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

  • Xiaoyong Xu EMAIL logo and Fengying Zhou
Published/Copyright: March 5, 2020
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Open Mathematics
From the journal Open Mathematics Volume 18 Issue 1

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.

Keywords: diffusion-wave equation; Crank-Nicolson method; weighted and shifted Grünwald integral operator; orthogonal spline collocation method; Caputo derivative
MSC 2010: 65M12; 26M33

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] (1.1)

subject to the initial condition

u(x,y,0)=ϕ(x,y),∂u(x,y,0)∂t=φ(x,y),(x,y)∈Ω, (1.2)

and the boundary condition

u(x,y,t)=0,(x,y,t)∈∂Ω×(0,T], (1.3)

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(2−γ)∫0t∂2u(x,y,s)∂s2(t−s)1−γds,

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=xi−xi−1, Iix=(xi−1,xi),1≤i≤Nx,  and  hjy=yj−yj−1, Ijy=(yj−1,yj),1≤j≤Ny , h=max(max1≤i≤Nxhix,max1≤j≤Nyhjy). Let Mr(δx) and Mr(δy) be the space of piecewise polynomial of degree at most r ≥ 3, defined by

Mr(δx)={v∈C1[0,1]:v|Iix∈Pr,1≤i≤Nx,v(0)=v(1)=0},Mr(δy)={v∈C1[0,1]:v|Ijy∈Pr,1≤j≤Ny,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 = δx ⊗ δy be a quasi-uniform partition of Ω, and Mr(δ) = Mr(δx) ⊗ Mr(δy) with the dimension of M>x × My. Let {λj}j=1r−1 denotes the nodes for the {r − 1}-point Gaussian quadrature rule on the interval I with corresponding weights {ωj}j=1r−1 . Denote by

Gx={ξi,lx}i,l=1Nx,r−1andGy={ξj,my}j,m=1Ny,r−1

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

ξi,lx=xi−1+hixλl,ξj,my=yj−1+hjyλm,1≤l,m≤r−1.

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 ∥v∥Mr are respectively defined by

〈u,v〉=∑i=1Nx∑j=1Nyhixhjy∑l=1r−1∑m=1r−1ωlωm(uv)(ξi,l,ξj,m),∥v∥Mr2=〈v,v〉.

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

∥v∥Hm=(∑l=0m∑i+j=l∥∂i+jv∂xi∂yj∥2)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(u−W)=0onG×[0,T], (2.1)

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

Lemma 2.1

[26] If ∂l u/∂ tl ∈ Hr+3−j, for all t ∈ [0, T], l = 0, 1, 2, j = 0, 1, 2, and W is defined by (2.1), then there exists a constant C such that

∥∂l(u−W)∂tl∥Hj≤Chr+1−j∥∂lu∂tl∥Hr+3−j. (2.2)

Lemma 2.2

[26] If ∂i u/∂ ti ∈ Hr+3, for t ∈ [0, T], i = 0, 1, then

∥∂l+i(u−W)∂xl1∂yl2∂ti∥Mr≤Chr+1−l∥∂iu∂ti∥Hr+3, (2.3)

where 0 ≤ l = l1 + l2 ≤ 4.

Lemma 2.3

[27] If u, v ∈ Mr(δ), then

〈−Δu,v〉=〈u,−Δv〉, (2.4)

and there exists a positive constant C such that

〈−Δu,u〉≥C∥∇u∥2≥0. (2.5)

Lemma 2.4

[28] The norms ∥⋅∥Mr and ∥⋅∥ are equivalent on Mr(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,

XY≤εX2+14εY2,X,Y∈R,ε>0. (2.6)

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)(t−s)1−αds, (2.7)

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). (2.8)

Let tk = kτ, k = 0, 1, ⋯, N, where τ = T/N is the time step size. For the convenience of description, we define Dtun+1=un+1−unτ,  and  un+12=un+1+un2, where un ≜ u(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=τα∑k=0nλk(α)un+1−k+E~≜0Itαun+1+E~, (2.9)

where Ẽ = O(τ2) and

λ0(α)=(1−α2)ω0(α),λk(α)=(1−α2)ωk(α)+α2ωk−1(α),k≥1, (2.10)

here

ωk(α)=(−1)k−αk,ω0(α)=1,ωk(α)=(1+α−1k)ωk−1(α),k≥1. (2.11)

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+1−0Itα△un+12+0Itαun+12=gn+12+En+12, (2.12)

where gn+12=0Itαfn+12+φ(x,y), En+12=E~+Ecn+12=O(τ2), Ecn+12=Dtun+12−ut(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=0N−1⊂Mr(δ) such that

uhn+1−uhnτ−τα∑k=0nλk(α)△uhn+12−k+τα∑k=0nλk(α)uhn+12−k=gn+12. (2.13)

For the needs of analysis, we give the following equivalent Galerkin weak formulation of the equation(2.12) by multiplying the equation with v ∈ H01 and integrating with respect to spatial domain Ω

(Dtun+1,v)+(0Itα▽un+12,▽v)+(0Itαun+12,v)=(gn+12,v)+(En+12,v). (2.14)

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

(uhn+1−uhnτ,vh)+τα∑k=0nλk(α)(▽uhn+12−k,▽vh)+τα∑k=0nλk(α)(uhn+12−k,vh)=(gn+12,vh),∀vh∈Mr(δ) (2.15)

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=0∞ defined in (2.10), then for any positive integer k and real vector (v1, v2, ⋯, vk)T ∈ 𝓡k, it holds that

∑n=0k−1(∑p=0nλp(α)vn+1−p)vn+1≥0.

Lemma 3.2

(Gronwall’s ineqality) [29] Assume that kn and pn are nonnegative sequence, and the sequence ϕn satisfies

ϕ0≤g0,ϕn≤ϕ0+∑l=0n−1pl+∑l=0n−1klpl,n≥1,

where, g0 ≥ 0. Then the sequence ϕn satisfies

ϕn≤(g0+∑l=0n−1pl)exp(∑l=0n−1kl),n≥1.

Theorem 3.1

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

||uhL+1||2≤C(||uh0||2+max0≤n≤N−1||gn+12||2),1≤L≤N−1. (3.1)

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)+τα∑k=0nλk(α)[(▽uhn+12−k,▽vh)+(uhn+12−k,vh)]≤12(||gn+12||2+||uhn+12||2). (3.2)

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

12τ∑n=0L(||uhn+1||2−||uhn||2)+τα∑n=0L∑k=0nλk(α)[(▽uhn+12−k,▽vh)+(uhn+12−k,vh)]≤12∑n=0L(||gn+12||2+||uhn+12||2). (3.3)

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

2τα+1∑n=0L∑k=0nλk(α)[(▽uhn+12−k,▽vh)+(uhn+12−k,vh)],

we have

||uhL+1||2≤||uh0||2+τ∑n=0L(||gn+12||2+||uhn+12||2)≤||uh0||2+Tmax0≤n≤N−1||gn+12||2+τ∑n=0L||uhn+12||2≤||uh0||2+Tmax0≤n≤N−1||gn+12||2+τ∑n=0L12(||uhn+1||2+||uhn||2). (3.4)

Then, it gives that,

(1−12τ)||uhL+1||2≤(1+12τ)||uh0||2+Tmax0≤n≤N−1||gn+12||2+τ∑n=1L||uhn||2. (3.5)

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

||uhL+1||2≤C(||uh0||2+Tmax0≤n≤N−1||gn+12||2+τ∑n=1L||uhn||2). (3.6)

Using Gronwall’s Lemma 3.2, we get

||uhL+1||2≤C(||uh0||2+max0≤n≤N−1||gn+12||2). (3.7)

The proof is complete.

Theorem 3.2

Suppose u is the exact solution of (1.1)-(1.3), and uhn (0 ≤ n ≤ N − 1) is the solution of the problem (2.13) with uh0 = W0, then there exists a positive constant C, independent of h and τ such that

∥u(tn)−uhn∥2≤C(τ2+hr+1). (3.8)

Proof

With W defined in (2.1), we set

ηn=Wn−un,ζn=uhn−Wn,0≤n≤N, (3.9)

thus we have

un−uhn=ηn+ζn. (3.10)

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

(ηn+1−ηnτ,vh)+τα∑k=0nλk(α)(▽ηn+12−k,▽vh)+τα∑k=0nλk(α)(ηn+12−k,vh)=−τα∑k=0nλk(α)(ζn+12−k,vh)−(ζn+1−ζnτ,vh)+(En+12,vh), (3.11)

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

(ηn+1−ηnτ,ηn+12)+τα∑k=0nλk(α)(▽ηn+12−k,▽ηn+12)+τα∑k=0nλk(α)(ηn+12−k,ηn+12)=−τα∑k=0nλk(α)(ζn+12−k,ηn+12)−(ζn+1−ζnτ,ηn+12)+(En+12,ηn+12). (3.12)

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

∑n=0L−1(||ηn+1||2−||ηn||2)+2τα+1∑n=0L−1∑k=0nλk(α)[(▽ηn+12−k,▽ηn+12)+(ηn+12−k,ηn+12)]=−2τα+1∑n=0L−1∑k=0nλk(α)(ζn+12−k,ηn+12)−2τ∑n=0L−1(ζn+1−ζnτ,ηn+12)+2τ∑n=0L−1(En+12,ηn+12)=I1+I2+I3. (3.13)

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

I1=−2τα+1∑n=0L−1∑k=0nλk(α)(ζn+12−k,ηn+12)=−2τα+1∑n=0L−1(0Itn+1αζ+0Itnαζ2−E~,ηn+12)=−2τα+1∑n=0L−1(1Γ(α)∫0tn+1ζ(x,y,s)(tn+1−s)1−αds+1Γ(α)∫0tnζ(x,y,s)(tn−s)1−αds−2E~,ηn+12)≤τ∑n=0L−1(−1Γ(α)α[(tn+1−s)α|0tn+1+(tn−s)α|0tn]max0≤s≤tn+1||ζ(x,y,s)||+||2E~||)||ηn+12||≤τΓ(α+1)∑n=0L−1(2Tαmax0≤t≤T||ζ(x,y,t)||+||E~||)||ηn+12||≤Cτ∑n=0L−1(2Tαmax0≤t≤T||ζ(x,y,t)||+||E~||)||ηn+12||≤Cτ∑n=0L−1(τ4+max0≤t≤T||ζ(x,y,t)||2+||ηn+12||2). (3.14)

Taking advantages of mean value theorem and Cauchy-Schwarz inequality as well as Young inequality, we have tn ≤ tn+θ ≤ tn+1

I2+I3=−2τ∑n=0L−1(ζn+1−ζnτ,ηn+12)+2τ∑n=0L−1(En+12,ηn+12)=τ∑n=0L−1(||ζt(x,y,tn+θ||2+||En+12||2+2||ηn+12||2). (3.15)

Using Lemma 1, we obtain

2τα+1∑n=0L∑k=0nλk(α)[(▽ηn+12−k,▽η)+(ηn+12−k,η)]≥0. (3.16)

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

||ηL||2≤||η0||2+Cτ∑n=0L−1(τ4+max0≤t≤T||ζ(x,y,t)||2+||ηn+12||2)+τ∑n=0L−1(||ζt(x,y,tn+θ||2+||En+12||2+2||ηn+12||2), (3.17)

that is

(1−Cτ)||ηL||2≤Cτ∑n=0L−1||ηn||2+Cτ∑n=0L−1(τ4+max0≤t≤T||ζ(x,y,t)||2+||ζt(x,y,tn+θ||2). (3.18)

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||2≤exp(CT).Cτ∑n=0L−1(τ4+Ch2r+2||u||Hr+32+Ch2r+2||ut||Hr+32)≤C(τ4+h2r+2) (3.19)

and

||u(tL)−uhL||2≤(||ηL||+||ζL||)2≤C(τ4+h2r+2) (3.20)

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+1−12τα+1λ0(α)Δuhn+1=−12τα+1∑k=1n+1λk(α)(−Δuhn+1−k+uhn+1−k)−12τα+1∑k=0nλk(α)(−Δuhn−k+uhn−k)+τgn+1+gn2+uhn (4.1)

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

(1+α0)uhn+1−α0Δuhn+1=β0∑k=1n+1λk(α)(Δuhn+1−k−uhn+1−k)+β0∑k=0nλk(α)(Δuhn−k−uhn−k)+τgn+1+gn2+uhn,n=0,⋯,N−1. (4.2)

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,⋯,ΦMx−1,ΦMx},Mr(δy)=span{Ψ1,Ψ2,⋯,ΨMy−1,ΨMy},

then

uhn(x,y)=∑j=1My∑i=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)(Bx⊗By)+α0(Ax⊗By+Bx⊗Ay)}u^n+1=−β0{Ax⊗By+Bx⊗Ay+Bx⊗By}(∑k=1n+1λk(α)u^n+1−k+∑k=0nλk(α)u^n−k)+(Bx⊗By)u^n+12τ(G1n+1+G2n),n=0,⋯,N−1, (4.3)

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), (4.4)

and

G1n+1=[gn+1(ξ1x,ξ1y),gn+1(ξ1x,ξ2y),⋯,gn+1(ξ1x,ξMyy),gn+1(ξ2x,ξ1y),⋯,gn+1(ξMxx,ξMyy)]T, (4.5)
G2n=[gn(ξ1x,ξ1y),gn(ξ1x,ξ2y),⋯,gn(ξ1x,ξMyy),gn(ξ2x,ξ1y),gn(ξ2x,ξ2y),⋯,gn(ξMxx,ξMyy)]T. (4.6)

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

××××××××××××××⋱⋱⋱⋱××××××. (4.7)

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 ≤ i ≤ K − 1, it follows that

ϕi(x)=−2(x−xi−1)3h3+3(x−xi−1)2h2,xi−1≤x≤xi,2(x−xi−1)3h3+3(x−xi+1)2h2,xi≤x≤xi+1,0,x<xi−1orx>xi+1, (4.8)

and

ψi(x)=(x−xi−1)2(x−xi)h2,xi−1≤x≤xi,(x−xi)(x−xi+1)2h2,xi≤x≤xi+1,0,x<xi−1orx>xi+1. (4.9)

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,⋯,ϕK−1,ψK−1,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)(1−uj)2,H2(uj)=uj(1−uj)2hk,H3(uj)=uj2(3−2uj),H4(uj)=uj2(uj−1)hk,I1(uj)=(12uj−6)/hk2,I2(uj)=(6uj−4)/hk,I3(uj)=(6−12uj)/hk2,I4(uj)=(6uj−2)/hk, (4.10)

where u1=(3−3)/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). (4.11)
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). (4.12)

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 order≈log(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)∂x2−u(x,t)+f(x,t),0<x<1,0<t≤1,u(x,0)=0,∂u(x,0)∂t=0,0≤x≤1,u(0,t)=u(1,t)=0,0≤t≤1, (5.1)

where f(x,t)=Γ(4)Γ(4−α)t3−γx2(1−x)2ex−2t3ex(1−4x+4x3). The analytical solution of this equation is u(x, t) = t3 x2(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 Numerical solution (a) and global error (b) for Example 1 with γ = 1.3, h = 1/32, τ = 1/32.
Figure 1

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

Table 1

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

γ τ L∞ error Convergence order L2 error Convergence order
1.1 110 7.0727×10−5 4.4681×10−5
120 1.7932×10−5 1.9798 1.1012×10−5 2.0206
140 4.5623×10−6 1.9747 2.7487×10−6 2.0022
180 1.1483×10−6 1.9903 6.8758×10−7 1.9992
1.3 110 2.6081×10−4 1.7238×10−4
120 6.6648×10−5 1.9684 4.2518×10−5 2.0194
140 1.6825×10−6 1.9860 1.0577×10−5 2.0072
180 4.2263×10−7 1.9931 2.6387×10−6 2.0030
1.5 110 4.1657×10−4 2.7911×10−4
120 1.0633×10−4 1.9701 6.8593×10−5 2.0247
140 2.6736×10−5 1.9916 1.7020×10−5 2.0108
180 6.7115×10−6 1.9941 4.2405×10−6 2.0050
1.7 110 5.3422×10−4 3.6265×10−4
120 1.3701×10−4 1.9632 8.9343×10−5 2.0212
140 3.4419×10−5 1.9930 2.2160×10−5 2.0114
180 8.6292×10−6 1.9959 5.5175×10−6 2.0059
1.9 110 5.7600×10−4 3.9339×10−4
120 1.4884×10−4 1.9523 9.7244×10−5 2.0163
140 3.7391×10−5 1.9930 2.4112×10−5 2.0119
180 9.3633×10−6 1.9976 5.9996×10−6 2.0068
1.95 110 5.6941×10−4 3.8862×10−4
120 1.4696×10−4 1.9540 9.6061×10−5 2.0163
140 3.6917×10−5 1.9931 2.3812×10−5 2.0123
180 9.2425×10−6 1.9979 5.9232×10−6 2.0072

Table 2

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

γ h L∞ error Convergence order L2 error Convergence order
1.1 110 2.4371×10−6 1.7740×10−6
120 1.5377×10−7 3.9863 1.0837×10−7 4.0329
140 9.6290×10−9 3.9972 6.6928×10−9 4.0172
180 6.0225×10−10 3.9989 4.1576×10−10 4.0088
1.3 110 3.8377×10−6 2.6750×10−6
120 2.4364×10−7 3.9774 1.6332×10−7 4.0338
140 1.5241×10−8 3.9987 1.0085×10−8 4.0174
180 9.5308×10−10 3.9992 6.2644×10−10 4.0089
1.5 110 4.7527×10−6 3.2535×10−6
120 3.0159×10−7 3.9781 1.9851×10−7 4.0347
140 1.8863×10−8 3.9990 1.2256×10−8 4.0177
180 1.1798×10−9 3.9990 7.6129×10−10 4.0089
1.7 110 5.1530×10−6 3.4857×10−6
120 3.2579×10−7 3.9834 2.1258×10−7 4.0354
140 2.0382×10−8 3.9986 1.3123×10−8 4.0178
180 1.2754×10−9 3.9982 8.1509×10−10 4.0090
1.9 110 4.6730×10−6 3.0735×10−6
120 2.9311×10−7 3.9948 1.8735×10−7 4.0361
140 1.8412×10−8 3.9927 1.1563×10−8 4.0181
180 1.1509×10−9 3.9999 7.1819×10−10 4.0090
1.95 110 4.3316×10−6 2.8280×10−6
120 2.7151×10−7 3.9958 1.7235×10−7 4.0364
140 1.7062×10−8 3.9922 1.0637×10−8 4.0182
180 1.0665×10−9 3.9999 6.6066×10−10 4.0091

Example 2

Consider the following one-dimensional fractional diffusion-wave equation

0cDtγu(x,t)=∂2u(x,t)∂x2−u(x,t)+f(x,t),0<x<1,0<t≤1,u(x,0)=0,∂u(x,0)∂t=−sin⁡πx,0≤x≤1,u(0,t)=u(1,t)=0,0≤t≤1, (5.2)

where f(x,t)=[2Γ(3−γ)t2−γ+(t2−t)π2+(t2−t)]sin⁡πx. The analytical solution of this equation is u(x, t) = (t2 − t)sinπ x.

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

Table 3

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

γ τ L∞ error Convergence order L2 error Convergence order
1.1 110 2.7779×10−5 1.8686×10−5
120 6.9405×10−6 2.0009 4.5452×10−6 2.0395
140 1.7225×10−6 2.0105 1.1135×10−6 2.0292
180 4.2704×10−7 2.0121 2.7427×10−7 2.0215
1.3 110 6.8399×10−5 4.6042×10−5
120 1.6912×10−5 2.0159 1.1079×10−5 2.0551
140 4.1818×10−6 2.0158 2.7032×10−6 2.0352
180 1.0358×10−6 2.0134 6.6503×10−7 2.0232
1.5 110 1.0251×10−4 6.9519×10−5
120 2.5114×10−5 2.0292 1.6555×10−5 2.0701
140 6.2025×10−6 2.0176 4.0327×10−6 2.0375
180 1.5384×10−6 2.0114 9.9335×10−7 2.0214
1.7 110 1.4424×10−4 9.9816×10−5
120 3.5642×10−5 2.0168 2.4001×10−5 2.0562
140 8.8717×10−6 2.0063 5.8868×10−6 2.0275
180 2.2116×10−6 2.0041 1.4572×10−6 2.0143
1.9 110 1.9061×10−4 1.2932×10−4
120 4.7290×10−5 2.0110 3.1561×10−5 2.0347
140 1.1810×10−5 2.0015 7.7937×10−6 2.0178
180 2.9519×10−6 2.0003 1.9367×10−6 2.0087
1.95 110 2.0110×10−4 1.3455×10−4
120 4.9930×10−5 2.0099 3.2930×10−5 2.0306
140 1.2482×10−5 2.0001 8.1369×10−6 2.0169
180 3.1218×10−6 1.9994 2.0222×10−6 2.0085

Table 4

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

γ h L∞ error Convergence order L2 error Convergence order
1.1 110 9.8873×10−8 7.9806×10−8
120 6.3013×10−9 3.9719 5.0124×10−9 3.9929
140 4.0052×10−10 3.9757 3.1726×10−10 3.9818
180 2.5425×10−11 3.9775 2.0139×10−11 3.9776
1.3 110 2.0590×10−7 1.1973×10−7
120 1.2348×10−8 4.0596 7.0248×10−9 4.0912
140 7.5090×10−10 4.0395 4.2273×10−10 4.0547
180 4.6084×10−11 4.0263 2.5826×10−11 4.0328
1.5 110 3.1378×10−7 1.8224×10−7
120 1.9140×10−8 4.0351 1.0838×10−8 4.0716
140 1.1827×10−9 4.0165 6.6114×10−10 4.0350
180 7.3513×10−11 4.0079 4.0836×10−11 4.0170
1.7 110 4.2637×10−7 2.5157×10−7
120 2.6414×10−8 4.0127 1.5170×10−8 4.0516
140 1.6453×10−9 4.0049 9.3246×10−10 4.0241
180 1.0270×10−10 4.0019 5.7825×10−11 4.0113
1.9 110 6.2873×10−7 3.5996×10−7
120 3.9276×10−8 4.0007 2.1898×10−8 4.0389
140 2.4548×10−9 4.0000 1.3510×10−9 4.0188
180 1.5342×10−10 4.0000 8.3898×10−11 4.0092
1.95 110 6.7414×10−7 3.9216×10−7
120 4.2262×10−8 3.9956 2.3894×10−8 4.0367
140 2.6423×10−9 3.9995 1.4745×10−9 4.0184
180 1.6515×10−10 3.9999 9.1572×10−11 4.0091

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], (5.3)

where Ω=[0,1]×[0,1], T=1, f(x,y,t)=[Γ(4)Γ(4−γ)t3−γxy(1−x)(1−y)+t3xy(7−3y−3x−xy)]ex+y. The exact solution of the equation is u(x, y, t) = t3 xy(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 Numerical solution (a) and global error (b) for Example 3 with γ = 1.7 at T = 1 (h = 1/32, τ = 1/32).
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).

Table 5

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

γ N L∞ error Convergence order L2 error Convergence order
1.1 10 1.6611×10−4 8.3486×10−5
15 7.5461×10−5 1.9461 3.7876×10−5 1.9493
20 4.2909×10−5 1.9624 2.1509×10−5 1.9669
25 2.7589×10−5 1.9792 1.3841×10−5 1.9754
1.3 10 5.1729×10−4 2.6164×10−4
15 2.3249×10−4 1.9724 1.1769×10−4 1.9704
20 1.3148×10−4 1.9813 6.6585×10−5 1.9799
25 8.4565×10−5 1.9779 4.2760×10−5 1.9847
1.5 10 7.7899×10−4 3.9475×10−4
15 3.4829×10−4 1.9853 1.7651×10−4 1.9850
20 1.9648×10−4 1.9899 9.9627×10−5 1.9882
25 1.2607×10−4 1.9886 6.3896×10−5 1.9905
1.7 10 9.8958×10−4 5.0659×10−4
15 4.3990×10−4 1.9995 2.2433×10−4 2.0090
20 2.4748×10−4 1.9995 1.2609×10−4 2.0028
25 1.5841×10−4 1.9994 8.0685×10−5 2.0006
1.9 10 1.1985×10−3 6.2808×10−4
15 5.3173×10−4 2.0044 2.7856×10−4 2.0052
20 2.9891×10−4 2.0022 1.5657×10−4 2.0026
25 1.9123×10−4 2.0015 1.0018×10−4 2.0014

Table 6

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

γ N L∞ error Convergence order L2 error Convergence order
1.1 10 1.7277×10−6 5.8164×10−7
15 3.7129×10−7 3.7921 1.1583×10−7 3.9799
20 1.2237×10−7 3.8582 3.6753×10−8 3.9902
25 5.1343×10−8 3.8922 1.5074×10−8 3.9942
1.3 10 4.5383×10−6 2.0806×10−6
15 8.9315×10−7 4.0091 4.1188×10−7 3.9946
20 2.8185×10−7 4.0092 1.3042×10−7 3.9974
25 1.1523×10−7 4.0083 5.3439×10−8 3.9984
1.5 10 7.1532×10−6 3.3974×10−6
15 1.4118×10−6 4.0021 6.7176×10−7 3.9976
20 4.4624×10−7 4.0036 2.1262×10−7 3.9988
25 1.8263×10−7 4.0037 8.7104×10−8 3.9993
1.7 10 9.2188×10−6 4.4527×10−6
15 1.8187×10−6 4.0031 8.7956×10−7 4.0000
20 5.7483×10−7 4.0038 2.7830×10−7 3.9999
25 2.3526×10−7 4.0036 1.1399×10−7 3.9999
1.9 10 1.1444×10−5 5.7230×10−6
15 2.2505×10−6 4.0110 1.1299×10−6 4.0011
20 7.1020×10−7 4.0091 3.5746×10−7 4.0005
25 2.9046×10−7 4.0068 1.4641×10−7 4.0003

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] (5.4)

where Ω=[0,1]×[0,1], T=1, f(x,y,t)=[Γ(3+γ)2+(2π2+1)tγ]t2sin⁡πxsin⁡πy. The exact solution of the equation is u(x, y, t) = t2+γ sinπ x sin π y.

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

Table 7

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

γ N L∞ error Convergence order L2 error Convergence order
1.1 10 8.8381×10−4 4.4449×10−4
15 3.9978×10−4 1.9566 2.0073×10−4 1.9607
20 2.2738×10−4 1.9615 1.1385×10−4 1.9711
25 1.4625×10−4 1.9775 7.3238×10−5 1.9771
1.3 10 3.1514×10−3 1.5847×10−3
15 1.4225×10−3 1.9617 7.1426×10−4 1.9654
20 8.0814×10−4 1.9656 4.0464×10−4 1.9752
25 5.1939×10−4 1.9811 2.6009×10−4 1.9807
1.5 10 5.3058×10−3 2.6680×10−3
15 2.3861×10−3 1.9709 1.1981×10−3 1.9745
20 1.3534×10−3 1.9711 6.7766×10−4 1.9808
25 8.6906×10−4 1.9851 4.3519×10−4 1.9847
1.7 10 7.2062×10−3 3.6236×10−3
15 3.2347×10−3 1.9755 1.6242×10−3 1.9792
20 1.8321×10−3 1.9760 9.1737×10−4 1.9857
25 1.1754×10−3 1.9893 5.8858×10−4 1.9889
1.9 10 8.0346×10−3 4.0402×10−3
15 3.6198×10−3 1.9665 1.8175×10−3 1.9701
20 2.0516×10−3 1.9736 1.0273×10−3 1.9833
25 1.3162×10−3 1.9893 6.5910×10−4 1.9888

Table 8

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

γ N L∞ error Convergence order L2 error Convergence order
1.1 10 1.2725×10−5 6.5169×10−5
15 2.5700×10−6 3.9453 1.2858×10−5 4.0029
20 8.0847×10−7 4.0202 4.0665×10−5 4.0014
25 3.3300×10−7 3.9751 1.6653×10−5 4.0009
1.3 10 3.6079×10−5 1.8230×10−5
15 7.1968×10−6 3.9758 3.6064×10−6 3.9964
20 2.2773×10−6 3.9997 1.1417×10−6 3.9982
25 9.3472×10−7 3.9907 4.6773×10−7 3.9989
1.5 10 5.7773×10−5 2.9133×10−5
15 1.1491×10−5 3.9829 5.7620×10−6 3.9968
20 3.6402×10−6 3.9959 1.8240×10−6 3.9984
25 1.4930×10−6 3.9942 7.4725×10−7 3.9991
1.7 10 7.6488×10−5 3.8541×10−5
15 1.5190×10−5 3.9868 7.6189×10−6 3.9981
20 4.8133×10−6 3.9948 2.4113×10−6 3.9991
25 1.9734×10−6 3.9959 9.8780×10−7 3.9994
1.9 10 8.4694×10−5 4.2666×10−5
15 1.6803×10−5 3.9892 8.4290×10−6 3.9997
20 5.3240×10−6 3.9951 2.6670×10−6 3.9999
25 2.1823×10−6 3.9968 1.0924×10−6 4.0000

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).

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Received: 2019-04-20
Accepted: 2020-01-18
Published Online: 2020-03-05

© 2020 Xiaoyong Xu and Fengying Zhou, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
  24. On the Malcev products of some classes of epigroups, I
  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
https://doi.org/10.1515/math-2020-0007
Keywords for this article
diffusion-wave equation; Crank-Nicolson method; weighted and shifted Grünwald integral operator; orthogonal spline collocation method; Caputo derivative
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Articles in the same Issue

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
  24. On the Malcev products of some classes of epigroups, I
  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
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