A Newton Linearized Crank-Nicolson Method for the Nonlinear Space Fractional Sobolev Equation

In this paper, one class of finite difference scheme is proposed to solve nonlinear space fractional Sobolev equation based on the Crank-Nicolson (CN) method. Firstly, a fractional centered finite difference method in space and the CN method in time are utilized to discretize the original equation. Next, the existence, uniqueness, stability, and convergence of the numerical method are analyzed at length, and the convergence orders are proved to be Oðτ2 + h2Þ in the sense of l2-norm, Hα/2-norm, and l∞ -norm. Finally, the extensive numerical examples are carried out to verify our theoretical results and show the effectiveness of our algorithm in simulating spatial fractional Sobolev equation.


Introduction
The main propose of this paper is to construct one class of the Newton linearized finite difference method based on CN discretization in temporal direction to efficiently solve the following spatial fractional Sobolev equation: where 1 < α, β ⩽ 2, μ and κ are given positive constants, u 0 ðxÞ and f ðuÞ are known sufficiently smooth functions. ∂ α x in (1) denotes the Riesz fractional derivative operator for 1 < α ⩽ 2 and is defined in [1] as follows: This type of equation is widely used as a mathematical model for fluid flow through thermodynamics [2], shear in second-order fluids [3], consolidation of clay [4], and so on. Note that some special forms of equation (1) are frequently encountered in many fields. For example, taking α, β = 2, (1) reduces to a one-dimensional integral-order Sobolev equation in the bounded domain [5]. When f ðuÞ = ∑ p i=1 γ i u p with integer p and given constants γ i ði = 1, 2,⋯,pÞ, then the equation is called a semiconductor equation [6]. When f ðuÞ = 0, it is reduced to a homogeneous space fractional Sobolev equation. When μ = 0, (1) is reduced to the classical nonlinear reaction-diffusion equations. Recently, many scholars are dedicated to the numerical investigation on fractional diffusion equations and Sobolev equations based on finite difference or finite element methods in the literature. For example, Çelik and Duman [7] investigated the CN method to approximate the fractional diffusion equation with the Riesz fractional derivative in a finite domain. Wang et al. [8] studied the finite difference method for the space fractional Schrödinger equations under the framework of the fractional Sobolev space. Ran and He [9] investigated the nonlinear multidelay fractional diffusion equation based on the CN method in time and the fractional centered difference in space. Chen et al. [5] proposed a Newton linearized compact finite difference scheme to numerically solve a class of Sobolev equations based on the CN method and proved the unique solvability, convergence, and stability of the proposed scheme. Wang and Huang [10] constructed a conservative linearized difference scheme for the nonlinear fractional Schrödinger equation. Zhang et al. [11] established the numerical asymptotic stability result of the compact θ -method for the generalized delay diffusion equation. More researches on delay fractional problems can be referred to [12,13] and the references therein.
The main work in this paper is to develop an efficient Newton linearized CN method to solve the nonlinear space fractional Sobolev problem (1). The existence, uniqueness, stability, and convergence of the proposed numerical scheme are demonstrated, and the convergent orders are obtained in the sense of l 2 -norm, H α/2 -norm, and l ∞ -norm. Besides, we also prove that the convergence orders of the constructed linearized numerical scheme are Oðτ 2 + h 2 Þ under three types of norms. The extensive numerical examples are proposed to argue a second-order accuracy in both temporal and spatial dimensions.
The organization of this paper is as follows. In Section 2, we define the fractional Sobolev norm and introduce the second-order centered finite difference approximation for the space Riesz derivative. In Section 3, we construct a CN finite difference scheme for the space fractional Sobolev equation. The existence, uniqueness, stability, and convergence of the proposed scheme in three classes of conventional norms are proved. Finally, the theoretical results are verified by several numerical examples.

Preliminaries
Firstly, we present some notations and lemmas which will be used to construct and analyze our numerical scheme.
2.1. Fractional Sobolev Norm. Firstly, we define the fractional Sobolev norm (cf. [14]). Let hℤ be denoted by the infinite grid with grid points x j = jh (j ∈ ℤ). For arbitrary grid functions u = fu j g, v = fv j g on hℤ, we define the discrete inner products and the corresponding l 2 -norm and l ∞ -norm Denote l 2 ≔ fu | u = fu j g,∥u∥ 2 <+∞g. For u ∈ l 2 , the semidiscrete Fourier transformationû is written aŝ It is easy to getû ∈ L 2 ½−π/h, π/h due to u ∈ l 2 . The inversion formula is defined by then we can easily check that Parseval's equality holds. Moreover, For the given constant 0 ⩽ σ ⩽ 1, the fractional Sobolev norm ∥·∥ H σ and seminorm j·j H σ are defined as follows: Obviously, ∥u∥ 2 H σ = ∥u∥+juj 2 H σ . 2.2. Second-Order Approximation of Spatial Riesz Fractional Derivative. In this section, we will review a second-order approximation for the Riesz fractional derivative. Introduce wheref ðωÞ = Ð ∞ −∞ e iωt f ðtÞdt denotes the Fourier transformation of f ðxÞ. Lemma 1. (cf. [7]). Suppose the function f ð·Þ ∈ C 2+α ðℝÞ and the fractional central difference is defined as follows: Then, it holds g ðαÞ k is defined as This is consistently established for arbitrary x ∈ ℝ.

Second-Order CN Method and Theoretical Analysis
In this section, we are concentrated on the derivation and theoretical analysis of the finite different scheme. In practical computation, it is necessary to truncate the whole space problem onto a finite interval (boundaries are usually chosen sufficient large such that the truncation error is negligible or the exact solution has compact support in the bounded domain [17]). Here, we will truncate (1) on the interval Ω = ða, bÞ as follows: 3.1. The Derivation of the Linearized Numerical Scheme. Take positive integers M, N and let τ = T/N, h = ðb − aÞ/M be the temporal and spatial step sizes, respectively. Denote Then, for a given grid function u ∈ V h , we introduce the following notations: Define the grid function Then, we consider (15) at the point ðx i , t k+ð1/2Þ Þ and have Utilizing the Taylor expansion, the first term on the left hand side (LHS) in (20) can be estimated as Noticing Lemma 1, for the second term on LHS in (20), we have For the first term on the right hand side (RHS) in (20), it yields Moreover, we have where c 0 is a positive constant.
Applying the Newton linearized method to the nonlinear term f on RHS in (20) and using Taylor expansion at the where . Plugging (21)-(23) and substituting (25) into (20), we have There exists a positive constant c 1 > 0 such that Omitting 3.2. The Unique Solvability of Finite Difference Scheme. This section is concerned with the solvability of scheme (28)-(30). Now, we give some lemmas which will be used in the demonstration of solvability.

Lemma 3. (cf. [7]). Let
It holds where for any |j | ⩽1, and 0 < λ i < 2g ðαÞ 0 ði ∈ ωÞ, λ i is the ith the eigenvalue of matrix A α . A β is given in a similar way. It implies that the matrices A α and A β are real symmetric positive definite matrices.
We will prove the above result by the mathematical induction. Obviously, (29) is true for k = 0. Now, we suppose u l ð0 ⩽ k ⩽ l ⩽ N − 1Þ has been uniquely determined; then, we only need to prove that u l+1 is uniquely determined by (28). We can rewrite (28) in the following matrix form whereG l+1 is a vector which depends only on the boundary value. By using Lemma 3, when τ is sufficiently small, it is easy to verify that the coefficient matrix of (39) is strictly diagonally dominant, which implies that there exists a unique solution u l+1 . This completes the proof.

The Convergence and Stability of the Finite Difference
Scheme. Firstly, we easily have the estimation of the local truncation error, according to (27).

Lemma 10.
Let uðx, ·Þ ∈ C ð2+αÞ ðx, ·Þ be the solution of the problem (15)- (17). Then, we have where c 1 is a positive constant independent of τ and h. Denote We will obtain the main convergence result.

Journal of Function Spaces
Proof. The mathematical induction will be employed. Firstly, it is obvious (42) is true for k = 0, via (29). Then, it assumes that (42) is true for 1 ⩽ k ⩽ m ⩽ N − 1. We will discuss that (42) holds for k = m + 1. According to the hypothesis, we can obtain the following estimation: where In the view of Lipschitz condition, we have where c 2 , c 3 , and c 4 are positive constants independent of τ and h. Now, subtracting (28) from (26), we can obtain the error equation where Firstly, we establish l 2 -error estimation. Taking the discrete inner product of (47) with e k+ð1/2Þ , we have Now, we estimate each term in (49). The first term on LHS in (49) can be estimated as Noticing Lemma 6, for the second term on the LHS in (49), we have Similarly, the first term on RHS in (49) can be obtained by According to (44)-(46), we have Using the Cauchy-Schwarz inequality and Young inequality, the second term on the RHS in (49) becomes The last term of RHS in (49) is estimated as Substituting (50)-(55) into (49), we get where c 5 = ð9c 2 2 /4Þ + ð9c 2 3 c 2 0 τ 2 /16Þ + ðð9/8Þ9/8c 2 4 Þ + ð1/3Þ:
Then, there exist positive constants τ 0 and h 0 , when τ < τ 0 and h < h 0 , we have where C 4 , C 5 , C 6 > 0 are positive constants independent of τ and h.

Numerical Examples
In this section, we will provide extensive numerical examples to testify the theoretical results. we will define the discrete l 2 -norm and l ∞ -norm separately and the corresponding convergence orders are defined as follows: Journal of Function Spaces Example 1. We firstly consider the following fractional Sobolev equation as The exact solution is The initial boundary conditions and gðx, tÞ are determined by (72). Table 1: l 2 -and l ∞ -errors and their convergence orders of (28)-(30) for 1 < α < 2 in the spatial direction for (72) with fixed time step τ = 1/2000 for Example 1.   Taking μ = 1, κ = 1, the linearized numerical scheme (28)-(30) with τ = h is applied to solve the above Sobolev equation. The global numerical errors and convergence orders with respect to different α and β are listed in the following tables. Table 1 lists the l 2 -norm and l ∞ -norm errors and spatial convergence orders with fixed time step τ = 1/2000. Table 2 tests the temporal convergence orders with fixed spatial step h = 1/2000. It demonstrates that the convergence orders of the scheme (28)-(30) is second-order accurate in both spatial and temporal directions which is consistent with Theorem 11.
All the data are referred to MATLAB codes in Example 1 in the supplementary files.
is oscillatory along with the temporal direction, where μ = 1 and κ = 1. And the initial boundary conditions and gðx, tÞ are determined by (73).
In this example, we examine the spatial convergence orders with the fixed time step τ = 1/2000 and the temporal convergence orders with the fixed spatial step h = 1/1000 in l 2 -norm and l ∞ -norm errors, respectively. All the numerical results in the example are listed in Tables 3 and 4. Similar results are observed. All the data are referred to MATLAB codes in Example 2 in the supplementary files.
Example 3. Then, we calculate the nonlinear fractional Sobolev equation as We choose the exact solution The initial boundary conditions and gðx, tÞ are determined by (75).
Similar to above example, Tables 5 and 6 list the l 2 -norm and l ∞ -norm errors and corresponding spatial and temporal convergence orders of (28)-(30), respectively. To testify the spatial convergence orders, we fixed the time step τ = 1/ 1000. Similarly, we take the fixed spatial step h = 1/1000 to obtain the temporal convergence orders. The numerical Table 7: l 2 -and l ∞ -errors and their convergence orders of (28)-(30) for 1 < α ⩽ 2 in the spatial direction with τ = 1/1000 for Example 4.
Example 4. We consider the following equation: with the exact solution is unknown.
In the computation, we take different spatial and temporal step sizes. The l 2 -norm, l ∞ -norm errors, and their convergence orders of (28)-(30) are listed in Table 7 with the fixed temporal step size τ = 1/1000. Similarly, the spatial step size fixed at h = 1/1000 in Table 8. Tables 7 and 8 show that the numerical results have second-order accurate in spatial and temporal directions. Figure 1 presents curves of uðx, tÞ with respect to x at different time with the step sizes h = 0:5 and τ = 0:02. All the data are referred to MATLAB codes in Example 4 in the supplementary files.

Conclusion
In the article, we establish an efficient finite difference scheme for nonlinear spatial fractional Sobolev equation based on Newton linearized technique. We have proved that the numerical solution of the scheme is unique solvable, stable, and convergent. The pointwise error estimate is proved with the convergence order Oðτ 2 + h 2 Þ. Extensive numerical examples are carried out to testify the numerical theoretical results. Extending the current work to high dimensional cases is possible, which will leave as our future work.

10
Journal of Function Spaces

Data Availability
All the data are available and referred to the supplementary file.

Conflicts of Interest
The authors declare that they have no competing interests.