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

Authors: Yifan Qin; Xiaocheng Yang; Yunzhu Ren; Yinghong Xu; Wahidullah Niazi
Journal: Journal of Function Spaces
 (2021)

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## Abstract

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 beOτ2+h2 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.


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## Body

## 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:(1)∂tu−μ∂xα∂tu=κ∂xβu+fu,inℝ×0,T,ux,0=u0x,inℝ×0.where 1<α,β⩽2, μ and κ are given positive constants, u0x and fu are known sufficiently smooth functions. ∂xα in (1) denotes the Riesz fractional derivative operator for 1<α⩽2 and is defined in [1] as follows:
(2)∂xαux,t=−12cosπα/2Γ2−α∂2∂x2∫−∞∞x−ξ1−αuξ,tdξ.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 fu=∑i=1pγiup with integer p and given constants γii=1,2,⋯,p, then the equation is called a semiconductor equation [6]. When fu=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 l2-norm, Hα/2-norm, and l∞-norm. Besides, we also prove that the convergence orders of the constructed linearized numerical scheme are Oτ2+h2 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 Section2, 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 hℤ be denoted by the infinite grid with grid points xj=jh (j∈ℤ). For arbitrary grid functions u=uj,v=vj on hℤ, we define the discrete inner products and the corresponding l2-norm and l∞-norm
(3)u,v=h∑j∈ℤujvj,∥u∥2=u,u,∥u∥l∞=supj∈ℤuj.Denote l2≔u∣u=uj,∥u∥2<+∞. For u∈l2, the semidiscrete Fourier transformation u^ is written as
(4)u^k≔12πh∑j∈ℤuje−ikxj.It is easy to getu^∈L2−π/h,π/h due to u∈l2. The inversion formula is defined by
(5)uj=12π∫−π/hπ/hu^keikxjdk,then we can easily check that Parseval’s equality
(6)u,v=∫−π/hπ/hu^kv^kdk,holds. Moreover, For the given constant 0⩽σ⩽1, the fractional Sobolev norm ∥·∥Hσ and seminorm ·Hσ are defined as follows:
(7)∥u∥Hσ2=∫−π/hπ/h1+k2σu∧k2dk,uHσ2=∫−π/hπ/hk2σu∧k2dk.Obviously,∥u∥Hσ2=∥u∥+uHσ2.


### 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(8)Cn+αR=f∫−∞∞1+∣ω∣n+α∣f^ω∣dω<∞,f∈L1ℝ,where f^ω=∫−∞∞eiωtftdt denotes the Fourier transformation of fx.Lemma 1.
(cf. [7]). Suppose the functionf·∈C2+αℝand the fractional central difference is defined as follows:(9)δxαfx=−h−α∑k=−∞+∞gkαfx−kh.
Then, it holds(10)δxαfx=∂xαfx+Oh2.gkα is defined as
(11)gkα=−1kΓα+1Γα/2−k+1Γα/2+k+1,k∈ℤ.
This is consistently established for arbitraryx∈ℝ.Remark 2.
(cf. [15, 16]). If we definef∗by(12)f∗x=fx,x∈a,b,0,x∉a,b,such that f∗x∈C2+αℝ. We get
(13)δxαfx=−h−α∑k=−b−x/hx−a/hgkαfx−kh+Oh2.
For anyt∈0,T, we define
(14)u∗x=ux,x∈a,b,0,x∉a,b,and suppose u∗x∈C2+αℝ.


## 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 xj=jh (j∈ℤ). For arbitrary grid functions u=uj,v=vj on hℤ, we define the discrete inner products and the corresponding l2-norm and l∞-norm
(3)u,v=h∑j∈ℤujvj,∥u∥2=u,u,∥u∥l∞=supj∈ℤuj.Denote l2≔u∣u=uj,∥u∥2<+∞. For u∈l2, the semidiscrete Fourier transformation u^ is written as
(4)u^k≔12πh∑j∈ℤuje−ikxj.It is easy to getu^∈L2−π/h,π/h due to u∈l2. The inversion formula is defined by
(5)uj=12π∫−π/hπ/hu^keikxjdk,then we can easily check that Parseval’s equality
(6)u,v=∫−π/hπ/hu^kv^kdk,holds. Moreover, For the given constant 0⩽σ⩽1, the fractional Sobolev norm ∥·∥Hσ and seminorm ·Hσ are defined as follows:
(7)∥u∥Hσ2=∫−π/hπ/h1+k2σu∧k2dk,uHσ2=∫−π/hπ/hk2σu∧k2dk.Obviously,∥u∥Hσ2=∥u∥+uHσ2.


## 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(8)Cn+αR=f∫−∞∞1+∣ω∣n+α∣f^ω∣dω<∞,f∈L1ℝ,where f^ω=∫−∞∞eiωtftdt denotes the Fourier transformation of fx.Lemma 1.
(cf. [7]). Suppose the functionf·∈C2+αℝand the fractional central difference is defined as follows:(9)δxαfx=−h−α∑k=−∞+∞gkαfx−kh.
Then, it holds(10)δxαfx=∂xαfx+Oh2.gkα is defined as
(11)gkα=−1kΓα+1Γα/2−k+1Γα/2+k+1,k∈ℤ.
This is consistently established for arbitraryx∈ℝ.Remark 2.
(cf. [15, 16]). If we definef∗by(12)f∗x=fx,x∈a,b,0,x∉a,b,such that f∗x∈C2+αℝ. We get
(13)δxαfx=−h−α∑k=−b−x/hx−a/hgkαfx−kh+Oh2.
For anyt∈0,T, we define
(14)u∗x=ux,x∈a,b,0,x∉a,b,and suppose u∗x∈C2+αℝ.


## 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 Ω=a,b as follows:
(15)∂tu−μ∂xα∂tu=κ∂xβu+fu,inΩ×0,T,(16)ux,0=u0x,inℝ×0,(17)ux,t=0,onℝ\Ω×0,T.


### 3.1. The Derivation of the Linearized Numerical Scheme

Take positive integersM, N and let τ=T/N, h=b−a/M be the temporal and spatial step sizes, respectively. Denote xi=a+ih, 0⩽i⩽M; tk=kτ, 0⩽k⩽N; tk+1/2=k+1/2τ, 0⩽k⩽N−1; Ωh=xi∣0⩽i⩽M, Ωτ=tk∣0⩽k⩽N. Define ω¯=j∣j=0,1,⋯,M,ω=j∣j=1,2,⋯,M−1, ∂ω=ω¯\ω. Let Vh=u∣u=uik∣0⩽i⩽M,0⩽k⩽N,u0k=uMk=0 be grid function space defined on Ωhτ=Ωh×Ωτ. Then, for a given grid function u∈Vh, we introduce the following notations:
(18)uik+1/2=12uik+1+uik,δtuik+1/2=1τuik+1−uik.Define the grid function(19)Uik=uxi,tk,i∈ω¯,0⩽k⩽N.Then, we consider (15) at the point xi,tk+1/2 and have
(20)∂tuxi,tk+1/2−μ∂xα∂tuxi,tk+1/2=κ∂xβuxi,tk+1/2+fuxi,tk+1/2,i∈ω,0⩽k⩽N−1.Utilizing the Taylor expansion, the first term on the left hand side (LHS) in (20) can be estimated as
(21)∂tuxi,tk+1/2=δtUik+1/2+Oτ2.Noticing Lemma1, for the second term on LHS in (20), we have
(22)∂xα∂tuxi,tk+1/2=−1hα∑j=0Mgi−jαδtUjk+1/2+Oτ2+h2=δtδxαUik+1/2+Oτ2+h2.For the first term on the right hand side (RHS) in (20), it yields
(23)∂xβuxi,tk+1/2=−1hβ∑j=0Mgi−jβUjk+1/2+Oτ2+h2=δxβUik+1/2+Oτ2+h2.Moreover, we have(24)uxi,tk+1/2=12Uik+1+Uik+Oτ2,uxi,tk+1−uxi,tk=Oτ⩽c0τ,where c0 is a positive constant.Applying the Newton linearized method to the nonlinear termf on RHS in (20) and using Taylor expansion at the point Uik, it yields
(25)fuxi,tk+1/2=fUik+Uik+1/2−Uikf′Uik+Oτ2=fUik+12Uik+1−Uikf′Uik+Oτ2,where f′Uik=∂UfU=Uik. Plugging (21)–(23) and substituting (25) into (20), we have
(26)δtUik+1/2−μδtδxαUik+1/2=κδxβUik+1/2+fUik+12f′UikUik+1−Uik+Rik,i∈ω,0⩽k⩽N−1.There exists a positive constantc1>0 such that
(27)Rik⩽c1τ2+h2,i∈ω,0⩽k⩽N−1.OmittingRik in (26), replacing Uik+1/2 with uik+1/2 in (26), then the finite difference scheme reads
(28)δtuik+1/2−μδtδxαuik+1/2=κδxβuik+1/2+fuik+12f′uikuik+1−uik,i∈ω,0⩽k⩽N−1,(29)ui0=u0xi,i∈ω¯,(30)uik=0,i∈∂ω,1⩽k⩽N.


### 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(31)Aα=g0αg−1α⋯g3−Mαg2−Mαg1αg0αg−1α⋯g3−Mα⋮⋱⋱⋱⋮gM−3α⋯g1αg0αg−1αgM−2αgM−3α⋯g1αg0α.It holds
(32)g0α=Γα+1Γ2α/2+1⩽0,∑j=−∞+∞gjα=0,gjα=g−jα⩽0,where for any ∣j∣⩽1, and 0<λi<2g0αi∈ω, λi is the ith the eigenvalue of matrix Aα. Aβ is given in a similar way. It implies that the matrices Aα and Aβ are real symmetric positive definite matrices.Lemma 4.
(discrete Sobolev inequality (cf. [14])) For every1/2<σ⩽1, there exists a constantCσ=Cσ>0, independent ofh>0, such that(33)∥v∥l∞⩽Cσ∥v∥Hσ.Lemma 5.
(cf. [8]). For any1<α⩽2and any grid functionv∈Vh, we have(34)CαvHα/22⩽−δxαv,v⩽vHα/22,where Cα=π/2α.Lemma 6.
(cf. [17]). For any grid functionv∈Vh, there exists a fractional symmetric positive quotient operatorδxα/2, such that(35)−δxαv,v=δxα/2v,δxα/2v.Lemma 7.
(cf. [18]) (discrete uniform Sobolev inequality). For every1/2<σ⩽1, there exists a constantCσ=Cσ>0independent ofh>0such that(36)∥u∥l∞⩽∥u∥Hσ.Lemma 8.
(cf. [19]). SupposeFkk=0∞be nonnegative sequence and satisfy(37)Fk⩽cτ∑l=0k−1Fl+g,k=0,1,2,⋯,
Then, we have(38)Fk⩽geckτ,k=0,1,2,⋯,where c and g are nonnegative constants.Theorem 9.
The linearized finite difference scheme (28)–(30) is uniquely solvable.Proof.
Denoteuk=u1k,u2k,⋯,uM−1kT. We will prove the above result by the mathematical induction. Obviously, (29) is true for k=0. Now, we suppose ul0⩽k⩽l⩽N−1 has been uniquely determined; then, we only need to prove that ul+1 is uniquely determined by (28). We can rewrite (28) in the following matrix form
(39)I+μhαAα+τκ2hβAβ−τ2diagf′ulul+1=I+μhαAα−τκ2hβAβ−τ2diagf′ulul+τful+G~l+1,where G~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 ul+1. 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.
Letux,·∈C2+αx,·be the solution of the problem (15)–(17). Then, we have(40)∥Rk∥2⩽b−ac12τ2+h22,0⩽k⩽N−1,where c1 is a positive constant independent of τ and h.Denote(41)eik=Uik−uik,i∈ω,0⩽k⩽N.We will obtain the main convergence result.Theorem 11.
Letux,·∈C2+αx,·be the solution of the problem (15)–(17). Then, there exist positive constantsτ0andh0, whenτ<τ0andh<h0, for0⩽k⩽N, we have(42)∥ek∥⩽C1τ2+h2,ekHα/2⩽C2τ2+h2,∥ek∥l∞⩽C3τ2+h2,where C1,C2,C3>0 are positive constants independent of τ and h.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:
(43)∥uk∥l∞⩽∥ek∥l∞+∥Uk∥l∞⩽C3τ2+h2+c~0⩽1+c~0,1⩽k⩽m,where τ<τ0=2C3−1/2, h<h0=2C3−1/2, and c~0=maxx,t∈Ω×0,T∣Ux,t∣.In the view of Lipschitz condition, we have(44)∣fUik−fuik∣⩽c2eik,i∈ω¯,0⩽k⩽N,(45)∣f′Uik−f′uik∣⩽c3eik,i∈ω¯,0⩽k⩽N,(46)f′uik⩽c4,i∈ω¯,0⩽k⩽N,where c2, c3, and c4 are positive constants independent of τ and h.Now, subtracting (28) from (26), we can obtain the error equation
(47)δteik+1/2−μδtδxαeik+1/2=κδxβeik+1/2+Pik+Rik,i∈ω,0⩽k⩽N−1,where
(48)Pik=fUik−fuik+12f′UikUik+1−Uik−f′uikuik+1−uik.Firstly, we establishl2-error estimation. Taking the discrete inner product of (47) with ek+1/2, we have
(49)δtek+1/2,ek+1/2−μδtδxαek+1/2,ek+1/2=κδxβek+1/2,ek+1/2+Pk,ek+1/2+Rk,ek+1/2.Now, we estimate each term in (49). The first term on LHS in (49) can be estimated as
(50)δtek+1/2,ek+1/2=12τ∥ek+1∥2−∥ek∥2.Noticing Lemma6, for the second term on the LHS in (49), we have
(51)δtδxαek+1/2,ek+1/2=−δxα/2ek+1−ekτ,δxα/2ek+1+ek2=−12τ∥δxα/2ek+1∥2−∥δxα/2ek∥2.Similarly, the first term on RHS in (49) can be obtained by
(52)δxβek+1/2,ek+1/2=−δxβ/2ek+1+ek2,δxβ/2ek+1+ek2=−14∥δxβ/2ek+1+ek∥2.According to (44)–(46), we have
(53)∣Pik∣=fUik−fuik+12f′Uik−f′uikUik+1−Uik+f′uikeik+1−eik⩽c2∣eik∣+12c3c0τ∣eik∣+c4∣eik+1−eik∣.Using the Cauchy-Schwarz inequality and Young inequality, the second term on the RHS in (49) becomes
(54)Pk,ek+1/2⩽∥Pk∥∥ek+1/2∥⩽34∥Pk∥2+13∥ek+1/2∥2⩽94c22∥ek∥2+14c32c02τ2∥ek∥2+14c42∥ek+1−ek∥2+16∥ek+1∥2+∥ek∥2⩽9c224+9c32c02τ216+98c42+16∥ek∥2+9c428+16∥ek+1∥2.The last term of RHS in (49) is estimated as
(55)Rk,ek+1/2⩽∥Rk∥∥ek+1/2∥⩽34∥Rk∥2+13∥ek+1/2∥2⩽34b−ac12τ2+h22+16∥ek+1∥2+∥ek∥2.Substituting (50)–(55) into (49), we get
(56)∥ek+1∥2−∥ek∥22τ+μ∥δxα/2ek+1∥2−∥δxα/2ek∥22τ+κ∥δxβ/2ek+1+ek∥24⩽c5∥ek+1∥2+∥ek∥2+34b−ac12τ2+h22,wherec5=9c22/4+9c32c02τ2/16+9/89/8c42+1/3.Summing fork from 0 to m, we have
(57)∥em+1∥2−∥e0∥22τ+μ∥δxα/2em+1∥2−∥δxα/2e0∥22τ+κ4∑k=0m∥δxβ/2ek+1+ek∥2⩽c5∑k=0m∥ek+1∥2+∥ek∥2+34b−ac12∑k=0mτ2+h22.Noticing thate0=0 and κ>0, we have
(58)∥em+1∥2+μ∥δxα/2em+1∥2⩽4τc5∑k=0m∥ek∥2+2τc5∥em+1∥2+3τ2b−ac12∑k=0mτ2+h22.LetFm=∥em∥2+μ∥δxα/2em∥2, we have
(59)Fm+1⩽4c5τ∑k=0mFk+2c5τFm+1+3τ2b−ac12∑k=0mτ2+h22.It implies whenτ⩽τ0=1/3c5, we have
(60)Fm+1⩽12τc5∑k=0mFk+92τb−ac12∑k=0mτ2+h22.Using Gronwall Lemma8, we have
(61)Fm+1⩽exp12c5mτ92τb−ac12∑k=0mτ2+h22.Therefore, we have(62)∥em+1∥⩽C1τ2+h2,where C1≔9b−ac12Texp12c5T/2.Similarly, applying Lemma5 yields
(63)em+1Hα/2⩽C2τ2+h2,where C2≔9b−ac12Texp12c5T/2Cαμ.Finally, we can establishl∞-error estimate by combining (62) with (63). Denoting C3≔CσC12+C22, it follows from Lemma 7 that
(64)∥em+1∥l∞⩽C3τ2+h2.We complete the proof.Next, we will analyze the stability of the scheme (28)–(30). Let vik∣0⩽i⩽M,0⩽k⩽N be the solution of the fractional Sobolev equation
(65)δtvik+1/2−μδtδxαvik+1/2=κδxβvik+1/2+fvik+12f′vikvik+1−vik,i∈ω,0⩽k⩽N−1,(66)vi0=u0xi+ϕi0,i∈ω¯,(67)vik=0,i∈∂ω,1⩽k⩽N,where ϕi0 is the perturbation of the initial value. Subtracting (65)–(67) from (28)–(30) and denoting ρik=vik−uik, we have
(68)δtρik+1/2−μδtδxαρik+1/2=κδxβρik+1/2+fvik−fuik+12f′vikvik+1−vik−f′uikuik+1−uik,i∈ω,0⩽k⩽N−1,ρi0=ϕi0,i∈ω¯,ρik=0,i∈∂ω,1⩽k⩽N.Similar to the proof of Theorem11, we have the following result.Theorem 12.
Denoteρik=vik−uik,i∈ω¯,0⩽k⩽N. Then, there exist positive constantsτ0andh0, whenτ<τ0andh<h0, we have(69)∥ρk∥⩽C4∥ρ0∥,ρkHα/2⩽C5ρ0Hα/2,∥ρk∥l∞⩽C6∥ρ0∥l∞,where C4,C5,C6>0 are positive constants independent of τ and h.


## 3.1. The Derivation of the Linearized Numerical Scheme

Take positive integersM, N and let τ=T/N, h=b−a/M be the temporal and spatial step sizes, respectively. Denote xi=a+ih, 0⩽i⩽M; tk=kτ, 0⩽k⩽N; tk+1/2=k+1/2τ, 0⩽k⩽N−1; Ωh=xi∣0⩽i⩽M, Ωτ=tk∣0⩽k⩽N. Define ω¯=j∣j=0,1,⋯,M,ω=j∣j=1,2,⋯,M−1, ∂ω=ω¯\ω. Let Vh=u∣u=uik∣0⩽i⩽M,0⩽k⩽N,u0k=uMk=0 be grid function space defined on Ωhτ=Ωh×Ωτ. Then, for a given grid function u∈Vh, we introduce the following notations:
(18)uik+1/2=12uik+1+uik,δtuik+1/2=1τuik+1−uik.Define the grid function(19)Uik=uxi,tk,i∈ω¯,0⩽k⩽N.Then, we consider (15) at the point xi,tk+1/2 and have
(20)∂tuxi,tk+1/2−μ∂xα∂tuxi,tk+1/2=κ∂xβuxi,tk+1/2+fuxi,tk+1/2,i∈ω,0⩽k⩽N−1.Utilizing the Taylor expansion, the first term on the left hand side (LHS) in (20) can be estimated as
(21)∂tuxi,tk+1/2=δtUik+1/2+Oτ2.Noticing Lemma1, for the second term on LHS in (20), we have
(22)∂xα∂tuxi,tk+1/2=−1hα∑j=0Mgi−jαδtUjk+1/2+Oτ2+h2=δtδxαUik+1/2+Oτ2+h2.For the first term on the right hand side (RHS) in (20), it yields
(23)∂xβuxi,tk+1/2=−1hβ∑j=0Mgi−jβUjk+1/2+Oτ2+h2=δxβUik+1/2+Oτ2+h2.Moreover, we have(24)uxi,tk+1/2=12Uik+1+Uik+Oτ2,uxi,tk+1−uxi,tk=Oτ⩽c0τ,where c0 is a positive constant.Applying the Newton linearized method to the nonlinear termf on RHS in (20) and using Taylor expansion at the point Uik, it yields
(25)fuxi,tk+1/2=fUik+Uik+1/2−Uikf′Uik+Oτ2=fUik+12Uik+1−Uikf′Uik+Oτ2,where f′Uik=∂UfU=Uik. Plugging (21)–(23) and substituting (25) into (20), we have
(26)δtUik+1/2−μδtδxαUik+1/2=κδxβUik+1/2+fUik+12f′UikUik+1−Uik+Rik,i∈ω,0⩽k⩽N−1.There exists a positive constantc1>0 such that
(27)Rik⩽c1τ2+h2,i∈ω,0⩽k⩽N−1.OmittingRik in (26), replacing Uik+1/2 with uik+1/2 in (26), then the finite difference scheme reads
(28)δtuik+1/2−μδtδxαuik+1/2=κδxβuik+1/2+fuik+12f′uikuik+1−uik,i∈ω,0⩽k⩽N−1,(29)ui0=u0xi,i∈ω¯,(30)uik=0,i∈∂ω,1⩽k⩽N.


## 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(31)Aα=g0αg−1α⋯g3−Mαg2−Mαg1αg0αg−1α⋯g3−Mα⋮⋱⋱⋱⋮gM−3α⋯g1αg0αg−1αgM−2αgM−3α⋯g1αg0α.It holds
(32)g0α=Γα+1Γ2α/2+1⩽0,∑j=−∞+∞gjα=0,gjα=g−jα⩽0,where for any ∣j∣⩽1, and 0<λi<2g0αi∈ω, λi is the ith the eigenvalue of matrix Aα. Aβ is given in a similar way. It implies that the matrices Aα and Aβ are real symmetric positive definite matrices.Lemma 4.
(discrete Sobolev inequality (cf. [14])) For every1/2<σ⩽1, there exists a constantCσ=Cσ>0, independent ofh>0, such that(33)∥v∥l∞⩽Cσ∥v∥Hσ.Lemma 5.
(cf. [8]). For any1<α⩽2and any grid functionv∈Vh, we have(34)CαvHα/22⩽−δxαv,v⩽vHα/22,where Cα=π/2α.Lemma 6.
(cf. [17]). For any grid functionv∈Vh, there exists a fractional symmetric positive quotient operatorδxα/2, such that(35)−δxαv,v=δxα/2v,δxα/2v.Lemma 7.
(cf. [18]) (discrete uniform Sobolev inequality). For every1/2<σ⩽1, there exists a constantCσ=Cσ>0independent ofh>0such that(36)∥u∥l∞⩽∥u∥Hσ.Lemma 8.
(cf. [19]). SupposeFkk=0∞be nonnegative sequence and satisfy(37)Fk⩽cτ∑l=0k−1Fl+g,k=0,1,2,⋯,
Then, we have(38)Fk⩽geckτ,k=0,1,2,⋯,where c and g are nonnegative constants.Theorem 9.
The linearized finite difference scheme (28)–(30) is uniquely solvable.Proof.
Denoteuk=u1k,u2k,⋯,uM−1kT. We will prove the above result by the mathematical induction. Obviously, (29) is true for k=0. Now, we suppose ul0⩽k⩽l⩽N−1 has been uniquely determined; then, we only need to prove that ul+1 is uniquely determined by (28). We can rewrite (28) in the following matrix form
(39)I+μhαAα+τκ2hβAβ−τ2diagf′ulul+1=I+μhαAα−τκ2hβAβ−τ2diagf′ulul+τful+G~l+1,where G~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 ul+1. 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.
Letux,·∈C2+αx,·be the solution of the problem (15)–(17). Then, we have(40)∥Rk∥2⩽b−ac12τ2+h22,0⩽k⩽N−1,where c1 is a positive constant independent of τ and h.Denote(41)eik=Uik−uik,i∈ω,0⩽k⩽N.We will obtain the main convergence result.Theorem 11.
Letux,·∈C2+αx,·be the solution of the problem (15)–(17). Then, there exist positive constantsτ0andh0, whenτ<τ0andh<h0, for0⩽k⩽N, we have(42)∥ek∥⩽C1τ2+h2,ekHα/2⩽C2τ2+h2,∥ek∥l∞⩽C3τ2+h2,where C1,C2,C3>0 are positive constants independent of τ and h.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:
(43)∥uk∥l∞⩽∥ek∥l∞+∥Uk∥l∞⩽C3τ2+h2+c~0⩽1+c~0,1⩽k⩽m,where τ<τ0=2C3−1/2, h<h0=2C3−1/2, and c~0=maxx,t∈Ω×0,T∣Ux,t∣.In the view of Lipschitz condition, we have(44)∣fUik−fuik∣⩽c2eik,i∈ω¯,0⩽k⩽N,(45)∣f′Uik−f′uik∣⩽c3eik,i∈ω¯,0⩽k⩽N,(46)f′uik⩽c4,i∈ω¯,0⩽k⩽N,where c2, c3, and c4 are positive constants independent of τ and h.Now, subtracting (28) from (26), we can obtain the error equation
(47)δteik+1/2−μδtδxαeik+1/2=κδxβeik+1/2+Pik+Rik,i∈ω,0⩽k⩽N−1,where
(48)Pik=fUik−fuik+12f′UikUik+1−Uik−f′uikuik+1−uik.Firstly, we establishl2-error estimation. Taking the discrete inner product of (47) with ek+1/2, we have
(49)δtek+1/2,ek+1/2−μδtδxαek+1/2,ek+1/2=κδxβek+1/2,ek+1/2+Pk,ek+1/2+Rk,ek+1/2.Now, we estimate each term in (49). The first term on LHS in (49) can be estimated as
(50)δtek+1/2,ek+1/2=12τ∥ek+1∥2−∥ek∥2.Noticing Lemma6, for the second term on the LHS in (49), we have
(51)δtδxαek+1/2,ek+1/2=−δxα/2ek+1−ekτ,δxα/2ek+1+ek2=−12τ∥δxα/2ek+1∥2−∥δxα/2ek∥2.Similarly, the first term on RHS in (49) can be obtained by
(52)δxβek+1/2,ek+1/2=−δxβ/2ek+1+ek2,δxβ/2ek+1+ek2=−14∥δxβ/2ek+1+ek∥2.According to (44)–(46), we have
(53)∣Pik∣=fUik−fuik+12f′Uik−f′uikUik+1−Uik+f′uikeik+1−eik⩽c2∣eik∣+12c3c0τ∣eik∣+c4∣eik+1−eik∣.Using the Cauchy-Schwarz inequality and Young inequality, the second term on the RHS in (49) becomes
(54)Pk,ek+1/2⩽∥Pk∥∥ek+1/2∥⩽34∥Pk∥2+13∥ek+1/2∥2⩽94c22∥ek∥2+14c32c02τ2∥ek∥2+14c42∥ek+1−ek∥2+16∥ek+1∥2+∥ek∥2⩽9c224+9c32c02τ216+98c42+16∥ek∥2+9c428+16∥ek+1∥2.The last term of RHS in (49) is estimated as
(55)Rk,ek+1/2⩽∥Rk∥∥ek+1/2∥⩽34∥Rk∥2+13∥ek+1/2∥2⩽34b−ac12τ2+h22+16∥ek+1∥2+∥ek∥2.Substituting (50)–(55) into (49), we get
(56)∥ek+1∥2−∥ek∥22τ+μ∥δxα/2ek+1∥2−∥δxα/2ek∥22τ+κ∥δxβ/2ek+1+ek∥24⩽c5∥ek+1∥2+∥ek∥2+34b−ac12τ2+h22,wherec5=9c22/4+9c32c02τ2/16+9/89/8c42+1/3.Summing fork from 0 to m, we have
(57)∥em+1∥2−∥e0∥22τ+μ∥δxα/2em+1∥2−∥δxα/2e0∥22τ+κ4∑k=0m∥δxβ/2ek+1+ek∥2⩽c5∑k=0m∥ek+1∥2+∥ek∥2+34b−ac12∑k=0mτ2+h22.Noticing thate0=0 and κ>0, we have
(58)∥em+1∥2+μ∥δxα/2em+1∥2⩽4τc5∑k=0m∥ek∥2+2τc5∥em+1∥2+3τ2b−ac12∑k=0mτ2+h22.LetFm=∥em∥2+μ∥δxα/2em∥2, we have
(59)Fm+1⩽4c5τ∑k=0mFk+2c5τFm+1+3τ2b−ac12∑k=0mτ2+h22.It implies whenτ⩽τ0=1/3c5, we have
(60)Fm+1⩽12τc5∑k=0mFk+92τb−ac12∑k=0mτ2+h22.Using Gronwall Lemma8, we have
(61)Fm+1⩽exp12c5mτ92τb−ac12∑k=0mτ2+h22.Therefore, we have(62)∥em+1∥⩽C1τ2+h2,where C1≔9b−ac12Texp12c5T/2.Similarly, applying Lemma5 yields
(63)em+1Hα/2⩽C2τ2+h2,where C2≔9b−ac12Texp12c5T/2Cαμ.Finally, we can establishl∞-error estimate by combining (62) with (63). Denoting C3≔CσC12+C22, it follows from Lemma 7 that
(64)∥em+1∥l∞⩽C3τ2+h2.We complete the proof.Next, we will analyze the stability of the scheme (28)–(30). Let vik∣0⩽i⩽M,0⩽k⩽N be the solution of the fractional Sobolev equation
(65)δtvik+1/2−μδtδxαvik+1/2=κδxβvik+1/2+fvik+12f′vikvik+1−vik,i∈ω,0⩽k⩽N−1,(66)vi0=u0xi+ϕi0,i∈ω¯,(67)vik=0,i∈∂ω,1⩽k⩽N,where ϕi0 is the perturbation of the initial value. Subtracting (65)–(67) from (28)–(30) and denoting ρik=vik−uik, we have
(68)δtρik+1/2−μδtδxαρik+1/2=κδxβρik+1/2+fvik−fuik+12f′vikvik+1−vik−f′uikuik+1−uik,i∈ω,0⩽k⩽N−1,ρi0=ϕi0,i∈ω¯,ρik=0,i∈∂ω,1⩽k⩽N.Similar to the proof of Theorem11, we have the following result.Theorem 12.
Denoteρik=vik−uik,i∈ω¯,0⩽k⩽N. Then, there exist positive constantsτ0andh0, whenτ<τ0andh<h0, we have(69)∥ρk∥⩽C4∥ρ0∥,ρkHα/2⩽C5ρ0Hα/2,∥ρk∥l∞⩽C6∥ρ0∥l∞,where C4,C5,C6>0 are positive constants independent of τ and h.


## 4. Numerical Examples

In this section, we will provide extensive numerical examples to testify the theoretical results. we will define the discretel2-norm and l∞-norm separately and the corresponding convergence orders are defined as follows:
(70)Eh,τ=h∑i=0MUiN−uiN2,E∞h,τ=max0⩽i⩽M,0⩽k⩽NUik−uik,Ord2=log2∥Eh,τ∥∥Eh/2,τ/2∥,Ord∞=log2∥Eh,τ∥∞∥Eh/2,τ/2∥∞.Example 1.
We firstly consider the following fractional Sobolev equation as(71)∂tu−μ∂xα∂tu=κ∂xβu+sinu+gx,t,x,t∈0,1×0,1,The exact solution is(72)ux,t=t3x21−x2.The initial boundary conditions andgx,t are determined by (72).Takingμ=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 l2-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.Table 1
l2- 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.
α,βh∥e∥Ord2∥e∥∞Ord∞1.2,1.81/107.5025e−4−1.0862e−3−1/201.7359e−42.11172.5686e−42.08021/404.0650e−52.07026.1167e−52.07021.5,1.51/107.7557e−4−1.1391e−3−1/201.7445e−42.15242.6326e−42.11331/403.9827e−52.13106.1202e−52.10481.8,1.21/101.1777e−3−1.6525e−3−1/202.7908e−42.07733.9673e−42.05841/406.6203e−52.07579.5326e−52.0572Table 2
l2- and l∞-errors and their convergence orders of (28)–(30) for 1<α<2 in the temporal direction for (72) with fixed spatial step h=1/2000 for Example 1.
α,βτ∥e∥Ord2∥e∥∞Ord∞1.2,1.81/101.9481e−4−3.1163e−4−1/204.8708e−51.99987.7916e−51.99981/401.2169e−52.00101.9465e−52.00101.5,1.51/101.5223e−4−2.4163e−4−1/203.8057e−52.00006.0406e−52.00011/409.5069e−62.00111.5088e−52.00131.8,1.21/101.2416e−4−1.9507e−4−1/203.1025e−52.00064.8747e−52.00061/407.7412e−62.00281.2163e−52.0028All the data are referred to MATLAB codes in Example1 in the supplementary files.Example 2.
Next, we consider the nonlinear fractional Sobolev equation as
The exact solution(73)ux,t=sint+12+x22−x2
is oscillatory along with the temporal direction, whereμ=1 and κ=1. And the initial boundary conditions and gx,t are determined by (73).In this example, we examine the spatial convergence orders with the fixed time stepτ=1/2000 and the temporal convergence orders with the fixed spatial step h=1/1000 in l2-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(74)∂tu−∂xα∂tu=∂xβu+u2−u4+gx,t,x,t∈−1,1×0,1,
We choose the exact solution(75)ux,t=t+t31+x21−x2.
The initial boundary conditions andgx,t are determined by (75).Table 3
l2-and l∞-errors and their convergence orders of (28)–(30) for 1<α<2 in the spatial direction for (73) with τ=1/2000 for Example 2.
α,βh∥e∥Ord2∥e∥∞Ord∞1.1,1.91/1001.3736e−4−2.9683e−4−1/2004.9959e−51.45919.9718e−51.57371/4001.6497e−51.59853.1155e−51.67841.5,1.51/1004.6568e−4−6.5750e−4−1/2001.3983e−41.73572.0089e−41.71061/4004.0066e−51.80325.8340e−51.78391.9,1.11/1001.7483e−4−2.1547e−4−1/2004.6475e−51.91155.7682e−51.90131/4001.2052e−51.94711.5024e−51.9408Table 4
l2-and l∞-errors and their convergence orders of (28)–(30) for 1<α<2 in the temporal direction for (73) with h=1/1000 for Example 2.
α,βτ∥e∥Ord2∥e∥∞Ord∞1.1,1.91/1002.0697e−2−5.6643e−2−1/2008.1358e−31.34711.3366e−22.08331/4002.0509e−31.98802.9698e−32.17011.5,1.51/1002.0526e−2−5.6522e−2−1/2008.1446e−31.33361.3303e−22.08711/4002.0523e−31.98862.9423e−32.17671.9,1.11/1002.0734e−2−5.6310e−2−1/2008.1558e−31.34611.3202e−22.09271/4002.0536e−31.98972.9033e−32.1850Similar to above example, Tables5 and 6 list the l2-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 results show that (28)–(30) is close to second-order accurate in spatial and temporal directions.Table 5
l2-and l∞-errors and their convergence orders of (28)–(30) for 1<α<2 in the spatial direction for (75) with τ=1/1000 for Example 3.
α,βh∥e∥Ord2∥e∥∞Ord∞1.3,1.71/101.4761e−2−1.6303e−2−1/203.3659e−32.13273.8460e−32.08371/408.0373e−42.06629.0995e−42.07951.5,1.51/101.6167e−2−1.8043e−2−1/203.6642e−32.14144.2293e−32.09301/408.7053e−42.07359.9262e−42.09111.7,1.31/102.0933e−2−2.2257e−2−1/204.8190e−32.11905.3128e−32.06671/401.1227e−32.10181.2648e−32.0705Table 6
l2-and l∞-errors and their convergence orders of (28)–(30) for 1<α<2 in the temporal direction for (75) with h=1/1000 for Example 3.
α,βτ∥e∥Ord2∥e∥∞Ord∞1.3,1.71/109.1725e−3−1.0470e−2−1/202.5218e−31.86292.8729e−31.86571/406.6380e−41.92567.5586e−41.92631.5,1.51/108.7813e−3−9.8734e−3−1/202.4228e−31.85782.7208e−31.85951/406.3844e−41.92407.1689e−41.92421.7,1.31/108.2127e−3−9.1056e−3−1/202.2771e−31.85072.5228e−31.85171/406.0134e−41.92096.6620e−41.9210All the data are referred to MATLAB codes in Example3 in the supplementary files.In the following model, the exact solution is unknown, we test convergence orders using the posterior error estimation(76)Ord2=log2∥uh,τ−uh,τ/2∥∥uh,τ/2−uh,τ/2∥,Ord∞=log2∥uh,τ−uh,τ/2∥∞∥uh,τ/2−uh,τ/4∥∞.Example 4.
We consider the following equation:(77)∂tu−∂xα∂tu=∂xβu+u−u2,x,t∈−25,25×0,0.1,ux,0=2sechx+5cos4/x,x∈−25,25,u−25,t=u25,t=0,t∈0,0.1,with the exact solution is unknown.In the computation, we take different spatial and temporal step sizes. Thel2-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 ux,t 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.Table 7
l2-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.
α,βh∥e∥Ord2∥e∥∞Ord∞1.2,1.81/401.7548e−2−1.1092e−2−1/803.9216e−32.16182.3542e−32.23621/1609.5790e−42.05585.6623e−42.05581.5,1.51/401.2035e−2−7.0740e−3−1/802.5833e−32.21991.4349e−32.30161/1606.2934e−42.03733.4473e−42.05741.8,1.21/407.8751e−3−3.5547e−3−1/801.6022e−32.29728.7242e−42.02661/1603.9110e−42.03452.1876e−41.99572.0,2.01/401.4624e−2−8.1939e−3−1/803.1088e−32.23391.5922e−32.36351/1607.5817e−42.03583.8083e−42.0638Table 8
l2-and l∞-errors and their convergence orders of (28)–(30) for 1<α⩽2 in the temporal direction with h=1/1000 for Example 4.
α,βτ∥e∥Ord2∥e∥∞Ord∞1.2,1.81/409.2300e−9−5.5579e−9−1/802.3078e−91.99981.3897e−91.99981/1605.7564e−102.00333.4662e−102.00341.5,1.51/407.5499e−9—3.2739e−9−1/801.8868e−92.00058.1803e−102.00081/1604.7057e−102.00342.0378e−102.00511.8,1.21/406.7080e−9−2.7409e−9−1/801.6769e−92.00016.8522e−102.00001/1604.1960e−101.99871.7116e−102.00122.0,2.01/409.2576e−9−4.4312e−9−1/802.3143e−92.00011.1077e−92.00011/1605.7857e−102.00002.7694e−102.0000Figure 1
Using scheme (28)–(30), curves of ux,t with respect to x at different time with h=0.5, τ=0.02 (a) and the evolutionary surfaces of ux,t at T=0.1 (b), T=1 (c), and T=2 (d) in Example 4.
(a)(b)(c)(d)


## 5. 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 orderOτ2+h2. 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.



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