Linear θ-Method and Compact θ-Method for Generalised Reaction-Diffusion Equation with Delay

Copyright © 2018 Fengyan Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper is concerned with the analysis of the linear θ-method and compact θ-method for solving delay reaction-diffusion equation. Solvability, consistence, stability, and convergence of the two methods are studied. When θ ∈ [0, 1/2), sufficient and necessary conditions are given to show that the two methods are asymptotically stable. When θ ∈ [1/2, 1], the two methods are proven to be unconditionally asymptotically stable. Finally, several examples are carried out to confirm the theoretical results.


Introduction
Partial functional differential equations (PFDEs) are widely used to model many natural phenomena in various scientific fields [1][2][3][4][5][6][7][8][9].In order to gain a better understanding of the complicated dynamics, numerous researchers have investigated PFDEs.For instance, Garrido-Atienza and Real discussed the existence and uniqueness of solutions for delay evolution equations [10].Mei et al. analysed the stability of travelling waves for nonlocal time-delayed reactiondiffusion equations [11].Polyanin and Zhurov constructed exact solutions for delay reaction-diffusion equations and more complex nonlinear equations by the functional constraints method [12].
However, the exact solutions are difficult to be obtained [1].Most researchers have to seek efficient and effective numerical methods to numerically solve PFDEs.Jackiewicz and Zubik-Kowal utilised spectral collocation and waveform relaxation methods to study nonlinear delay partial differential equations [13].Chen and Wang utilised the variational iteration method to solve a neutral functionaldifferential equation with proportional delays [14].Li et al. used the discontinuous Galerkin methods to solve the delay differential equations [15][16][17].Bhrawy et al. applied an accurate Chebyshev pseudospectral scheme to study the multidimensional parabolic problems with time delays [18].
Aziz and Amin employed the Haar wavelet to study the numerical solution of a class of delay differential and delay partial differential equations [19].
When it comes to solving PFDEs numerically, here comes one question, that is, whether the numerical solution approximates the exact solution in a stable manner, especially for a long time.In this study, we use the following model as the test equation for analysing stability of the numerical method, which is an extension to the previous work [20][21][22].
For the case where  3 =  4 = 0, model (1) has been studied by many researchers [2,[20][21][22][23][24][25][26][27].In this work, we will examine the case where  1 > 0,  2 > 0,  3 ∈ R, and  4 ∈ R, which is a generalisation of above mentioned work, International Journal of Differential Equations and analyse the stability condition of the numerical method.The standard second-order central difference method and compact finite difference method are utilised to discrete the diffusion operator, respectively, and the linear -method is utilised to discrete the temporal direction.For convenience, we name the standard second-order central difference version as linear -method and the compact finite difference method version as compact -method.With the spectral radius condition, we consider the stability of the linear -method and compact -method, respectively.
The rest of this paper is organized as follows.In Section 2, we give a sufficient delay-independent condition for Problem (1) to be asymptotically stable.In Section 3, we propose the linear -method for solving Problem (1); solvability, stability, and convergence of the method are discussed.In Section 4, we extend the compact -method to solve Problem (1).In Section 5, several numerical tests are performed to validate the theoretical results.

Stability of PFDE (1)
In this section, based on Tian's work [20], we give a sufficient condition for the trivial solution of Problem (1) to be asymptotically stable.

Theorem 3. Assume that the solution of Problem
and  ≥ 0.
Then the sufficient condition for the trivial solution of Problem (1) to be asymptotically stable is that where Proof.Let  = [0, ] denote the Banach space equipped with the maximum norm, and (A) = { ∈  :   ∈ , (0) = () = 0}, and A =   for  ∈ (A).
Let  1  2 ( = 1, 2, ⋅ ⋅ ⋅ ) be the eigenvalues of −A.According to [1,28], if all zeros of the characteristic equations have negative real part, then the trivial solution is asymptotically stable.Meanwhile, if at least one zero has positive real part, then it is unstable.Let () = 0; that is, Multiplying by   , we have Setting  = , we get Denote  = ( 3 −  1  2 ), and  = ( 4 −  2  2 ), and rewrite the above equation as Applying Lemma 2, if then the real parts of all zeros of the characteristic equations are negative.Therefore, the trivial solution of Problem (1) is asymptotically stable.Otherwise, there exists a zero  0 whose real part is positive such that ( 0 ) = 0. Hence, the trivial solution is unstable.It completes the proof.

Linear 𝜃-Method
In this section, the linear -method is presented to solve Problem (1).

Solvability of Linear 𝜃-Method
Theorem 4. The linear -method (8) is solvable and has a unique solution.
Proof.The mathematical induction is utilised to prove it.We can obtain the solution of  1 according to the initial condition.Now, assume that the solution of   has been determined.Then we can derive the solution of  +1 with (9).It follows from ( 9) that the coefficient matrix of the linear system is It is easy to verify that the matrix  0 () is symmetric positive definite.Therefore, the solution of  +1 is determined uniquely.By mathematical induction, the existence and uniqueness of the solution of difference system (8) are obtained immediately.

Asymptotic Stability of
In order to prove that a polynomial is a Schur polynomial, the following lemma is needed.
Lemma 6 (cf.[29]).Let   () = ()  − () be a polynomial, where () and () are polynomials of constant degree.Then, the polynomial   () is a Schur polynomial for any  ≥ 1 if and only if the following conditions hold: Taking the analytical technique in [20][21][22], we know that the linear -method (8) is asymptotically stable about the trivial solution if and only if is a Schur polynomial for any  ≥ 1.

Basic calculations give
where With the help of Lemma 6, we obtain the following theorem when  ∈ [0,1/2), which offers a sufficient and necessary condition of asymptotic stability for the linear method.
and then, after some basic calculations, we arrive at This signifies that condition () of Lemma 6 does not hold.Therefore, the linear -method (8) is not asymptotically stable.
Then, we know that ( 16) is a necessary condition for asymptotic stability.This completes the proof.
Remark 8.When  3 =  4 = 0, the sufficient and necessary condition (16) in Theorem 7 is simplified to which is consistent with the previous work [20].
Next, when  ∈ [1/2, 1], we will prove that the linear -method (8) is unconditionally asymptotically stable with respect to the trivial solution.Proof.We will prove the theorem with Lemma 6.First, it follows from   () = 0 that Similar to the proof of Theorem 7, we get || < 1.
Then, we check items (ii) and (iii) of Lemma 6.To do that, we introduce the following complex variable function: According to Lemma 6, we conclude that the linear method ( 8) is asymptotically stable about the trivial solution.This completes the proof of the theorem.
Theorem 10.Assume that the assumptions in Theorems 7 and 9 hold.Then, for  = 1, 2, ⋅ ⋅ ⋅ , we have the following convergent result: where Proof.It follows from Theorem 4 that difference system (8) is solvable and has a unique solution.Moreover, the assumptions in Theorems 7 and 9 hold, signifying that the method is stable.Together with the consistence of the method, we derive that (33) holds by the Lax equivalence theorem [30,31].

Extension to Compact 𝜃-Method
In this section, we would like to use the compact -method with a higher convergence order in space to extend our work.We introduce the compact difference operator, and an important lemma below, which will be needed to construct and prove our main results.
Lemma 11 (cf.[32]).Assume that V() where   ∈ ( −1 ,  +1 ).Now, applying the compact difference operator (34) to discrete the diffusion operator, we have the compact method: Unconditionally stable 4 Linear -method for problem (1) Unconditionally stable 2 Compact -method for problem (1) Unconditionally stable 4 The compact -method (36) can be rewritten in the following matrix form: where Similarly, the solvability, asymptotic stability, and convergence of the compact -method (36) can also be obtained.For conciseness, we merely list our main results and omit the details.
Remark 13.When  3 =  4 = 0, the sufficient and necessary condition (39) in Theorem 12 is reduced to which is consistent with the previous work [22].
where C is a constant that is independent of temporal and spatial stepsizes.
Remark 16.The comparison of linear -method and compact -method applied to problem in [20] and Problem (1) is presented in Table 1.

5.1.1.
Linear -Method.First, to verify the effectiveness of the sufficient and necessary condition (16) of the linear method, here we choose the case where  = 0 to illustrate that.Noting that Δ = /, where  > 0 is an integer, and substituting parameters  = 0,  = 1, Δ = /10,  1 = 1.0,  2 = 0.5,  3 = −1.0,and  4 = −0.5 into condition (16), we derive that the proposed method is asymptotically stable if  > 30.4025.In other words, if  ≥ 31, then the method is asymptotically stable.Meanwhile, if  ≤ 30, then the method is not asymptotically stable.In Figure 1, we can get a pictorial understanding of that.
Next, when  = 1/2 or 1, we apply the linear method and use different stepsizes to solve problem (42).Theoretically, the numerical solution is asymptotically stable by Theorem 9. Numerically, we know that the numerical solution is asymptotically stable from the plots of Figure 3, which is consistent with the theoretical result.

5.1.2.
Compact -Method.First, we choose the case  = 0 to verify the effectiveness of the sufficient and necessary condition (39) of the compact -method.Under the case that  = 0,  = 1, Δ = /10,  1 =  3 = 1.0, and  2 =  4 = 0.5, and noting that Δ = /, by condition (39), the proposed method is asymptotically stable if and only if  > 45.052.In other words, if  ≥ 46, then the proposed method is asymptotically stable.Meanwhile, if  ≤ 45, then the proposed method is not asymptotically stable.In order to validate it, we give a pictorial understanding of that in Figure 4.
From Figures 4(a) and 4(b), it is easily seen that the numerical solution is unstable as time goes on for  = 40 and 45.The numerical solution is asymptotically stable for  = 46 and 50 in Figures 4(c) and 4(d), respectively.Furthermore, denote the left-hand side of (39) as LHS = 1/6+(1−2)[+− (+)/2(1+cos(Δ))], and denote the right-hand side of (39) as RHS = 1/(1 + cos(Δ)), and Ind = LHS − RHS.According to the analysis of above paragraph, we know that Ind > 0 for  ≤ 45, and Ind < 0 for  ≥ 46.The schematic presentation of the relationship between Ind and positive integer  is given in Figures 5(a) and 5(b).All the numerical results agree well with the findings in Theorem 12.
Then, for  = 1/2 or 1, we apply the compact method and choose different stepsizes to solve problem (42).The numerical results are shown in Figure 6.Theoretically, according to Theorem 14, the numerical solution is asymptotically stable.Numerically, from these figures, we know that the numerical solution is asymptotically stable, which confirms the theoretical result.Here, the added term ℎ(, ) and the initial condition  0 (, ) are specified so that the exact solution is (, ) = e − sin().
For  = 0, we let Δ = 1 − 5 to guarantee that the linear -method (8) is asymptotically stable.When the compact method (36) is applied to solve problem (43), we let Δ ≈ Δ 4 .The numerical errors and corresponding orders in different sense of norms are displayed in Table 2. Clearly, these results confirm the convergence of the two methods.
For  = 1/2, when the linear -method (8) is applied to solve problem (43), we let Δ ≈ Δ, and for the compact method (36), we let Δ ≈ Δ 2 .For  = 1, when method (8) is applied to solve problem (43), we let Δ ≈ Δ 2 , and for method (36), we let Δ ≈ Δ 4 .The numerical errors and corresponding orders in different sense of norms are listed in Tables 3 and 4, respectively.It is readily found that these results confirm the convergence of the two methods.Obviously, the compact -method gives a better convergence result in the space.
Linear -Method.Section 2 gives the sufficient condition for the trivial solution of Problem (1) to be asymptotically stable.Next, we will analyse the numerical stability of the linear -method (8) under this condition.Definition 5.A numerical method applied to Problem (1) is called asymptotically stable about the trivial solution if its approximate solution    corresponding to a sufficiently differentiable function  0 (, ) with  0 −  that (2 −   ) −  > (2 −   ) −  > 0.Then, for all  ∈ C, || = 1, we find that             ()

Table 1 :
Stability and convergence order of different methods.