Journal of Function Spaces

Journal of Function Spaces / 2021 / Article
Special Issue

Fractional Delay Differential Equations and their Numerical Solutions

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Research Article | Open Access

Volume 2021 |Article ID 9979791 |

Yifan Qin, Xiaocheng Yang, Yunzhu Ren, Yinghong Xu, Wahidullah Niazi, "A Newton Linearized Crank-Nicolson Method for the Nonlinear Space Fractional Sobolev Equation", Journal of Function Spaces, vol. 2021, Article ID 9979791, 11 pages, 2021.

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

Academic Editor: Maria Alessandra Ragusa
Received16 Mar 2021
Accepted10 Apr 2021
Published27 Apr 2021


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 in the sense of -norm, -norm, and -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.

1. 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 , and are given positive constants, and are known sufficiently smooth functions. in (1) denotes the Riesz fractional derivative operator for 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 , (1) reduces to a one-dimensional integral-order Sobolev equation in the bounded domain [5]. When with integer and given constants , then the equation is called a semiconductor equation [6]. When , it is reduced to a homogeneous space fractional Sobolev equation. When , (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 -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 -norm, -norm, and -norm. Besides, we also prove that the convergence orders of the constructed linearized numerical scheme are 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.

2. 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 be denoted by the infinite grid with grid points (). For arbitrary grid functions on , we define the discrete inner products and the corresponding -norm and -norm Denote . For , the semidiscrete Fourier transformation is written as

It is easy to get due to . The inversion formula is defined by then we can easily check that Parseval’s equality holds. Moreover, For the given constant , the fractional Sobolev norm and seminorm are defined as follows:

Obviously, .

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 where denotes the Fourier transformation of .

Lemma 1. (cf. [7]). Suppose the functionand the fractional central difference is defined as follows:Then, it holds is defined as This is consistently established for arbitrary .

Remark 2. (cf. [15, 16]). If we definebysuch that . We get For any , we define and suppose .

3. 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 as follows:

3.1. The Derivation of the Linearized Numerical Scheme

Take positive integers , and let , be the temporal and spatial step sizes, respectively. Denote , ; , ; , ; , . Define , Let be grid function space defined on . Then, for a given grid function , we introduce the following notations:

Define the grid function

Then, we consider (15) at the point 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 is a positive constant.

Applying the Newton linearized method to the nonlinear term on RHS in (20) and using Taylor expansion at the point , it yields where . Plugging (21)–(23) and substituting (25) into (20), we have

There exists a positive constant such that

Omitting in (26), replacing with in (26), then the finite difference scheme reads

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]). LetIt holds where for any , and , is the th the eigenvalue of matrix . is given in a similar way. It implies that the matrices and are real symmetric positive definite matrices.

Lemma 4. (discrete Sobolev inequality (cf. [14])) For every, there exists a constant, independent of, such that

Lemma 5. (cf. [8]). For anyand any grid function, we havewhere .

Lemma 6. (cf. [17]). For any grid function, there exists a fractional symmetric positive quotient operator, such that

Lemma 7. (cf. [18]) (discrete uniform Sobolev inequality). For every, there exists a constantindependent ofsuch that

Lemma 8. (cf. [19]). Supposebe nonnegative sequence and satisfyThen, we have where and are nonnegative constants.

Theorem 9. The linearized finite difference scheme (28)–(30) is uniquely solvable.

Proof. Denote . We will prove the above result by the mathematical induction. Obviously, (29) is true for . Now, we suppose has been uniquely determined; then, we only need to prove that is uniquely determined by (28). We can rewrite (28) in the following matrix form where is a vector which depends only on the boundary value. By using Lemma 3, when 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 . This completes the proof.

3.3. 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. Letbe the solution of the problem (15)–(17). Then, we havewhere is a positive constant independent of and .


We will obtain the main convergence result.

Theorem 11. Letbe the solution of the problem (15)–(17). Then, there exist positive constantsand, whenand, for, we havewhere are positive constants independent of and .

Proof. The mathematical induction will be employed. Firstly, it is obvious (42) is true for , via (29). Then, it assumes that (42) is true for . We will discuss that (42) holds for . According to the hypothesis, we can obtain the following estimation: where , , and .

In the view of Lipschitz condition, we have where , , and are positive constants independent of and .

Now, subtracting (28) from (26), we can obtain the error equation where

Firstly, we establish -error estimation. Taking the discrete inner product of (47) with , 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


Summing for from to , we have

Noticing that and , we have

Let , we have

It implies when , we have

Using Gronwall Lemma 8, we have

Therefore, we have where .

Similarly, applying Lemma 5 yields where .

Finally, we can establish -error estimate by combining (62) with (63). Denoting , it follows from Lemma 7 that

We complete the proof.

Next, we will analyze the stability of the scheme (28)–(30). Let be the solution of the fractional Sobolev equation where is the perturbation of the initial value. Subtracting (65)–(67) from (28)–(30) and denoting , we have

Similar to the proof of Theorem 11, we have the following result.

Theorem 12. Denote. Then, there exist positive constantsand, whenand, we havewhere are positive constants independent of and .

4. Numerical Examples

In this section, we will provide extensive numerical examples to testify the theoretical results. we will define the discrete -norm and -norm separately and the corresponding convergence orders are defined as follows:

Example 1. We firstly consider the following fractional Sobolev equation as

The exact solution is

The initial boundary conditions and are determined by (72).

Taking , , the linearized numerical scheme (28)–(30) with is applied to solve the above Sobolev equation. The global numerical errors and convergence orders with respect to different and are listed in the following tables. Table 1 lists the -norm and -norm errors and spatial convergence orders with fixed time step . Table 2 tests the temporal convergence orders with fixed spatial step . 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.