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

Xiaoyong Xu and Fengying Zhou
From the journal Open Mathematics

# 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.

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) . 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 , spline collocation method [10, 11, 12] and so forth.

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

0 C D t γ u ( x , y , t ) = Δ u ( 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 0 C D t γ denotes the Caputo derivative of order γ (1 < γ < 2), which reads as follows:

0 C D t γ u ( x , y , t ) = 1 Γ ( 2 γ ) 0 t 2 u ( x , y , s ) s 2 ( t s ) 1 γ d s ,

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.  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 , Liu et al developed a high-order local discontinuous Galerkin method combined with WSGD approximation for a Caputo time-fractional sub-diffusion equation. In , 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 , 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  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. 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 . Another feature of OSC method lies in its super-convergence .

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:

δ x : 0 = x 0 < x 1 < < x N x = 1 , δ y : 0 = y 0 < y 1 < < y N y = 1 ,

and h i x = x i x i 1 , I i x = ( x i 1 , x i ) , 1 i N x , and h j y = y j y j 1 , I j y = ( y j 1 , y j ) , 1 j N y , h = max ( max 1 i N x h i x , max 1 j N y h j y ) . Let Mr(δx) and Mr(δy) be the space of piecewise polynomial of degree at most r ≥ 3, defined by

M r ( δ x ) = { v C 1 [ 0 , 1 ] : v | I i x P r , 1 i N x , v ( 0 ) = v ( 1 ) = 0 } , M r ( δ y ) = { v C 1 [ 0 , 1 ] : v | I j y P r , 1 j N y , 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 { λ j } j = 1 r 1 denotes the nodes for the {r − 1}-point Gaussian quadrature rule on the interval I with corresponding weights { ω j } j = 1 r 1 . Denote by

G x = { ξ i , l x } i , l = 1 N x , r 1 a n d G y = { ξ j , m y } j , m = 1 N y , r 1

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

ξ i , l x = x i 1 + h i x λ l , ξ j , m y = y j 1 + h j y λ m , 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 ∥vMr are respectively defined by

u , v = i = 1 N x j = 1 N y h i x h j y l = 1 r 1 m = 1 r 1 ω l ω m ( u v ) ( ξ i , l , ξ j , m ) , v M r 2 = v , v .

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

v H m = ( l = 0 m i + j = l i + j v x i y j 2 ) 1 2 ,

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 ) = 0 o n G × [ 0 , T ] , (2.1)

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

 If l u/∂ tlHr+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 ) t l H j C h r + 1 j l u t l H r + 3 j . (2.2)

### Lemma 2.2

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

l + i ( u W ) x l 1 y l 2 t i M r C h r + 1 l i u t i H r + 3 , (2.3)

where 0 ≤ l = l1 + l2 ≤ 4.

### Lemma 2.3

 If u, vMr(δ), then

Δ u , v = u , Δ v , (2.4)

and there exists a positive constant C such that

Δ u , u C u 2 0. (2.5)

### Lemma 2.4

 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,

X Y ε X 2 + 1 4 ε Y 2 , 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

0 I t α u ( x , y , t ) = 1 Γ ( α ) 0 t u ( x , y , s ) ( t s ) 1 α d s , (2.7)

where 0 < α = γ − 1 < 1.

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

u t ( x , y , t ) 0 I t α Δ u ( x , y , t ) + 0 I t α u ( x , y , t ) = 0 I t α f ( x , y , t ) + φ ( x , y ) . (2.8)

Let tk = , k = 0, 1, ⋯, N, where τ = T/N is the time step size. For the convenience of description, we define D t u n + 1 = u n + 1 u n τ , and u n + 1 2 = u n + 1 + u n 2 , where unu(x, y, tn). Based on the idea of weighted and shifted Grünwald difference operator, Wang and Vong () established the second order accuracy approximation formula of the Riemann-Liouville fractional integral operator 0 I t α u n + 1 , which is called as WSGI approximation,

0 I t α u n + 1 = τ α k = 0 n λ k ( α ) u n + 1 k + E ~ 0 I t α u n + 1 + E ~ , (2.9)

where = O(τ2) and

λ 0 ( α ) = ( 1 α 2 ) ω 0 ( α ) , λ k ( α ) = ( 1 α 2 ) ω k ( α ) + α 2 ω k 1 ( α ) , k 1 , (2.10)

here

ω k ( α ) = ( 1 ) k α k , ω 0 ( α ) = 1 , ω k ( α ) = ( 1 + α 1 k ) ω k 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

D t u n + 1 0 I t α u n + 1 2 + 0 I t α u n + 1 2 = g n + 1 2 + E n + 1 2 , (2.12)

where g n + 1 2 = 0 I t α f n + 1 2 + φ ( x , y ) , E n + 1 2 = E ~ + E c n + 1 2 = O ( τ 2 ) , E c n + 1 2 = D t u n + 1 2 u t ( t n + 1 2 ) = O ( τ 2 ) . Then by using (2.9), (2.12), the fully discrete WSGI-OSC scheme for Eqs(1.1) consists in finding { u h n } n = 0 N 1 M r ( δ ) such that

u h n + 1 u h n τ τ α k = 0 n λ k ( α ) u h n + 1 2 k + τ α k = 0 n λ k ( α ) u h n + 1 2 k = g n + 1 2 . (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 H 0 1 and integrating with respect to spatial domain Ω

( D t u n + 1 , v ) + ( 0 I t α u n + 1 2 , v ) + ( 0 I t α u n + 1 2 , v ) = ( g n + 1 2 , v ) + ( E n + 1 2 , v ) . (2.14)

We take the space Mr(δ) ⊂ H 0 1 and obtain the fully discrete scheme as follows:

( u h n + 1 u h n τ , v h ) + τ α k = 0 n λ k ( α ) ( u h n + 1 2 k , v h ) + τ α k = 0 n λ k ( α ) ( u h n + 1 2 k , v h ) = ( g n + 1 2 , v h ) , v h M r ( 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

 Let { λ k ( 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 = 0 k 1 ( p = 0 n λ p ( α ) v n + 1 p ) v n + 1 0.

## Lemma 3.2

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

ϕ 0 g 0 , ϕ n ϕ 0 + l = 0 n 1 p l + l = 0 n 1 k l p l , n 1 ,

where, g0 ≥ 0. Then the sequence ϕn satisfies

ϕ n ( g 0 + l = 0 n 1 p l ) e x p ( l = 0 n 1 k l ) , n 1.

## Theorem 3.1

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

| | u h L + 1 | | 2 C ( | | u h 0 | | 2 + max 0 n N 1 | | g n + 1 2 | | 2 ) , 1 L N 1. (3.1)

## Proof

Taking v h = u h n + 1 2 = u n + 1 + u n 2 in (2.15) and applying the Cauchy-Schwarz inequality and Young inequality, it gives that

1 2 τ ( | | u h n + 1 | | 2 | | u h n | | 2 ) + τ α k = 0 n λ k ( α ) [ ( u h n + 1 2 k , v h ) + ( u h n + 1 2 k , v h ) ] 1 2 ( | | g n + 1 2 | | 2 + | | u h n + 1 2 | | 2 ) . (3.2)

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

1 2 τ n = 0 L ( | | u h n + 1 | | 2 | | u h n | | 2 ) + τ α n = 0 L k = 0 n λ k ( α ) [ ( u h n + 1 2 k , v h ) + ( u h n + 1 2 k , v h ) ] 1 2 n = 0 L ( | | g n + 1 2 | | 2 + | | u h n + 1 2 | | 2 ) . (3.3)

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

2 τ α + 1 n = 0 L k = 0 n λ k ( α ) [ ( u h n + 1 2 k , v h ) + ( u h n + 1 2 k , v h ) ] ,

we have

| | u h L + 1 | | 2 | | u h 0 | | 2 + τ n = 0 L ( | | g n + 1 2 | | 2 + | | u h n + 1 2 | | 2 ) | | u h 0 | | 2 + T max 0 n N 1 | | g n + 1 2 | | 2 + τ n = 0 L | | u h n + 1 2 | | 2 | | u h 0 | | 2 + T max 0 n N 1 | | g n + 1 2 | | 2 + τ n = 0 L 1 2 ( | | u h n + 1 | | 2 + | | u h n | | 2 ) . (3.4)

Then, it gives that,

( 1 1 2 τ ) | | u h L + 1 | | 2 ( 1 + 1 2 τ ) | | u h 0 | | 2 + T max 0 n N 1 | | g n + 1 2 | | 2 + τ n = 1 L | | u h n | | 2 . (3.5)

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

| | u h L + 1 | | 2 C ( | | u h 0 | | 2 + T max 0 n N 1 | | g n + 1 2 | | 2 + τ n = 1 L | | u h n | | 2 ) . (3.6)

Using Gronwall’s Lemma 3.2, we get

| | u h L + 1 | | 2 C ( | | u h 0 | | 2 + max 0 n N 1 | | g n + 1 2 | | 2 ) . (3.7)

The proof is complete.

## Theorem 3.2

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

u ( t n ) u h n 2 C ( τ 2 + h r + 1 ) . (3.8)

## Proof

With W defined in (2.1), we set

η n = W n u n , ζ n = u h n W n , 0 n N , (3.9)

thus we have

u n u h n = η n + ζ n . (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 u h n . Firstly, from(1.1), (2.1), (2.13), and(2.15), then for vhMr(δ), we obtain

( η n + 1 η n τ , v h ) + τ α k = 0 n λ k ( α ) ( η n + 1 2 k , v h ) + τ α k = 0 n λ k ( α ) ( η n + 1 2 k , v h ) = τ α k = 0 n λ k ( α ) ( ζ n + 1 2 k , v h ) ( ζ n + 1 ζ n τ , v h ) + ( E n + 1 2 , v h ) , (3.11)

where E n + 1 2 is defined in (2.12). Taking v h = η n + 1 2 in (3.11), we have

( η n + 1 η n τ , η n + 1 2 ) + τ α k = 0 n λ k ( α ) ( η n + 1 2 k , η n + 1 2 ) + τ α k = 0 n λ k ( α ) ( η n + 1 2 k , η n + 1 2 ) = τ α k = 0 n λ k ( α ) ( ζ n + 1 2 k , η n + 1 2 ) ( ζ n + 1 ζ n τ , η n + 1 2 ) + ( E n + 1 2 , η n + 1 2 ) . (3.12)

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

n = 0 L 1 ( | | η n + 1 | | 2 | | η n | | 2 ) + 2 τ α + 1 n = 0 L 1 k = 0 n λ k ( α ) [ ( η n + 1 2 k , η n + 1 2 ) + ( η n + 1 2 k , η n + 1 2 ) ] = 2 τ α + 1 n = 0 L 1 k = 0 n λ k ( α ) ( ζ n + 1 2 k , η n + 1 2 ) 2 τ n = 0 L 1 ( ζ n + 1 ζ n τ , η n + 1 2 ) + 2 τ n = 0 L 1 ( E n + 1 2 , η n + 1 2 ) = I 1 + I 2 + I 3 . (3.13)

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

I 1 = 2 τ α + 1 n = 0 L 1 k = 0 n λ k ( α ) ( ζ n + 1 2 k , η n + 1 2 ) = 2 τ α + 1 n = 0 L 1 ( 0 I t n + 1 α ζ + 0 I t n α ζ 2 E ~ , η n + 1 2 ) = 2 τ α + 1 n = 0 L 1 ( 1 Γ ( α ) 0 t n + 1 ζ ( x , y , s ) ( t n + 1 s ) 1 α d s + 1 Γ ( α ) 0 t n ζ ( x , y , s ) ( t n s ) 1 α d s 2 E ~ , η n + 1 2 ) τ n = 0 L 1 ( 1 Γ ( α ) α [ ( t n + 1 s ) α | 0 t n + 1 + ( t n s ) α | 0 t n ] max 0 s t n + 1 | | ζ ( x , y , s ) | | + | | 2 E ~ | | ) | | η n + 1 2 | | τ Γ ( α + 1 ) n = 0 L 1 ( 2 T α max 0 t T | | ζ ( x , y , t ) | | + | | E ~ | | ) | | η n + 1 2 | | C τ n = 0 L 1 ( 2 T α max 0 t T | | ζ ( x , y , t ) | | + | | E ~ | | ) | | η n + 1 2 | | C τ n = 0 L 1 ( τ 4 + max 0 t T | | ζ ( x , y , t ) | | 2 + | | η n + 1 2 | | 2 ) . (3.14)

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

I 2 + I 3 = 2 τ n = 0 L 1 ( ζ n + 1 ζ n τ , η n + 1 2 ) + 2 τ n = 0 L 1 ( E n + 1 2 , η n + 1 2 ) = τ n = 0 L 1 ( | | ζ t ( x , y , t n + θ | | 2 + | | E n + 1 2 | | 2 + 2 | | η n + 1 2 | | 2 ) . (3.15)

Using Lemma 1, we obtain

2 τ α + 1 n = 0 L k = 0 n λ k ( α ) [ ( η n + 1 2 k , η ) + ( η n + 1 2 k , η ) ] 0. (3.16)

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

| | η L | | 2 | | η 0 | | 2 + C τ n = 0 L 1 ( τ 4 + max 0 t T | | ζ ( x , y , t ) | | 2 + | | η n + 1 2 | | 2 ) + τ n = 0 L 1 ( | | ζ t ( x , y , t n + θ | | 2 + | | E n + 1 2 | | 2 + 2 | | η n + 1 2 | | 2 ) , (3.17)

that is

( 1 C τ ) | | η L | | 2 C τ n = 0 L 1 | | η n | | 2 + C τ n = 0 L 1 ( τ 4 + max 0 t T | | ζ ( x , y , t ) | | 2 + | | ζ t ( x , y , t n + 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

| | η L | | 2 e x p ( C T ) . C τ n = 0 L 1 ( τ 4 + C h 2 r + 2 | | u | | H r + 3 2 + C h 2 r + 2 | | u t | | H r + 3 2 ) C ( τ 4 + h 2 r + 2 ) (3.19)

and

| | u ( t L ) u h L | | 2 ( | | η L | | + | | ζ L | | ) 2 C ( τ 4 + h 2 r + 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 + 1 2 τ α + 1 λ 0 ( α ) ) u h n + 1 1 2 τ α + 1 λ 0 ( α ) Δ u h n + 1 = 1 2 τ α + 1 k = 1 n + 1 λ k ( α ) ( Δ u h n + 1 k + u h n + 1 k ) 1 2 τ α + 1 k = 0 n λ k ( α ) ( Δ u h n k + u h n k ) + τ g n + 1 + g n 2 + u h n (4.1)

We denote α 0 = 1 2 τ α + 1 λ 0 ( α ) , β 0 = 1 2 τ α + 1 , then the above equation can be rewritten as

( 1 + α 0 ) u h n + 1 α 0 Δ u h n + 1 = β 0 k = 1 n + 1 λ k ( α ) ( Δ u h n + 1 k u h n + 1 k ) + β 0 k = 0 n λ k ( α ) ( Δ u h n k u h n k ) + τ g n + 1 + g n 2 + u h n , n = 0 , , N 1. (4.2)

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

M r ( δ x ) = s p a n { Φ 1 , Φ 2 , , Φ M x 1 , Φ M x } , M r ( δ y ) = s p a n { Ψ 1 , Ψ 2 , , Ψ M y 1 , Ψ M y } ,

then

u h n ( x , y ) = j = 1 M y i = 1 M x u ^ i , j n Φ i ( x ) Ψ j ( y ) ,

where { u ^ i , j n } i , j = 1 M x , M y are unknown coefficients to be determined. Setting

u ^ = [ u ^ 1 , 1 n , u ^ 1 , 2 n , , u ^ 1 , M y n , u ^ 2 , 1 n , u ^ 2 , 2 n , , u ^ M x , M y n ] T ,

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

{ ( 1 + α 0 ) ( B x B y ) + α 0 ( A x B y + B x A y ) } u ^ n + 1 = β 0 { A x B y + B x A y + B x B y } ( k = 1 n + 1 λ k ( α ) u ^ n + 1 k + k = 0 n λ k ( α ) u ^ n k ) + ( B x B y ) u ^ n + 1 2 τ ( G 1 n + 1 + G 2 n ) , n = 0 , , N 1 , (4.3)

where

A x = ( a i , j x ) i , j = 1 M x , a i , j x = Φ j ( ξ i x ) , B x = ( b i , j x ) i , j = 1 M x , b i , j x = Φ j ( ξ i x ) , A y = ( a i , j y ) i , j = 1 M y , a i , j y = Ψ j ( ξ i y ) , B y = ( b i , j y ) i , j = 1 M y , b i , j y = Ψ j ( ξ i y ) , (4.4)

and

G 1 n + 1 = [ g n + 1 ( ξ 1 x , ξ 1 y ) , g n + 1 ( ξ 1 x , ξ 2 y ) , , g n + 1 ( ξ 1 x , ξ M y y ) , g n + 1 ( ξ 2 x , ξ 1 y ) , , g n + 1 ( ξ M x x , ξ M y y ) ] T , (4.5)

G 2 n = [ g n ( ξ 1 x , ξ 1 y ) , g n ( ξ 1 x , ξ 2 y ) , , g n ( ξ 1 x , ξ M y y ) , g n ( ξ 2 x , ξ 1 y ) , g n ( ξ 2 x , ξ 2 y ) , , g n ( ξ M x x , ξ M y y ) ] 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 , namly, for 1 ≤ iK − 1, it follows that

ϕ i ( x ) = 2 ( x x i 1 ) 3 h 3 + 3 ( x x i 1 ) 2 h 2 , x i 1 x x i , 2 ( x x i 1 ) 3 h 3 + 3 ( x x i + 1 ) 2 h 2 , x i x x i + 1 , 0 , x < x i 1 o r x > x i + 1 , (4.8)

and

ψ i ( x ) = ( x x i 1 ) 2 ( x x i ) h 2 , x i 1 x x i , ( x x i ) ( x x i + 1 ) 2 h 2 , x i x x i + 1 , 0 , x < x i 1 o r x > x i + 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

{ ψ 0 , ϕ 1 , ψ 1 , ϕ 2 , , ϕ K 1 , ψ K 1 , ψ K } = { Φ 1 , Φ 2 , Φ 3 , , Φ 2 K } ,

then

M 3 ( δ x ) = s p a n { Φ 1 , Φ 2 , Φ 3 , , Φ 2 K } , M 3 ( δ y ) = s p a n { Φ 1 , Φ 2 , Φ 3 , , Φ 2 K } .

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:

H 1 ( u j ) = ( 1 + 2 u j ) ( 1 u j ) 2 , H 2 ( u j ) = u j ( 1 u j ) 2 h k , H 3 ( u j ) = u j 2 ( 3 2 u j ) , H 4 ( u j ) = u j 2 ( u j 1 ) h k , I 1 ( u j ) = ( 12 u j 6 ) / h k 2 , I 2 ( u j ) = ( 6 u j 4 ) / h k , I 3 ( u j ) = ( 6 12 u j ) / h k 2 , I 4 ( u j ) = ( 6 u j 2 ) / h k , (4.10)

where u 1 = ( 3 3 ) / 6 , u 2 = ( 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

A x = I 2 ( u 1 ) I 3 ( u 1 ) I 4 ( u 1 ) 0 0 0 0 0 0 0 I 2 ( u 2 ) I 3 ( u 2 ) I 4 ( u 2 ) 0 0 0 0 0 0 0 0 I 1 ( u 1 ) I 2 ( u 1 ) I 3 ( u 1 ) I 4 ( u 1 ) 0 0 0 0 0 0 I 1 ( u 2 ) I 2 ( u 2 ) I 3 ( u 2 ) I 4 ( u 2 ) 0 0 0 0 0 0 0 0 I 1 ( u 1 ) I 2 ( u 1 ) I 3 ( u 1 ) I 4 ( u 1 ) 0 0 0 0 0 0 I 1 ( u 2 ) I 2 ( u 2 ) I 3 ( u 2 ) I 4 ( u 2 ) 0 0 0 0 0 0 0 0 I 1 ( u 1 ) I 2 ( u 1 ) I 3 ( u 1 ) I 4 ( u 1 ) 0 0 0 0 0 0 I 1 ( u 2 ) I 2 ( u 2 ) I 3 ( u 2 ) I 4 ( u 2 ) 0 0 0 0 0 0 0 0 I 1 ( u 1 ) I 2 ( u 1 ) I 4 ( u 1 ) 0 0 0 0 0 0 0 I 1 ( u 2 ) I 2 ( u 2 ) I 4 ( u 2 ) . (4.11)

B x = H 2 ( u 1 ) H 3 ( u 1 ) H 4 ( u 1 ) 0 0 0 0 0 0 0 H 2 ( u 2 ) H 3 ( u 2 ) H 4 ( u 2 ) 0 0 0 0 0 0 0 0 H 1 ( u 1 ) H 2 ( u 1 ) H 3 ( u 1 ) H 4 ( u 1 ) 0 0 0 0 0 0 H 1 ( u 2 ) H 2 ( u 2 ) H 3 ( u 2 ) H 4 ( u 2 ) 0 0 0 0 0 0 0 0 H 1 ( u 1 ) H 2 ( u 1 ) H 3 ( u 1 ) H 4 ( u 1 ) 0 0 0 0 0 0 H 1 ( u 2 ) H 2 ( u 2 ) H 3 ( u 2 ) H 4 ( u 2 ) 0 0 0 0 0 0 0 0 H 1 ( u 1 ) H 2 ( u 1 ) H 3 ( u 1 ) H 4 ( u 1 ) 0 0 0 0 0 0 H 1 ( u 2 ) H 2 ( u 2 ) H 3 ( u 2 ) H 4 ( u 2 ) 0 0 0 0 0 0 0 0 H 1 ( u 1 ) H 2 ( u 1 ) H 4 ( u 1 ) 0 0 0 0 0 0 0 H 1 ( u 2 ) H 2 ( u 2 ) H 4 ( u 2 ) . (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 .

## 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 l o g ( e m / e m + 1 ) l o g ( h m / h m + 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

0 c D t γ u ( x , t ) = 2 u ( x , t ) x 2 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 ) t 3 γ x 2 ( 1 x ) 2 e x 2 t 3 e x ( 1 4 x + 4 x 3 ) . 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.

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 1 10 7.0727×10−5 4.4681×10−5
1 20 1.7932×10−5 1.9798 1.1012×10−5 2.0206
1 40 4.5623×10−6 1.9747 2.7487×10−6 2.0022
1 80 1.1483×10−6 1.9903 6.8758×10−7 1.9992
1.3 1 10 2.6081×10−4 1.7238×10−4
1 20 6.6648×10−5 1.9684 4.2518×10−5 2.0194
1 40 1.6825×10−6 1.9860 1.0577×10−5 2.0072
1 80 4.2263×10−7 1.9931 2.6387×10−6 2.0030
1.5 1 10 4.1657×10−4 2.7911×10−4
1 20 1.0633×10−4 1.9701 6.8593×10−5 2.0247
1 40 2.6736×10−5 1.9916 1.7020×10−5 2.0108
1 80 6.7115×10−6 1.9941 4.2405×10−6 2.0050
1.7 1 10 5.3422×10−4 3.6265×10−4
1 20 1.3701×10−4 1.9632 8.9343×10−5 2.0212
1 40 3.4419×10−5 1.9930 2.2160×10−5 2.0114
1 80 8.6292×10−6 1.9959 5.5175×10−6 2.0059
1.9 1 10 5.7600×10−4 3.9339×10−4
1 20 1.4884×10−4 1.9523 9.7244×10−5 2.0163
1 40 3.7391×10−5 1.9930 2.4112×10−5 2.0119
1 80 9.3633×10−6 1.9976 5.9996×10−6 2.0068
1.95 1 10 5.6941×10−4 3.8862×10−4
1 20 1.4696×10−4 1.9540 9.6061×10−5 2.0163
1 40 3.6917×10−5 1.9931 2.3812×10−5 2.0123
1 80 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 1 10 2.4371×10−6 1.7740×10−6
1 20 1.5377×10−7 3.9863 1.0837×10−7 4.0329
1 40 9.6290×10−9 3.9972 6.6928×10−9 4.0172
1 80 6.0225×10−10 3.9989 4.1576×10−10 4.0088
1.3 1 10 3.8377×10−6 2.6750×10−6
1 20 2.4364×10−7 3.9774 1.6332×10−7 4.0338
1 40 1.5241×10−8 3.9987 1.0085×10−8 4.0174
1 80 9.5308×10−10 3.9992 6.2644×10−10 4.0089
1.5 1 10 4.7527×10−6 3.2535×10−6
1 20 3.0159×10−7 3.9781 1.9851×10−7 4.0347
1 40 1.8863×10−8 3.9990 1.2256×10−8 4.0177
1 80 1.1798×10−9 3.9990 7.6129×10−10 4.0089
1.7 1 10 5.1530×10−6 3.4857×10−6
1 20 3.2579×10−7 3.9834 2.1258×10−7 4.0354
1 40 2.0382×10−8 3.9986 1.3123×10−8 4.0178
1 80 1.2754×10−9 3.9982 8.1509×10−10 4.0090
1.9 1 10 4.6730×10−6 3.0735×10−6
1 20 2.9311×10−7 3.9948 1.8735×10−7 4.0361
1 40 1.8412×10−8 3.9927 1.1563×10−8 4.0181
1 80 1.1509×10−9 3.9999 7.1819×10−10 4.0090
1.95 1 10 4.3316×10−6 2.8280×10−6
1 20 2.7151×10−7 3.9958 1.7235×10−7 4.0364
1 40 1.7062×10−8 3.9922 1.0637×10−8 4.0182
1 80 1.0665×10−9 3.9999 6.6066×10−10 4.0091

## Example 2

Consider the following one-dimensional fractional diffusion-wave equation

0 c D t γ u ( x , t ) = 2 u ( x , t ) x 2 u ( x , t ) + f ( x , t ) , 0 < x < 1 , 0 < t 1 , u ( x , 0 ) = 0 , u ( x , 0 ) t = sin π x , 0 x 1 , u ( 0 , t ) = u ( 1 , t ) = 0 , 0 t 1 , (5.2)

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

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 1 10 2.7779×10−5 1.8686×10−5
1 20 6.9405×10−6 2.0009 4.5452×10−6 2.0395
1 40 1.7225×10−6 2.0105 1.1135×10−6 2.0292
1 80 4.2704×10−7 2.0121 2.7427×10−7 2.0215
1.3 1 10 6.8399×10−5 4.6042×10−5
1 20 1.6912×10−5 2.0159 1.1079×10−5 2.0551
1 40 4.1818×10−6 2.0158 2.7032×10−6 2.0352
1 80 1.0358×10−6 2.0134 6.6503×10−7 2.0232
1.5 1 10 1.0251×10−4 6.9519×10−5
1 20 2.5114×10−5 2.0292 1.6555×10−5 2.0701
1 40 6.2025×10−6 2.0176 4.0327×10−6 2.0375
1 80 1.5384×10−6 2.0114 9.9335×10−7 2.0214
1.7 1 10 1.4424×10−4 9.9816×10−5
1 20 3.5642×10−5 2.0168 2.4001×10−5 2.0562
1 40 8.8717×10−6 2.0063 5.8868×10−6 2.0275
1 80 2.2116×10−6 2.0041 1.4572×10−6 2.0143
1.9 1 10 1.9061×10−4 1.2932×10−4
1 20 4.7290×10−5 2.0110 3.1561×10−5 2.0347
1 40 1.1810×10−5 2.0015 7.7937×10−6 2.0178
1 80 2.9519×10−6 2.0003 1.9367×10−6 2.0087
1.95 1 10 2.0110×10−4 1.3455×10−4
1 20 4.9930×10−5 2.0099 3.2930×10−5 2.0306
1 40 1.2482×10−5 2.0001 8.1369×10−6 2.0169
1 80 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 1 10 9.8873×10−8 7.9806×10−8
1 20 6.3013×10−9 3.9719 5.0124×10−9 3.9929
1 40 4.0052×10−10 3.9757 3.1726×10−10 3.9818
1 80 2.5425×10−11 3.9775 2.0139×10−11 3.9776
1.3 1 10 2.0590×10−7 1.1973×10−7
1 20 1.2348×10−8 4.0596 7.0248×10−9 4.0912
1 40 7.5090×10−10 4.0395 4.2273×10−10 4.0547
1 80 4.6084×10−11 4.0263 2.5826×10−11 4.0328
1.5 1 10 3.1378×10−7 1.8224×10−7
1 20 1.9140×10−8 4.0351 1.0838×10−8 4.0716
1 40 1.1827×10−9 4.0165 6.6114×10−10 4.0350
1 80 7.3513×10−11 4.0079 4.0836×10−11 4.0170
1.7 1 10 4.2637×10−7 2.5157×10−7
1 20 2.6414×10−8 4.0127 1.5170×10−8 4.0516
1 40 1.6453×10−9 4.0049 9.3246×10−10 4.0241
1 80 1.0270×10−10 4.0019 5.7825×10−11 4.0113
1.9 1 10 6.2873×10−7 3.5996×10−7
1 20 3.9276×10−8 4.0007 2.1898×10−8 4.0389
1 40 2.4548×10−9 4.0000 1.3510×10−9 4.0188
1 80 1.5342×10−10 4.0000 8.3898×10−11 4.0092
1.95 1 10 6.7414×10−7 3.9216×10−7
1 20 4.2262×10−8 3.9956 2.3894×10−8 4.0367
1 40 2.6423×10−9 3.9995 1.4745×10−9 4.0184
1 80 1.6515×10−10 3.9999 9.1572×10−11 4.0091

## Example 3

Consider the following two-dimensional fractional diffusion-wave equation

0 c D t γ u ( x , y , t ) Δ u ( 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 ) t 3 γ x y ( 1 x ) ( 1 y ) + t 3 x y ( 7 3 y 3 x x y ) ] e x + 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).

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

0 c D t γ u ( x , y , t ) Δ u ( 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 + ( 2 π 2 + 1 ) t γ ] t 2 sin π x sin π y . 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).

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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