Y.F. Tang, L. Vázquez, and J.Y. Yang, Two finite difference schemes for time fractional diffusion-wave equation, Numer. Algorithms 64 (2013), no. 4, 707–720,

    • Crossref
    • Export Citation
  • [6]

    Y.N. Zhang, Z.Z. Sun, and X. Zhao, Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation, SIAM J. Numer. Anal. 50 (2012), no. 3, 1535–1555,

    • Crossref
    • Export Citation
  • [7]

    M. Aslefallah and E. Shivanian, An efficient meshless method based on RBFs for the time fractional diffusion-wave equation, Afr. Mat. 29 (2018), no. 7-8, 1203–1214,

    • Crossref
    • Export Citation
  • [8]

    M. Dehghan, M. Abbaszadeh, and A. Mohebbi, Analysis of a meshless method for the time fractional diffusion-wave equation, Numer. Algorithms 73 (2016), no. 2, 445–476,

    • Crossref
    • Export Citation
  • [9]

    M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini, and C. Cattanic, Wavelets method for the time fractional diffusion-wave equation, Phys. Lett. A 379 (2015), no. 3, 71–76,

    • Crossref
    • Export Citation
  • [10]

    G. Fairweather, X.H. Yang, D. Xu, and H.X. Zhang, An ADI Crank-Nicolson orthogonal spline collocation method for the two-dimensional fractional diffusion-wave equation, J. Sci. Comput. 65 (2015), no. 3, 1217–1239,

    • Crossref
    • Export Citation
  • [11]

    M. Yaseen, M. Abbas, T. Nazir, and D. Baleanu, A finite difference scheme based on cubic trigonometric B-splines for a time fractional diffusion-wave equation, Adv. Differ. Equ. 2017 (2017), 274,

    • Crossref
    • Export Citation
  • [12]

    A. Esen, O. Tasbozan, Y. Ucar, and N.M. Yagmurlu, A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations, Tbilisi Math. J. 8 (2015), no. 2, 181–193,

    • Crossref
    • Export Citation
  • [13]

    W.Y. Tian, H. Zhou, and W.H. Deng, A class of second order difference approximations for solving space fractional diffusion equations, Math. Comput. 84 (2015), no. 294, 1298–1314,

    • Crossref
    • Export Citation
  • [14]

    Y. Liu, M. Zhang, H. Li, and J.C. Li, High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional subdiffusion equation, Comput. Math. Appl. 73 (2017), no. 6, 1298–1314,

    • Crossref
    • Export Citation
  • [15]

    H. Chen, S. Lü, and W. Chen, A unified numerical scheme for the multi-term time fractional diffusion and diffusion-wave equations with variable coefficients, J. Comput. Appl. Math. 330 (2018), 380–397,

    • Crossref
    • Export Citation
  • [16]

    X. Yang, H. Zhang, and D. Xu, WSGD-OSC Scheme for two-dimensional distributed order fractional reaction-diffusion equation, J. Sci. Comput. 76 (2018), no. 3, 1502–1520,

    • Crossref
    • Export Citation
  • [17]

    Z. Wang and S. Vong, Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation, J. Comput. Phys. 277 (2014), 1–15,

    • Crossref
    • Export Citation
  • [18]

    Y. Cao, B.L. Yin, Y. Liu, and H. Li, Crank-Nicolson WSGI difference scheme with finite element method for multi-dimensional time-fractional wave problem, Comput. Appl. Math. 37 (2018), no. 4, 5126–5145,

    • Crossref
    • Export Citation
  • [19]

    B. Bialecki and G. Fairweather, Orthogonal spline collocation methods for partial differential equations, J. Comput. Appl. Math. 128 (2001), no. 1-2, 55–82,

    • Crossref
    • Export Citation
  • [20]

    C.E. Greenwell-Yanik and G. Fairweather, Analyses of spline collocation methods for parabolic and hyperbolic problems in two space variables, SIAM J. Numer. Anal. 23 (1986), no. 2, 282–296,

    • Crossref
    • Export Citation
  • [21]

    C. Li, T.G. Zhao, W.H. Deng, and Y.J. Wu, Orthogonal spline collocation methods for the subdiffusion equation, J. Comput. Appl. Math. 255 (2014), 517–528,

    • Crossref
    • Export Citation
  • [22]

    L. Qiao and D. Xu, Orthogonal spline collocation scheme for the multi-term time-fractional diffusion equation, Int. J. Comput. Math. 95 (2018), no. 8, 1478–1493,

    • Crossref
    • Export Citation
  • [23]

    H.X. Zhang, X.H. Yang, and D. Xu, A high-order numerical method for solving the 2D fourth-order reaction-diffusion equation, Numer. Algorithms 80 (2019), no. 3, 849–877,

    • Crossref
    • Export Citation
  • [24]

    G. Fairweather and I. Gladwell, Algorithms for almost block diagonal linear systems, SIAM Rev. 46 (2004), no. 1, 49–58,

    • Crossref
    • Export Citation
  • [25]

    A.K. Pani, G. Fairweather, and R.I. Fernandes, ADI orthogonal spline collocation methods for parabolic partial integro-differential equations, IMA J. Numer. Anal. 30 (2010), no. 1, 248–276,

    • Crossref
    • Export Citation
  • [26]

    C.E. Greenwell-Yanik and G. Fairweather, Analyses of spline collocation methods for parabolic and hyperbolic problems in two space variables, SIAM J. Numer. Anal. 23 (1996), no. 2, 282–296,

    • Crossref
    • Export Citation
  • [27]

    M.P. Robinson and G. Fairweather, Orthogonal spline collocation methods for Schrödinger-type equations in one space variable, Numer. Math. 68 (1994), no. 3, 355–376,

    • Crossref
    • Export Citation
  • [28]

    B. Li, G. Fairweather, and B. Bialecki, Discrete-time orthogonal spline collocation methods for Schrödinger equations in two space variables, SIAM J. Numer. Anal. 35 (1998), no. 2, 453–477,

    • Crossref
    • Export Citation
  • [29]

    A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer, Berlin, 1997.

  • [30]

    S. Arora, I. Kaur, H. Kumar, and V.K. Kukreja, A robust technique of cubic hermite collocation for solution of two phase nonlinear model, Journal of King Saud University – Engineering Sciences 29 (2017), no. 2, 159–165,

    • Crossref
    • Export Citation
OPEN ACCESS

Journal + Issues

Open Mathematics is a fully peer-reviewed, open access, electronic journal that publishes significant and original works in all areas of mathematics. The journal publishes both research papers and comprehensive and timely survey articles. Open Math aims at presenting high-impact and relevant research on topics across the full span of mathematics.

Search