# 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

subject to the initial condition

and the boundary condition

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 *γ* (1 < *γ* < 2), which reads as follows:

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 *L*_{1} 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 *N _{x}* and

*N*,

_{y}*δ*and

_{x}*δ*are two uniform partitions of

_{y}*I*= [0, 1] which are defined as follows:

and *M _{r}*(

*δ*) and

_{x}*M*(

_{r}*δ*) be the space of piecewise polynomial of degree at most

_{y}*r*≥ 3, defined by

where *P _{r}* denotes the set of polynomial of degree at most

*r*. It is easy to know that the dimension of the spaces

*M*(

_{x}*δ*) and

_{x}*M*(

_{y}*δ*) are (

_{y}*r*− 1)

*N*:=

_{x}*M*and (

_{x}*r*− 1)

*N*:=

_{y}*M*, respectively.

_{y}Let *δ* = *δ _{x}* ⊗

*δ*be a quasi-uniform partition of

_{y}*Ω*, and

*M*(

_{r}*δ*) =

*M*(

_{r}*δ*) ⊗

_{x}*M*(

_{r}*δ*) with the dimension of

_{y}*M>*×

_{x}*M*. Let

_{y}*r*− 1}-point Gaussian quadrature rule on the interval

*I*with corresponding weights

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

Let 𝓖 = {*ξ* = (*ξ ^{x}*,

*ξ*) :

^{y}*ξ*∈ 𝓖

^{x}_{x},

*ξ*∈ 𝓖

^{y}_{y}}. For the functions

*u*and

*v*defined on 𝓖, the inner product 〈

*u*,

*v*〉 and norm ∥

*v*∥

_{Mr}are respectively defined by

For *m* a nonnegative integer, let *H ^{m}*(

*Ω*) denotes the usual Sobolev space with norm

where the norm ∥⋅∥ denotes the usual *L*_{2} 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*] → *M _{r}*(

*δ*) by

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/∂ t^{l}* ∈

*H*

^{r+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*

### Lemma 2.2

*[26] If* *∂ ^{i} u/∂ t^{i}* ∈

*H*

^{r+3},

*for*

*t*∈ [0,

*T*],

*i*= 0, 1,

*then*

where 0 ≤ *l* = *l*_{1} + *l*_{2} ≤ 4.

and there exists a positive constant *C* such that

### Lemma 2.4

*[28] The norms* ∥⋅∥_{Mr} *and* ∥⋅∥ *are equivalent on* *M _{r}*(

*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,

### 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

where 0 < *α* = *γ* − 1 < 1.

We integrate the equation(1.1) using Riemann-Liouville fractional integral operator

Let *t _{k}* =

*kτ*,

*k*= 0, 1, ⋯,

*N*, where

*τ*=

*T/N*is the time step size. For the convenience of description, we define

*u*≜

^{n}*u*(

*x*,

*y*,

*t*). 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

_{n}where *Ẽ* = *O*(*τ*^{2}) and

here

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

where

For the needs of analysis, we give the following equivalent Galerkin weak formulation of the equation(2.12) by multiplying the equation with *v* ∈ *Ω*

We take the space *M _{r}*(

*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* *defined in (2.10)*, *then for any positive integer* *k* *and real vector* (*v*_{1}, *v*_{2}, ⋯, *v _{k}*)

^{T}∈ 𝓡

^{k},

*it holds that*

## Lemma 3.2

*(Gronwall’s ineqality) [29] Assume that* *k _{n}*

*and*

*p*

_{n}*are nonnegative sequence*,

*and the sequence*

*ϕ*

_{n}*satisfies*

*where*, *g*_{0} ≥ 0. *Then the sequence* *ϕ _{n} satisfies*

## Theorem 3.1

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

## Proof

Taking

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

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

we have

Then, it gives that,

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

Using Gronwall’s Lemma 3.2, we get

The proof is complete.

## Theorem 3.2

*Suppose* *u* *is the exact solution of (1.1)*-*(1.3)*, *and* *n* ≤ *N* − 1) *is the solution of the problem (2.13) with* *W*^{0}, *then there exists a positive constant* *C*, *independent of* *h* *and* *τ* *such that*

## Proof

With *W* defined in (2.1), we set

thus we have

Because the estimate of *η ^{n}* are provided by Lemma 2.2, it is sufficient to bound

*ζ*, then use the triangle inequality to bound

^{n}*u*−

^{n}*v*∈

_{h}*M*(

_{r}*δ*), we obtain

where

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

Next, we will give the estimate of *I*_{1}, *I*_{2} and *I*_{3}, respectively.

Taking advantages of mean value theorem and Cauchy-Schwarz inequality as well as Young inequality, we have *t _{n}* ≤

*t*

_{n+θ}≤

*t*

_{n+1}

Using Lemma 1, we obtain

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

that is

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

and

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

We denote

For applying the numerical schemes, firstly, we usually represent *M _{r}*(

*δ*), then solve the coefficients of the representation formula. Letting

then

where

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

where

and

The matrices *A ^{x}*,

*B*,

^{x}*A*and

^{y}*B*are

^{y}*M*×

_{x}*M*having the following structure,

_{y}We carry out the WSGI-OSC scheme in piecewise Hermite cubic spline space *M*_{3}(*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

and

Note that functions *ϕ _{i}*(

*x*),

*ψ*(

_{i}*x*) satisfy zero boundary conditions

*ϕ*(0) =

_{i}*ϕ*(1) =

_{i}*ψ*(0) =

_{i}*ψ*(1) = 0. Renumber the basis functions and let

_{i}then

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:

where *H _{i}* and

*I*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

_{i}*A*and

^{x}*B*in the case of

^{x}*N*=

_{x}*N*= 5 and

_{y}*h*= 1/

_{k}*N*. We have

_{x}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 *N _{x}* =

*N*=

_{y}*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

*L*

_{2}errors at

*T*= 1 and the corresponding convergence order defined by

where *h _{m}* = 1/

*K*is the time step size and

*e*is the norm of the corresponding error.

_{m}## Example 1

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

where *u*(*x*, *t*) = *t*^{3} *x*^{2}(1 − *x*)^{2}*e ^{x}*.

From the theoretical analysis, the numerical convergence order of WSGI-OSC (4.2) is expected to be *O*(*τ*^{2} + *h*^{4}) 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 *L*_{2} 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* = *h*^{2} is fixed. Numerical solution and global error for *γ* = 1.3, *h* = 1/32, *τ* = 1/32 are shown in Figure 1.

### Figure 1

### Table 1

γ |
τ |
L_{∞} error |
Convergence order | L_{2} error |
Convergence order |
---|---|---|---|---|---|

1.1 | 7.0727×10^{−5} |
4.4681×10^{−5} |
|||

1.7932×10^{−5} |
1.9798 | 1.1012×10^{−5} |
2.0206 | ||

4.5623×10^{−6} |
1.9747 | 2.7487×10^{−6} |
2.0022 | ||

1.1483×10^{−6} |
1.9903 | 6.8758×10^{−7} |
1.9992 | ||

1.3 | 2.6081×10^{−4} |
1.7238×10^{−4} |
|||

6.6648×10^{−5} |
1.9684 | 4.2518×10^{−5} |
2.0194 | ||

1.6825×10^{−6} |
1.9860 | 1.0577×10^{−5} |
2.0072 | ||

4.2263×10^{−7} |
1.9931 | 2.6387×10^{−6} |
2.0030 | ||

1.5 | 4.1657×10^{−4} |
2.7911×10^{−4} |
|||

1.0633×10^{−4} |
1.9701 | 6.8593×10^{−5} |
2.0247 | ||

2.6736×10^{−5} |
1.9916 | 1.7020×10^{−5} |
2.0108 | ||

6.7115×10^{−6} |
1.9941 | 4.2405×10^{−6} |
2.0050 | ||

1.7 | 5.3422×10^{−4} |
3.6265×10^{−4} |
|||

1.3701×10^{−4} |
1.9632 | 8.9343×10^{−5} |
2.0212 | ||

3.4419×10^{−5} |
1.9930 | 2.2160×10^{−5} |
2.0114 | ||

8.6292×10^{−6} |
1.9959 | 5.5175×10^{−6} |
2.0059 | ||

1.9 | 5.7600×10^{−4} |
3.9339×10^{−4} |
|||

1.4884×10^{−4} |
1.9523 | 9.7244×10^{−5} |
2.0163 | ||

3.7391×10^{−5} |
1.9930 | 2.4112×10^{−5} |
2.0119 | ||

9.3633×10^{−6} |
1.9976 | 5.9996×10^{−6} |
2.0068 | ||

1.95 | 5.6941×10^{−4} |
3.8862×10^{−4} |
|||

1.4696×10^{−4} |
1.9540 | 9.6061×10^{−5} |
2.0163 | ||

3.6917×10^{−5} |
1.9931 | 2.3812×10^{−5} |
2.0123 | ||

9.2425×10^{−6} |
1.9979 | 5.9232×10^{−6} |
2.0072 |

### Table 2

γ |
h |
L_{∞} error |
Convergence order | L_{2} error |
Convergence order |
---|---|---|---|---|---|

1.1 | 2.4371×10^{−6} |
1.7740×10^{−6} |
|||

1.5377×10^{−7} |
3.9863 | 1.0837×10^{−7} |
4.0329 | ||

9.6290×10^{−9} |
3.9972 | 6.6928×10^{−9} |
4.0172 | ||

6.0225×10^{−10} |
3.9989 | 4.1576×10^{−10} |
4.0088 | ||

1.3 | 3.8377×10^{−6} |
2.6750×10^{−6} |
|||

2.4364×10^{−7} |
3.9774 | 1.6332×10^{−7} |
4.0338 | ||

1.5241×10^{−8} |
3.9987 | 1.0085×10^{−8} |
4.0174 | ||

9.5308×10^{−10} |
3.9992 | 6.2644×10^{−10} |
4.0089 | ||

1.5 | 4.7527×10^{−6} |
3.2535×10^{−6} |
|||

3.0159×10^{−7} |
3.9781 | 1.9851×10^{−7} |
4.0347 | ||

1.8863×10^{−8} |
3.9990 | 1.2256×10^{−8} |
4.0177 | ||

1.1798×10^{−9} |
3.9990 | 7.6129×10^{−10} |
4.0089 | ||

1.7 | 5.1530×10^{−6} |
3.4857×10^{−6} |
|||

3.2579×10^{−7} |
3.9834 | 2.1258×10^{−7} |
4.0354 | ||

2.0382×10^{−8} |
3.9986 | 1.3123×10^{−8} |
4.0178 | ||

1.2754×10^{−9} |
3.9982 | 8.1509×10^{−10} |
4.0090 | ||

1.9 | 4.6730×10^{−6} |
3.0735×10^{−6} |
|||

2.9311×10^{−7} |
3.9948 | 1.8735×10^{−7} |
4.0361 | ||

1.8412×10^{−8} |
3.9927 | 1.1563×10^{−8} |
4.0181 | ||

1.1509×10^{−9} |
3.9999 | 7.1819×10^{−10} |
4.0090 | ||

1.95 | 4.3316×10^{−6} |
2.8280×10^{−6} |
|||

2.7151×10^{−7} |
3.9958 | 1.7235×10^{−7} |
4.0364 | ||

1.7062×10^{−8} |
3.9922 | 1.0637×10^{−8} |
4.0182 | ||

1.0665×10^{−9} |
3.9999 | 6.6066×10^{−10} |
4.0091 |

## Example 2

Consider the following one-dimensional fractional diffusion-wave equation

where *u*(*x*, *t*) = (*t*^{2} − *t*)sin*π* *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*_{∞}, *L*_{2} 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* = *h*^{2}, and *γ* = 1.1, 1.3, 1.5, 1.7, 1.9, 1.95. Table 4 presents the maximum *L*_{∞}, *L*_{2} 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

### Table 3

γ |
τ |
L_{∞} error |
Convergence order | L_{2} error |
Convergence order |
---|---|---|---|---|---|

1.1 | 2.7779×10^{−5} |
1.8686×10^{−5} |
|||

6.9405×10^{−6} |
2.0009 | 4.5452×10^{−6} |
2.0395 | ||

1.7225×10^{−6} |
2.0105 | 1.1135×10^{−6} |
2.0292 | ||

4.2704×10^{−7} |
2.0121 | 2.7427×10^{−7} |
2.0215 | ||

1.3 | 6.8399×10^{−5} |
4.6042×10^{−5} |
|||

1.6912×10^{−5} |
2.0159 | 1.1079×10^{−5} |
2.0551 | ||

4.1818×10^{−6} |
2.0158 | 2.7032×10^{−6} |
2.0352 | ||

1.0358×10^{−6} |
2.0134 | 6.6503×10^{−7} |
2.0232 | ||

1.5 | 1.0251×10^{−4} |
6.9519×10^{−5} |
|||

2.5114×10^{−5} |
2.0292 | 1.6555×10^{−5} |
2.0701 | ||

6.2025×10^{−6} |
2.0176 | 4.0327×10^{−6} |
2.0375 | ||

1.5384×10^{−6} |
2.0114 | 9.9335×10^{−7} |
2.0214 | ||

1.7 | 1.4424×10^{−4} |
9.9816×10^{−5} |
|||

3.5642×10^{−5} |
2.0168 | 2.4001×10^{−5} |
2.0562 | ||

8.8717×10^{−6} |
2.0063 | 5.8868×10^{−6} |
2.0275 | ||

2.2116×10^{−6} |
2.0041 | 1.4572×10^{−6} |
2.0143 | ||

1.9 | 1.9061×10^{−4} |
1.2932×10^{−4} |
|||

4.7290×10^{−5} |
2.0110 | 3.1561×10^{−5} |
2.0347 | ||

1.1810×10^{−5} |
2.0015 | 7.7937×10^{−6} |
2.0178 | ||

2.9519×10^{−6} |
2.0003 | 1.9367×10^{−6} |
2.0087 | ||

1.95 | 2.0110×10^{−4} |
1.3455×10^{−4} |
|||

4.9930×10^{−5} |
2.0099 | 3.2930×10^{−5} |
2.0306 | ||

1.2482×10^{−5} |
2.0001 | 8.1369×10^{−6} |
2.0169 | ||

3.1218×10^{−6} |
1.9994 | 2.0222×10^{−6} |
2.0085 |

### Table 4

γ |
h |
L_{∞} error |
Convergence order | L_{2} error |
Convergence order |
---|---|---|---|---|---|

1.1 | 9.8873×10^{−8} |
7.9806×10^{−8} |
|||

6.3013×10^{−9} |
3.9719 | 5.0124×10^{−9} |
3.9929 | ||

4.0052×10^{−10} |
3.9757 | 3.1726×10^{−10} |
3.9818 | ||

2.5425×10^{−11} |
3.9775 | 2.0139×10^{−11} |
3.9776 | ||

1.3 | 2.0590×10^{−7} |
1.1973×10^{−7} |
|||

1.2348×10^{−8} |
4.0596 | 7.0248×10^{−9} |
4.0912 | ||

7.5090×10^{−10} |
4.0395 | 4.2273×10^{−10} |
4.0547 | ||

4.6084×10^{−11} |
4.0263 | 2.5826×10^{−11} |
4.0328 | ||

1.5 | 3.1378×10^{−7} |
1.8224×10^{−7} |
|||

1.9140×10^{−8} |
4.0351 | 1.0838×10^{−8} |
4.0716 | ||

1.1827×10^{−9} |
4.0165 | 6.6114×10^{−10} |
4.0350 | ||

7.3513×10^{−11} |
4.0079 | 4.0836×10^{−11} |
4.0170 | ||

1.7 | 4.2637×10^{−7} |
2.5157×10^{−7} |
|||

2.6414×10^{−8} |
4.0127 | 1.5170×10^{−8} |
4.0516 | ||

1.6453×10^{−9} |
4.0049 | 9.3246×10^{−10} |
4.0241 | ||

1.0270×10^{−10} |
4.0019 | 5.7825×10^{−11} |
4.0113 | ||

1.9 | 6.2873×10^{−7} |
3.5996×10^{−7} |
|||

3.9276×10^{−8} |
4.0007 | 2.1898×10^{−8} |
4.0389 | ||

2.4548×10^{−9} |
4.0000 | 1.3510×10^{−9} |
4.0188 | ||

1.5342×10^{−10} |
4.0000 | 8.3898×10^{−11} |
4.0092 | ||

1.95 | 6.7414×10^{−7} |
3.9216×10^{−7} |
|||

4.2262×10^{−8} |
3.9956 | 2.3894×10^{−8} |
4.0367 | ||

2.6423×10^{−9} |
3.9995 | 1.4745×10^{−9} |
4.0184 | ||

1.6515×10^{−10} |
3.9999 | 9.1572×10^{−11} |
4.0091 |

## Example 3

Consider the following two-dimensional fractional diffusion-wave equation

where *u*(*x*, *y*, *t*) = *t*^{3} *xy*(1 − *x*)(1 − *y*)*e*^{x+y}.

Similar to the selection of parameters in Examples 1 and 2, Tables 5 and 6 list the maximum *L*_{∞}, *L*_{2} 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

### Table 5

γ |
N |
L_{∞} error |
Convergence order | L_{2} 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

γ |
N |
L_{∞} error |
Convergence order | L_{2} 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

where *u*(*x*, *y*, *t*) = *t*^{2+γ} sin*π* *x* sin *π* *y*.

Tables 7 and 8 display *L*_{∞} and *L*_{2} 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

### Table 7

γ |
N |
L_{∞} error |
Convergence order | L_{2} 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

γ |
N |
L_{∞} error |
Convergence order | L_{2} 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*(*τ*^{3−γ}) of general *L*_{1} 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.