# High-Order Iterative Scheme for a Viscoelastic Wave Equation and Numerical Results

Authors: Doan Thi Nhu Quynh; Bui Duc Nam; Le Thi Mai Thanh; Tran Trinh Manh Dung; Nguyen Huu Nhan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we consider a Robin problem for a viscoelastic wave equation. First, by the high-order iterative method coupled with the Galerkin method, the existence of a recurrent sequence via anN-order iterative scheme is established, and then the N-order convergent rate of the obtained sequence to the unique weak solution of the proposed model is also proved. Next, with N=2, a numerical algorithm given by the finite-difference method is constructed to approximate the solution via the 2-order iterative scheme. Moreover, the same algorithm for the single-iterative scheme generated by the 2-order iterative scheme is also considered. Finally, comparison with errors of the numerical solutions obtained by the single-iterative scheme and the 2-order iterative scheme shows that the convergent rate of the 2-order iterative scheme is faster than that of the single-iterative scheme.


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## Body

## 1. Introduction

In this paper, we consider the following initial boundary value problem for a viscoelastic wave equation with nonlinear damping:(1)utt−uxx+λutq−2ut+∫0tgt−suxxx,sds=fx,t,u,0<x<1,0<t<T,(2)ux0,t−h0u0,t=ux1,t+h1u1,t=0,(3)ux,0=u˜0x,utx,0=u˜1x,where λ>0,q≥2,h0≥0,andh1≥0 are constants, with h0+h1>0, and f,g, u˜0, and u˜1 are given functions.Equation (1) arises naturally within frameworks of mathematical models in engineering and physical sciences. The left-hand integral of equation (1) stands for the characters of viscoelastic materials. Many researchers have paid attention to viscoelastic materials for a quite long time, especially in the last two decades, and have made a lot of progress, taking into account viscoelastic fluid, which achieved major attention due to its application in different physiological and industrial processes. In the same content, nanofluid has become an interesting objective which describes various phenomena such as electrical conductivity, especially in the bubble electrospinning [1–3], heat transfer on solid particle motion [4], biologically inspired peristaltic transport [5], and rheology controlled by the concentration of the added particles (such as SiC) [6]. In addition to studying the specific properties of viscoelastic materials, numerous researchers have considered the extensions of the mathematical model for viscoelastic problems and have obtained many interesting properties of solutions such as global existence, decay, and blow-up result. One of the problems similar to problems (1)–(3) was considered by Messaoudi [7], in which the blow-up result of solutions with negative initial energy was established. After that, Li and He [8] proved, under suitable conditions, the global existence and the general decay of solutions for the same model. Recently, the results obtained in [8] have also been investigated by Mezouar and Boulaaras [9, 10] for the proposed nonlocal viscoelastic problems.Consider a recurrent sequenceum associated with equation (1) and defined by(4)u0≡0,um″−Δum+λum′q−2um′+∫0tgt−sΔumx,sds=∑i=0N−11i!D3ifx,t,um−1um−um−1i,0<x<1,0<t<T, with um satisfying (2) and (3). If the sequence um converges to the weak solution u of problems (1)–(3) and satisfies an estimate of N order in the form of um−uX≤Cum−1−uXN, for some C>0, all integer numbers N≥2, and X is a certain Banach space, such a method for finding the solution of problems (1)–(3) is called high-order iterative method. The original idea of this method is based on investigating the recurrent relations of a Newton-like method in Banach spaces [11]. After that, this approach was also applied successfully to [12–17]. In [17], Truong et al. considered the nonlinear wave equation of Kirchhoff-Carrier type as follows:(5)utt−μt,ut2,uxt2uxx=fu,0<x<1,0<t<T,ux0,t−hu0,t=gt,u1,t=0,ux,0=u˜0x,utx,0=u˜1x,where μ,f,u˜0, and u˜1 are given functions, h≥0 is a given constant, and μt,ut2,uxt2 depends on the integrals ut2=∫01u2x,tdx and uxt2=∫01ux2x,tdx. The authors associated the first equation in (5) with a recurrent sequence um defined by(6)∂2um∂t2−μt,umt2,umxt2umxx=∑i=0N−11i!∂if∂uium−1um−um−1i,0<x<1,0<t<T,where um satisfies second and third equations in (5), and proved that um converges to the unique weak solution of problem (5). Moreover, with N=3, the 3-order iterative scheme was established, and some numerical results of finite-difference approximate solutions were presented. In [15], the authors investigated the Dirichlet problem for a wave equation with linear damping and nonlinear integral as follows:(7)utt−∂∂xμx,tux+λut=fx,t,u+∫0tgx,t,s,ux,sds,0<x<1,0<t<T,u0,t=u1,t=0,ux,0=u˜0x,utx,0=u˜1x,where μ, f, g, u˜0, and u˜1 are given functions and λ≠0 is a given constant. If μ∈C10,1×ℝ+, f∈CN0,1×ℝ+×ℝ, and g∈CN−10,1×Δ×ℝ, with Δ=t,s∈ℝ+2:s≤t, by the high-order iterative method coupled with the Galerkin method, the existence of a recurrent sequence um associated with equation (7) and defined by(8)u0≡0,∂2um∂t2−∂∂xμx,t∂um∂x+λ∂um∂t=∑i=0N−11k!∂kf∂ukx,t,um−1um−um−1k+∑k=0N−11k!∫0t∂kg∂ukx,t,s,um−1x,sumx,s−um−1x,skds,0<x<1, 0<t<T, with um satisfying second and third equations in (7), was proved, and um convergence with the N-order rate to the unique weak solution of problem (7) was also confirmed. Some other iteration methods can be widely used to solve various nonlinear problems, for example, the variational iteration method and the homotopy perturbation method. Based on the basic idea of the enhanced perturbation method and the homotopy technology, Li and He [18] constructed a modified homotopy perturbation method, making a very high accuracy of the obtained approximate solution. Recently, Ji et al. [19] successfully applied Li and He’s modified homotopy perturbation method coupled with the energy method for the dropping shock response of a tangent nonlinear packaging system.Wheng=0andf≡bup−2u, equation (1) is reduced to the following nonlinear wave equation:(9)utt−Δu+autq−2ut=bup−2u,where a,b>0 and p,q≥2, which has been extensively studied and obtained many interesting results such as global existence, exponential decay, and finite-time blow-up result. If a=0, it is well known that the source term bup−2u causes a finite-time blow-up phenomenon of the solution with negative initial energy; see [20–23]. If b=0, the damped term autq−2ut assures the global existence for suitably initial data; see [24–29]. The interactions between the damping and the source terms give the results that the solutions with any initial data continue to exist globally in time if q≥p and blow up in finite time if q<p and the initial energy is sufficiently negative.In the presence of the viscoelastic termg≠0 and a=b=1 in (9), Messaoudi [7] considered a Dirichlet problem for a nonlinear wave equation as follows:(10)utt−Δu+∫0tgt−sΔux,sds+utm−2ut=up−2u,inΩ×0,∞,ux,t=0,x∈∂Ω,t≥0,ux,0=u0x,utx,0=u1x,x∈Ω,where Ω is a bounded domain of ℝn (n≥1) with a smooth boundary ∂Ω, p>2, m≥1, and g:ℝ+⟶ℝ+ is a positive nonincreasing function. With suitable conditions on g, he proved that the solutions with initial negative energy blow up in finite time if p>m and continue to exist if p≤m satisfied the condition(11)maxm,p≤2n−1n−2,withn≥3.Later, in case ofm=2 and x∈ℝn, Kafini and Messaoudi [30] also established the finite-time blow-up result with suitable conditions on the initial data and the relaxation function g. In the presence of the strong damping −Δut and the linear damping utm=2, Li and He [8] proved the global existence of solutions and established a general decay rate estimate for problem (10). Moreover, they showed the finite-time blow-up results of solutions with both negative initial energy and positive initial energy.Although there are many studies of solution properties of viscoelastic problems, however, it seems that few works related to numerical algorithms for this type were published. In [31], Long et al. proved the global existence and exponential decay of equation (1) associated with a mixed nonhomogeneous condition(12)ux0,t=u0,t,ux1,t+ηu1,t=ht,and initial condition (3). With q=4 and f=u2+Fx,t, the derivatives were first approximated by finite-difference schemes. Then, a linear recursive scheme generated by the nonlinear difference equation was constructed. Finally, the exact solution and the approximate solution were illustrated numerically. In [14], Ngoc et al. also obtained the same results given in [31] for the following nonlinear wave equation associated with nonlocal boundary conditions:(13)utt−uxx+u+λut=aup−2u+fx,t,0<x<1,t>0,ux0,t=g0t−∫0tH0t−su0,sds+∫0tk0x,tux,tdx,−ux1,t=g1t−∫0tH1t−su1,sds+∫01k1x,tux,tdx,ux,0=u˜0x,utx,0=u˜1x,0<x<1,where a=0, λ>0, and p>2 are given constants and u˜0, u˜1, f, gi, ki, and Hii=0,1 are given functions satisfying conditions specified later.In some recent literature studies, various difference methods have been applied to studying the consistency, stability, efficiency, and convergence of the proposed schemes such as Boulaaras [32] used finite element methods to prove the existence and uniqueness of the discrete solution for an evolutionary implicit 2-sided obstacle problem. Boulaaras and Haiour [33] analysed the convergence and regularity of the proposed algorithm via the finite element methods coupled with a theta time discretization scheme for evolutionary Hamilton–Jacobi–Bellman equations. Mohanty and Gopal [34] used a cubic spline finite difference method of Numerov type with an accuracy of order two in time and four in space directions for the solution of the telegraphic equation with the forcing function:(14)utt+2αut+β2u=uxx+fx,t,0<x<1,t>0,where α>0andβ≥0 are real parameters. In [35], Alsuyuti et al. applied a new modified Galerkin algorithm based on the shifted Jacobi polynomials to solve fractional differential equations and a system of fractional differential equations governed by homogeneous and nonhomogeneous initial and boundary conditions. The new algorithm was also applied for fractional partial differential equations with Robin boundary conditions and the time-fractional telegraph equation. In addition, some illustrative examples were presented to confirm the validity and efficiency of the proposed algorithm. Recently, capability of the meshless methods has been proved by using them for many different problems; see [36, 37] and the references therein. In [36], Oruç used two meshless methods based on the local radial basis function and barycentric rational interpolation for solving a two-dimensional viscoelastic wave equation. The stability of the methods, the accuracy, and the computational cost were considered. From the comparisons, it was shown that the performance of the barycentric rational interpolation in the sense of accuracy is slightly better than the performance of the local radial basis function; however, the computational cost of the local radial basis function is less than the computational cost of the barycentric rational interpolation. Before, a local meshless method was proposed in [37] for convection-dominated steady and unsteady partial differential equations in which numerical results have confirmed that the new approach is accurate and efficient for solving a wide class of one- and two-dimensional convection-dominated problems having sharp corners and jump discontinuities.The first goal of our present paper is devoted to studying the existence and theN-order convergence of the high-order iterative scheme defined by (4). In case N=2, λ=1, and q=2, the second goal is mentioned to building a numerical algorithm in order to approximate the successive solutions of the 2-order iterative scheme as follows:(15)u0≡0,um″t+um′t−Δumt+∫0tgt−sΔumsds−bmx,tumx,t=Fmx,t,0<x<1,0<t<T,umx0,t−um0,t=umx1,t+um1,t=0,umx,0=u˜0x,um′x,0=u˜1x,where(16)Fmx,t=fx,t,um−1x,t−bmx,tum−1x,t,bmx,t=D3fx,t,um−1x,t.Furthermore, in a specific case of (15) with bmx,t=0, the corresponding scheme called the single-iterative scheme is considered. Then, the numerical algorithms of both schemes are established, and the results of errors are presented to compare their convergent rates also.For the first purpose, by using the high-order iterative method coupled with the Galerkin method, we shall prove the existence of a recurrent sequenceum associated with equation (1) and defined by (4), and then um convergence with the N-order rate to the unique weak solution of problems (1)–(3) will also be claimed. For the second purpose, first, we shall use the uniform spatial partition xi=ih, h=1/N, i=0,1,…, N, and the forward difference formulas (see [38], pages 36 and 43) to approximate the kth derivatives. Then, problems (15) and (16) will be changed into a system of second-order integrodifferential equations (in the time variable) of the unknown functions uimt=umxi,t, i=0,1,…, N. Normally, this system will be converted into a system of 2N first-order integrodifferential equations by using the auxiliary functions v˙imt=uimt; see the same transformations in [14, 17, 23, 26, 29, 31]; however, these transformations will increase computations. To reduce the computations, the above second-order integrodifferential system will be transformed into a system of N first-order integrodifferential equations by integrating in time on interval 0,t. After that, to approximate double integrals, the trapezoidal formula will be successively used twice. It can be said with much confidence that this technique has never been used before. Next, by using uniform partition tj=jΔt, Δt=T/M, j=0,1,…, M, for discretization in time variable t, we will obtain an algorithm to determine the finite-difference approximate solutions of um via 2-order iterative schemes (15) and (16) given by the following difference equation (formula (149) below):(17)u⟶j+1m=u⟶jm+ΔtΨj∗,m,where Ψj∗,m is defined by (147) and (148). Finally, we will construct the algorithm to find the finite-difference approximate solutions of um given by the single-iterative scheme (formulas (157)–(159) below) and present a numerical example to compare convergent rates of two schemes. The errors of computations of the numerical solutions given by two schemes show that the convergent rate of the 2-order iterative scheme is faster than that of the single-iterative scheme.Our paper is organized as follows. In Section2, we introduce some notations and modified lemmas. In Section 3, by using the Galerkin method and standard arguments of compactness, we prove the existence and convergence of the high-order sequence defined by (4). In Section 4, by using the finite-difference method and some new techniques to reduce computations and to approximate a double integral, we construct a numerical algorithm to determine the finite-difference approximate solutions of um via 2-order iterative schemes (15) and (16). Moreover, a concrete example is numerically illustrated to compare the convergent rate of the single-iterative scheme with that of the 2-order iterative scheme. In Section 5, we summarize the main outcomes of our paper.


## 2. Preliminaries

First, we putΩ=0,1 and denote the usual function spaces used in this paper by the notations Lp=LpΩ and Hm=HmΩ. Let ⋅,⋅ be either the scalar product in L2 or the dual pairing of a continuous linear functional and an element of a function space. The notation ⋅ stands for the norm in L2, ⋅X is the norm in the Banach space X, and X′ is the dual space of X.We denote byLp0,T;X, 1≤p≤∞, for the Banach space of real functions u:0,T⟶X measurable such that(18)uLp0,T;X=∫0TutXpdt1/p<∞for1≤p<∞,uL∞0,T;X=esssup0<t<TutXforp=∞.Withf∈Ck0,1×ℝ+×ℝ, f=fx,t,u, we put D1f=∂f/∂x, D2f=∂f/∂t, D3f=∂f/∂u, and Dαf=D1α1D2α2D3α3f; α=α1,α2,α3∈ℤ+3, α=α1+α2+α3≤k, and D0,0,0f=D0f=f.Next, we put(19)au,v=∫01uxxvxxdx+h0u0v0+h1u1v1,for allu,v∈H1,va=av,v,for allv∈H1,(20)vi=v2i+∫01vx2xdx1/2,i=0,1.OnH1, three norms vH1,va, and vi are equivalent norms.The weak solution of problems (1)–(3) can be defined as follows. A function u=ux,t is a weak solution of problems (1)–(3) if(21)u∈L∞0,T;H2,u′∈L∞0,T;H1,u″∈L∞0,T;L2,and u satisfies the following variational equation:(22)u″t,w+aut,w+λu′tq−2u′t,w=∫0tgt−saus,wds+fx,t,u,w,for all w∈H1, and a.e., t∈0,T, together with the initial conditions(23)u0=u˜0,u′0=u˜1,where a⋅,⋅ is the symmetric bilinear form on H1 defined by (19).We now have the following lemmas, the proofs of which are straightforward, so we omit the details.Lemma 1.
The imbeddingH1↪C0Ω¯ is compact, and(24)ivC0Ω¯≤2vH1,iivC0Ω¯≤2vi,iii13vH1≤vi≤3vH1,for all v∈H1, i=0,1.Lemma 2.
Leth0≥0 and h1≥0 with h0+h1>0. Then, the symmetric bilinear form a⋅,⋅ defined by (19) is continuous on H1×H1 and coercive on H1, i.e.,(25)iau,v≤a1uH1vH1,for allu,v∈H1,iiav,v≥a0vH12,for allv∈H1,where a0=1/3min1,maxh0,h1 and a1=1+2h0+h1.Lemma 3.
There exists the Hilbert orthonormal basewj of L2 consisting of the eigenfunctions wj corresponding to the eigenvalue λj such that(26)0<λ1≤λ2≤⋯≤λj≤⋯,limj⟶+∞λj=+∞,awj,v=λjwj,vfor allv∈H1,j=1,2,….Furthermore, the sequencewj/λj is also a Hilbert orthonormal base of H1 with respect to the scalar product a⋅,⋅.On the contrary, we havewj satisfying the following boundary value problem:(27)−Δwj=λjwj,in0,1,wjx0−h0wj0=wjx1+h1wj1=0,wj∈C∞Ω¯.The proof of Lemma3 can be found in Theorem 7.7 of [39], p.87, with H=L2 and a⋅,⋅ as defined by (19).


## 3. The High-Order Iterative Method

First, we make the following assumptions:(28)H1u˜0,u˜1∈H2×H1,H2g∈H10,T∗,H3f∈C00,1×ℝ+×ℝsuch thatiD3if∈C00,1×ℝ+×ℝ,1≤i≤N,iiD1D3if∈C00,1×ℝ+×ℝ,0≤i≤N−1.FixT∗>0. For each M>0 given, we set the constant KMf as follows:(29)KMf=∑i=0ND3ifC0ΩM+∑i=1N−1D1D3ifC0ΩM,fC0ΩM=supfx,t,u:x,t,u∈ΩM,ΩM=0,1×0,T∗×−2M,2M.For everyT∈0,T∗ andM>0, we put(30)WM,T=v∈L∞0,T;H2:v′∈L∞0,T;H1,v″∈L2QT,withvL∞0,T;H2,v′L∞0,T;H1,v″L2QT≤M,W1M,T=v∈WM,T:v″∈L∞0,T;L2.Now, we establish a recurrent sequenceum in which the first term is chosen as u0≡0, and suppose that(31)um−1∈W1M,T.Themth iteration um can be defined by the following problem: find um∈W1M,Tm≥1 satisfying the nonlinear variational problem(32)um″t,w+aumt,w+λum′tq−2um′t,w=∫0tgt−saums,wds+Fmt,w,∀w∈H1,um0=u˜0,um′0=u˜1,where(33)Fmx,t=∑i=0N−11i!D3ifx,t,um−1um−um−1i.The existence of the above sequenceum is claimed by the following theorem.Theorem 1.
LetH1−H3 hold. Then, there exist a constant M>0 depending on u˜0, u˜1, h0, and h1 and a constant T>0 depending on u˜0, u˜1, g, f, h0, h1, q, and λ such that, for u0≡0, there exists a recurrent sequence um⊂W1M,T defined by (32) and (33).Proof.
The proof of Theorem1 consists of three steps.Step 1.
(Faedo–Galerkin approximation). Letwj be a basis ofH1 as in Lemma 3; we find an approximate solution of problems (32) and (33) in the form of(34)umkt=∑j=1kcmjktwj,where the coefficients cmjk satisfy the following system of nonlinear differential equations:(35)u¨mkt,wj+aumkt,wj+λu˙mktq−2u˙mkt,wj=∫0tgt−saumks,wjds+Fmkt,wj,1≤j≤k,umk0=u˜0k,u˙mk0=u˜1k,in which(36)u˜0k=∑j=1kαjkwj⟶u˜0strongly inH2,u˜1k=∑j=1kβjkwj⟶u˜1strongly inH1,and(37)Fmkx,t=∑i=0N−11i!D3ifx,t,um−1umk−um−1i=∑j=0N−1Ajx,t,um−1umkj,with(38)Ajx,t,um−1=∑i=jN−1−1i−jj!i−j!D3ifx,t,um−1um−1i−j.
Note that, by using (31) and standard methods in ordinary differential equations, we can prove that system (35) admits a unique solution cmjkt, 1≤j≤k, on interval 0,Tmk⊂0,T. The following estimates allow one to take Tmk=T independent of m and k.Step 2.
(a priori estimates). First, for allj=1,…,k, multiplying the first equation in (35) by c˙mjkt, summing on j, and integrating with respect to the time variable from 0 to t, we have(39)Xmkt=Xmk0+2∫0tds∫0sgs−τaumkτ,u˙mksdτ+2∫0tFmks,u˙mksds,where(40)Xmkt=u˙mkt2+umkta2+2λ∫0tu˙mksLqqds.
Next, by replacingwj in the first equation in (35) by −Δwj, we obtain that(41)au¨mkt,wj+Δumkt,Δwj+λau˙mktq−2u˙mkt,wj=∫0tgt−τΔumkτ,Δwjdτ+aFmkt,wj,1≤j≤k,similar to the first equation in (35), and it yields(42)Ymkt=Ymk0+2∫0tds∫0sgs−τΔumkτ,Δu˙mksdτ+2∫0taFmks,u˙mksds,=Ymk0+2∫0tgt−τΔumkτ,Δumktdτ−2g0∫0tΔumks2ds−2∫0tds∫0sg′s−τΔumkτ,Δumksdτ+2∫0taFmks,u˙mksds,with(43)Ymkt=u˙mkta2+Δumkt2+8λq2q−1∫0t∂∂xu˙mksq−2/2u˙mks2ds+2λh0∫0tu˙mk0,sqds+2λh1∫0tu˙mk1,sqds.
Putting(44)Smkt=Xmkt+Ymkt+∫0tu¨mks2ds=u˙mkt2+u˙mkta2+umkta2+Δumkt2+2λ∫0tu˙mksLqqds+8λq2q−1∫0t∂∂xu˙mksq−2/2u˙mks2ds+2λh0∫0tu˙mk0,sqds+2λh1∫0tu˙mk1,sqds+∫0tu¨mks2ds,it follows from (39), (42), and (44) that(45)Smkt=Smk0−2g0∫0tΔumks2ds+2∫0tgt−τΔumkτ,Δumktdτ+2∫0tds∫0sgs−τaumkτ,u˙mksdτ−2∫0tds∫0sg′s−τΔumkτ,Δumksdτ+∫0tu¨mks2ds+2∫0tFmks,u˙mksds+2∫0taFmks,u˙mksds=Smk0+∑j=17Ij.
We estimateSmk0 and the integrals on the right-hand side of (45) as follows.
First integralI1: it is clear that(46)I1=−2g0∫0tΔumks2ds≤2g0∫0tSmksds.
Second integralI2: by the inequality 2ab≤1/2a2+2b2, ∀a, b∈ℝ, we have(47)I2=2∫0tgt−τΔumkτ,Δumktdτ≤1/2Smkt+2gL20,T∗2∫0tSmksds.
Third integralI3: using the following inequality au,v≤au,uav,v=uava for all u, v∈H1, we obtain(48)I3=2∫0tds∫0sgs−τaumkτ,u˙mksdτ≤2∫0tds∫0sgs−τumkτau˙mksadτ≤2∫0tSmksds1/2∫0tds∫0sgs−τSmkτdτ21/2≤2T∗gL20,T∗∫0tSmksds.
Fourth integralI4:(49)I4=−2∫0tds∫0sg′s−τΔumkτ,Δumksdτ≤2T∗g′L20,T∗∫0tSmksds.
Fifth integralI5: note that the first equation in (35) can be written as follows:(50)u¨mkt,wj+λu˙mktq−2u˙mkt,wj=Δumkt,wj−∫0tgt−sΔumks,wjds+Fmkt,wj,1≤j≤k.
Hence, replacingwj with u¨mkt and using the inequality a+b+c2≤3a2+b2+c2 for all a, b, c∈ℝ, we obtain(51)u¨mkt2+λqddtu˙mktLqq=Δumkt,u¨mkt−∫0tgt−τΔumkτ,u¨mktdτ+Fmkt,u¨mkt≤Δumkt+∫0tgt−τΔumkτdτ+Fmktu¨mkt≤32Δumkt2+gL20,T∗2∫0tΔumkτ2dτ+Fmkt2+12u¨mkt2≤32Smkt+gL20,T∗2∫0tSmkτdτ+Fmkt2+12u¨mkt2.
This implies that(52)u¨mkt2+2λqddtu˙mktLqq≤3Smkt+gL20,T∗2∫0tSmkτdτ+Fmkt2.
Integrating int, we have(53)I5=∫0tu¨mks2ds≤∫0tu¨mks2ds+2λqu˙mktLqq≤2λqu˜1kLqq+3∫0tSmksds+3T∗gL20,T∗2∫0tSmkτdτ+3∫0tFmks2ds.
Sixth integralI6:(54)I6=2∫0tFmks,u˙mksds≤2∫0tFmksu˙mksds≤∫0tFmks2ds+∫0tSmksds.
Seventh integralI7:(55)I7=2∫0taFmks,u˙mksds≤2∫0tFmksau˙mksads≤∫0tFmksa2ds+∫0tSmksds.
Combining (45), (46)–(49), and (53)–(55), it leads to(56)Smkt≤2Smk0+4λqu˜1kLqq+DT∫0tSmksds+8∫0tFmks2ds+2∫0tFmksa2ds,where(57)DT=10+4g0+22+3T∗gL20,T∗2+4T∗gL20,T∗+g′L20,T∗.
To estimate integrals∫0tFmks2ds and ∫0tFmksa2ds, we use the following lemma.Lemma 4.
The following inequalities are valid:(58)iFmktL∞≤A˜M1+SmktN−1,iiFmkta≤a1A˜M+B˜M1+SmktN−1,where A˜M and B˜M are defined as follows:(59)A˜M=KMf∑i=0N−1a˜i,B˜M=KMf∑i=0N−1b˜i,a˜i=1+12∑i=1N−122ii!Mi,i=0,121i!22a0i,i≥1,b˜i=2i−1i!1+M2a0i+i2a0i−1,i≥1,b˜0=1+M+∑i=1N−1b˜iMi.Proof of Lemma4.proof.
of (i). Using the inequalitiesa+bi≤2i−1ai+bi, for all a, b≥0, i≥1, and si≤1+sq, ∀s≥0, ∀i,q, 0≤i≤q, we have(60)∑i=0N−11i!umk+um−1i≤1+∑i=1N−12ii!umktH1+Mi≤1+∑i=1N−12ii!2i−1umktH1i+Mi≤1+12∑i=1N−122ii!Mi+12∑i=1N−122ii!1a0iSmkti=∑i=0N−1a˜iSmkti≤∑i=0N−1a˜i1+SmktN−1,with a˜i, 0≤i≤N−1, defined by (59). Hence,(61)Fmkx,t≤∑i=0N−11i!D3ifx,t,um−1umk−um−1i≤KMf∑i=0N−11i!umk+um−1i≤KMf∑i=0N−1a˜i1+SmktN−1=A˜M1+SmktN−1,and it implies that equation (i) in (58) holds.proof.
of (ii). We also have(62)Fmxkx,t=D1fx,t,um−1+D3fx,t,um−1∇um−1+∑i=1N−11i!D1D3ifx,t,um−1+D3i+1fx,t,um−1∇um−1umk−um−1i+D3ifx,t,um−1iumk−um−1i−1umxk−∇um−1.
Hence,(63)Fmxkx,t≤KMf1+∇um−1+KMf∑i=1N−11i!1+∇um−1umk−um−1i+iumk−um−1i−1umxk−∇um−1.
It follows that(64)Fmxkt≤KMf1+∇um−1+KMf∑i=1N−11i!1+∇um−1umk−um−1C00,1i+iumk−um−1C00,1i−1umxk−∇um−1≤KMf1+M+∑i=1N−11i!1+M2a0i+i2a0i−1umk−um−1ai≤KMf1+M+∑i=1N−12i−1i!1+M2a0i+i2a0i−1umkai+um−1ai≤KMf1+M+∑i=1N−1b˜iSmkti+Mi=KMf1+M+∑i=1N−1b˜iMi+∑i=1N−1b˜iSmkti=KMf∑i=0N−1b˜iSmkti≤KMf∑i=0N−1b˜i1+SmktN−1=B˜M1+SmktN−1,with b˜i, 0≤i≤N−1, and B˜M defined by (59).
Hence,(65)Fmkta≤a1FmktH1=a1Fmkt2+Fmxkt21/2≤a1Fmkt+Fmxkt≤a1A˜M+B˜M1+SmktN−1.
Therefore, equation (ii) in (58) follows. Lemma 4 is proved.
Now, the integrals∫0tFmks2ds and ∫0tFmksa2ds are estimated as follows:(66)∫0tFmks2ds≤A˜M2∫0t1+SmksN−12ds≤2A˜M2∫0t1+SmksN−1ds,∫0tFmksa2ds≤2a1A˜M+B˜M2∫0t1+SmksN−1ds.
Combining (60) and (66), we have(67)Smkt≤2Smk0+4λqu˜1kLqq+TC1M,T+C1M,T∫0tSmksNds,where(68)C1M,T=DT+16A˜M2+4a1A˜M+B˜M2.
By means of convergences (36), we can deduce the existence of a constant M>0 independent of k and m such that(69)2Smk0+4λqu˜1kLqq≤M24,∀m,k∈ℕ.
Finally, it follows from (67) and (69) that(70)Smkt≤M24+TC1M,T+C1M,T∫0tSmksNds,0≤t≤Tmk≤T.
Then, by solving nonlinear Volterra integral inequality (70) (based on the methods in [40]), the following lemma is proved.Lemma 5.
There exists a constantT>0 independent of k and m such that(71)Smkt≤M2,∀t∈0,T,∀m,k∈ℕ.By Lemma5, we can take constant Tmk=T for all k and m∈ℕ. Thus, we have(72)umk∈W1M,T,∀m,k∈ℕ.Step 3.
(limiting process). Thanks to (72), there exists a subsequence umkj of umk, still denoted by umk, such that(73)umk⟶uminL∞0,T;H2weakly∗,u˙mk⟶um′inL∞0,T;H1weakly∗,u¨mk⟶um″inL2QTweakly,um∈WM,T.
Using the compactness lemma of Lions ([41], p.57) and applying the Fischer–Riesz theorem, from (73), there exists a subsequence of umk, denoted by the same symbol satisfying(74)umk⟶umstrongly inL20,T;H1and a.e.inQT,u˙mk⟶um′strongly inL2QTand a.e.inQT.limx⟶∞
By using the following inequality,(75)u˙mktq−2u˙mkt−um′tq−2um′t≤q−12Mq−2u˙mkt−um′t,q≥2,it follows from (44), (71), and (74)2 that(76)u˙mktq−2u˙mkt↪um′tq−2um′tstrongly inL2QT.
On the contrary, byL∞0,T;H1↪L∞QT and using the inequality(77)aj−bj≤jMj−1a−b,∀a,b∈−M,M,∀M>0,∀j∈ℕ,we deduce from (71) that(78)umkj−umj≤jMj−1umk−um,0≤j≤N−1.
Therefore, (74)1 and (78) lead to(79)umkj⟶umjstrongly inL2QT.
We note that(80)Ajx,t,um−1t≤KMf∑i=jN−11j!i−j!2Mi−j≡D¯jM.
By (33), (37), and (71), it gives(81)Fmkt−Fmt≤∑j=0N−1D¯jMumktj−umjt.
Hence, we have(82)Fmk−FmL2QT2≤N∑j=0N−1D¯j2Mumkj−umjL2QT2,so(83)Fmk⟶Fmstrongly inL2QT.
Passing to the limit in (35) and (36), we have um satisfying (32) and (33) in L20,T.
On the contrary, it follows from the first equation in (32) and the fourth equation in (73) that(84)um′′=Δum−λum′q−2um′−∫0tgt−sΔumsds+Fm∈L∞0,T;L2.
Hence,um∈W1M,T, and Theorem 1 is proved.
Next, the main result is also given by the following theorem. We consider the spaceW1T, defined by(85)W1T=C0,T;H1∩C10,T;L2,and then W1T is a Banach space with respect to the norm(86)vW1T=vC0,T;H1+v′C10,T;L2.Theorem 2.
LetH1−H3 hold. Then, there exist constants M>0 and T>0 such that problems (1)–(3) have a unique weak solution u∈W1M,T and the recurrent sequence um defined by (32) and (33) converges at the N-order rate to the solution u strongly in W1T in the sense(87)um−uW1T≤Cum−1−uW1TN,for all m≥1, where C is a suitable constant. Furthermore, the following estimate is fulfilled:(88)um−uW1T≤CTγTNm,for allm∈ℕ,where CT and 0<γT<1 are the constants only depending on T.Proof.
(i) Existence of solutions: we shall prove thatum is a Cauchy sequence in W1T.
Indeed, we putvm=um+1−um. Then, vm satisfies the variational problem(89)vm″t,w+avmt,w+λum+1′tq−2um+1′t−um′tq−2um′t,w=∫0tgt−savms,wds+Fm+1t−Fmt,w,∀w∈H1,vm0=vm′0=0.
Takingw=vm′ in (89), after integrating in t, we get(90)ρmt=−2λ∫0tum+1′sq−2um+1′s−um′sq−2um′s,vm′sds+2∫0tds∫0sgs−τavmτ,vm′sdτ+2∫0tFm+1s−Fms,vm′sds=−2λ∫0tum+1′sq−2um+1′s−um′sq−2um′s,vm′sds+2∫0tgt−τavmτ,vmtdτ−2∫0tg0avms,vmsds−2∫0tds∫0sg′s−τavmτ,vmsdτ+2∫0tFm+1s−Fms,vm′sds≡∑k=15Jk,with(91)ρmt=vm′t2+vmta2.
Next, we need to estimate the integrals on the right side of (90).
By the inequality(92)um+1′x,sq−2um+1′x,s−um′x,sq−2um′x,s≤q−12Mq−2vm′x,s,q≥2,we have(93)J1=−2λ∫0tum+1′sq−2um+1′s−um′sq−2um′s,vm′sds≤2λ∫0tum+1′sq−2um+1′s−um′sq−2um′svm′sds≤2λq−12Mq−2∫0tvm′s2ds≤2λq−12Mq−2∫0tρmsds.
It is not difficult to estimateJ2, J3, and J4 as follows:(94)J2=2∫0tgt−τavmτ,vmtdτ≤12ρmt+2gL20,T∗2∫0tρmsds,(95)J3=−2∫0tg0avms,vmsds≤2g0∫0tvmsa2ds≤2g0∫0tρmsds,(96)J4=−2∫0tds∫0sg′s−τavmτ,vmsdτ≤2T∗g′L20,T∗∫0tρmsds.
Using Taylor’s expansion of the functionfx,t,um=fx,t,um−1+vm−1 around the point um−1 up to order N, we obtain(97)fx,t,um−fx,t,um−1=∑i=1N−11i!D3ifx,t,um−1vm−1i+1N!D3Nfx,t,θ˜mvm−1N,where θ˜m=θ˜mx,t=um+θ1vm−1, 0<θ1<1.
Hence, it follows from (33) and (97) that(98)Fm+1x,t−Fmx,t=∑i=1N−11i!D3ifx,t,umvmi+1N!D3Nfx,t,θ˜mvm−1N,and then(99)Fm+1t−Fmt≤KMf∑i=1N−11i!2vmtH1i+1N!KMf2vm−1tH1N≤ηT1ρmt+ηT2vm−1W1TN,in which ηT1=KMf/a0∑i=1N−122Mi−1/i! and ηT2=2N/N!KMf.
So,(100)J5=2∫0tFm+1s−Fms,vm′sds≤2∫0tFm+1s−Fmsvm′sds≤2ηT1∫0tρmsds+TηT2vm−1W1T2N+ηT2∫0tρmsds.
Then, it implies from (90), (93)–(96), and (100) that(101)ρmt≤2TηT2vm−1W1T2N+ηT3∫0tρmsds,where ηT3=4λq−12Mq−2+g0+gL20,T∗2+T∗g′L20,T∗+ηT1+1/2ηT2.
By using Gronwall’s lemma, (101) gives(102)vmW1T≤μTvm−1W1TN,where μT=1+1/a02TηT2expTηT3.
ChoosingT>0 small enough such that γT=MμT1/N−1<1, it follows from (102) that(103)um−um+pW1T≤1−γT−1μT−1/N−1γTNm,for allmandp∈ℕ.
Hence,um is a Cauchy sequence in W1T. Then, there exists u∈W1T such that(104)um⟶ustrongly inW1T.
Note thatum∈W1M,T; then, there exists a subsequence umj of um such that(105)umj⟶uinL∞0,T;H2weakly∗,umj′⟶u′inL∞0,T;H1weakly∗,umj′′⟶u′′inL2QTweakly,u∈WM,T.
Moreover, by (104) and the inequality(106)um′tq−2um′t−u′tq−2u′t≤q−12Mq−2um′t−u′t≤q−12Mq−2um′−u′C10,T;L2≤q−12Mq−2um−uW1T,we have(107)um′tq−2um′t⟶u′tq−2u′tstrongly inL∞0,T;L2.
On the contrary,(108)Fm⋅,t−f⋅,t,ut≤f⋅,t,um−1t−f⋅,t,ut+∑i=1N−11i!D3if⋅,t,um−1um−um−1i≤KMfum−1−uW1T+∑i=1N−11i!um−um−1W1Ti.
Therefore, it implies from (104) and (38) that(109)Fmt⟶f⋅,t,utstrongly inL∞0,T;L2.
Finally, passing to the limit in (32) and (33) as m=mj⟶∞, there exists u∈WM,T satisfying the equation(110)u′′t,w+aut,w+λu′tq−2u′t,w+∫0tgt−saus,wds=f⋅,t,ut,w,for all w∈H1, and the initial condition(111)u0=u˜0,u′0=u˜1.
On the contrary, it follows from the fourth equation in (105) and (110) that(112)u″=Δu−λu′q−2u′+∫0tgt−sΔusds+fx,t,u∈L∞0,T;L2.
Hence,u∈W1M,T.Proof.
(ii) Uniqueness: applying the similar estimations used in the proof of Theorem1, it is easy to prove that u∈W1M,T is a unique local weak solution of problems (11)–(13).
Passing to the limit in (103) as p⟶∞ for fixed m, we get (88). Also, with a similar argument, (87) follows. Theorem 2 is proved completely.Remark 1.
In order to construct theN−order iterative scheme defined by (14), H3 is a necessary assumption. If we only consider the existence of solutions of problems (11)–(13), this condition can be weakened as follows:(113)f∈C10,1×ℝ+×ℝ,see [42].


## 4. Numerical Results

In this section, we shall construct a numerical algorithm for the 2-order iterative scheme obtained by (14) and (15). The finite-difference method and some techniques for approximating double integrals will be used to find the finite-difference approximate solutions of umx,t of this scheme. Moreover, a numerical algorithm for the single-iterative scheme obtained by (154) and (155) below will also be constructed, and a concrete example will be numerically illustrated to compare the convergent rates of the single-iterative scheme and the 2-order iterative scheme.Consider problems (11)–(13) with λ=h0=h1=1 and q=2; then, they are transformed into the following problem:(114)utt−Δu+ut+∫0tgt−sΔux,sds=fx,t,u,0<x<1,0<t<T,ux0,t−u0,t=ux1,t+u1,t=0,ux,0=u˜0x,utx,0=u˜1x,where the functions g, f, u˜0, and u˜1 are defined by(115)gt=e−2t,fx,t,u=u3+4e−2t−81+x−x23e−3t,u˜0x=21+x−x2,u˜1x=−21+x−x2.The exact solution of problem (114), with g, f, u˜0, and u˜1 defined by (115), is a function uex given by(116)uexx,t=21+x−x2e−t.With datum (115) and 0≤x≤1 and 0≤t≤0.5, Figure 1 describes the surface of the exact solution uexx,t of problem (114) below.Figure 1
The surface of the exact solutionuex of problem (114).We consider the 2-order iterative scheme of problem (114) as follows:(117)u0≡0,u¨mt+u˙mt−Δumt+∫0tgt−sΔumsds−bmx,tumx,t=Fmx,t,0<x<1,0<t<T,uxm0,t−um0,t=uxm1,t+um1,t=0,umx,0=u˜0x,u˙mx,0=u˜1x,where(118)Fmx,t=fx,t,um−1x,t−bmx,tum−1x,t,bmx,t=D3fx,t,um−1x,t.In order to solve problems (117) and (118) numerically, we first use the spatial difference grid uimt≡umxi,t, with xi=ih, h=1/N, i=0,N¯.Replacing the derivatives in spatial variablex of (117) by the following approximations,(119)∂um∂xxi,t≈uim−ui−1mmh,Δumxi,t=∂2um∂x2xi,t≈ui−1m−2uim+ui+1mh2,problems (117) and (118) are turned into the following system of intergodifferential equations:(120)u¨imt+u˙imt−ui−1m−2uim+ui+1mh2+∫0tgt−sui−1ms−2uims+ui+1msh2ds−bimtuimt=Fimt,i=1,N¯,u1m−u0mh−u0m=uN+1m−uNmh+uNm=0,uim0=u˜0xi≡u˜0i,u˙im0=u˜1xi≡u˜1i,i=1,N¯,where(121)bimt=bmxi,t=D3fxi,t,uim−1t,Fimt=Fmxi,t=fxi,t,uim−1t−bimtuim−1t,i=0,N¯.Boundary conditions in the second equation in (120) lead to(122)u0m=u1m1+h,uN+1m=1−huNm.Eliminating the unknown functionsu0m and uN+1m in the first equation (i=1) and the last equation (i=N), respectively, system (120) is equivalent to(123)u¨1mt+u˙1mt+a˜1u1m−u2mh2+∫0tgt−s−a˜1u1ms+u2mh2sds−b1mtu1mt=F1mt,i=1,u¨imt+u˙imt−ui−1m−2uim+ui+1mh2+∫0tgt−sui−1ms−2uims+ui+1msh2ds−bimtuimt=Fimt,i=2,N−1¯,u¨Nmt+u˙Nmt−uN−1m−1+huNmh2+∫0tgt−suN−1ms−1+huNmsh2ds−bNmtuNmt=FNmt,i=N,uim0=u˜0xi≡u˜0i,u˙im0=u˜1xi≡u˜1i,i=1,N¯,where a˜1=1+2h/h21+h.Rewrite (123) into a vector equation:(124)Ed2u⟶mdt2t+Edu⟶mdtt+Bmtu⟶mt−∫0tgt−sCu⟶msds=F⟶mt,u⟶m0=u1m0,…,uNm0T=u˜0x1,…,u˜0xNT≡X˜0∈ℝN,du⟶mdt0=u˙1m0,…,u˙Nm0T=u˜1x1,…,u˜1xNT≡X˜1∈ℝN,where(125)u⟶mt=u1mt,…,uNmtT=umx1,t,…,umxN,tT,F⟶mt=F1mt,…,FNmtT,and E, Bmt, and C∈MN (MN is the set of real N-order matrices) are defined by(126)E=111⋱⋱11,C=a˜1γγa˜γγa˜γ⋱⋱⋱⋱⋱⋱γa˜γγa˜N,B˜mt=b1mtb2mt⋱⋱bNmt,Bmt=C−B˜mt,a˜1=1+2hh21+h,a˜N=1+hh2,a˜=2h2,γ=−1h2.Integrating in time variablet, we have(127)du⟶mdtt=H⟶mt−e−t∫0tesBmsu⟶msds+e−t∫0tesds∫0tgs−rCu⟶mrdr,u⟶m0=X˜0,where(128)H⟶mt=X˜1e−t+e−t∫0tesF⟶msds,X˜0=u˜0x1,…,u˜0xNT,X˜1=u˜1x1,…,u˜1xNT.Approximating the derivativesdu⟶m/dttj by the following time differences,(129)du⟶mdttj≃u⟶j+1m−u⟶jmΔt,u⟶jm=u⟶mtj,tj=jΔt,j=0,…,M,Δt=TM,H⟶jm=H⟶mtj,Bjm=Bmtj,equation (127)1 is turned into(130)u⟶j+1m−u⟶jmΔt=H⟶jm−e−tj∫0tjesBmsu⟶msds+e−tj∫0tjesds∫0sgs−rCu⟶mrdr.Note that the integral∫0tjΦsds can be approximated by the trapezoidal formula:(131)∫0tjΦsds≃ΔtΦ0+Φ12,j=1,ΔtΦ0+Φj2+∑ν=1j−1Φν,2≤j≤M,Φj=Φtj.Applying (131) with Φs=esBmsu⟶ms and Φs=gtj−rCu⟶mrdr, respectively, we have the following approximations:(132)∫0tjesBmsu⟶msds≃Ψj1,m≡Δtet0B0mu⟶0m+et1B1mu⟶1m2,j=1,Δtet0B0mu⟶0m+etjBjmu⟶jm2+∑ν=1j−1etνBνmu⟶νm,2≤j≤M,∫0tjgtj−rCu⟶mrdr≃Δtg1Cu⟶0m+g0Cu⟶1m2,j=1,ΔtgjCu⟶0m+g0Cu⟶jm2+∑ν=1j−1gj−νCu⟶νm,2≤j≤M,gj=gtj,0≤j≤M.Again using (131) with Φs=es∫0sgs−rCu⟶mrdr, the double integral ∫0tjesds∫0sgs−rCu⟶mrdr can be approximated as follows:(133)∫0tjesds∫0sgs−rCu⟶mrdr=∫0tjes∫0sgs−rCu⟶mrdrds≃Ψj2,m≡Δt2et1∫0t1gt1−rCu⟶mrdr,j=1,Δt12etj∫0tjgtj−rCu⟶mrdr+∑ν=1j−1etν∫0tνgtν−rCu⟶mrdr,2≤j≤M.Similarly, we also obtain the following approximations:(134)H⟶jm=X˜1e−tj+e−tj∫0tjesF⟶msds≃Ψj3,m≡X˜1e−t1+e−t1Δtet0F⟶0m+et1F⟶1m2,j=1,X˜1e−tj+e−tjΔtet0F⟶0m+etjF⟶jm2+∑ν=1j−1etνF⟶νm,2≤j≤M,F⟶jm=F⟶mtj.So, equation (130) can be rewritten by(135)u⟶j+1m=u⟶jm+ΔtΨj∗,m,where(136)Ψj∗,m=−e−tjΨj1,m+e−tjΨj2,m+Ψj3,m.Note thatΨj1,m and Ψj2,m depend on u⟶0m,u⟶1m,…,u⟶jm and Ψj3,m on u⟶0m−1,u⟶1m−1,…,u⟶jm−1, so(137)Ψj∗,m=−e−tjΨj1,m+e−tjΨj2,m+Ψj3,m,where(138)Ψj3,m=X˜1e−tj+e−tjΔtet0F⟶0m+etjF⟶jm2+∑ν=1j−1etνF⟶νm,Ψj1,m=Δtet0B0mu⟶0m+etjBjmu⟶jm2+∑ν=1j−1etνBνmu⟶νm,Ψj2,m=Δt12etj∫0tjgtj−rCu⟶mrdr+∑ν=1j−1etν∫0tνgtν−rCu⟶mrdr.


## 5. Description of Finite-Difference Algorithm (135) and (136)

(i)
LetMandN be fixed constants. At the first iteration with m=0, we set up the given vector(139)u⟶j0=u⟶0tj=u10tj,…,uN0tjT=u0x1,tj,…,u0xN,tjT≡0,j=1,…,M.(ii)
At them−1th iteration, suppose that we can compute(140)u⟶jm−1=u⟶m−1tj=u1m−1tj,…,uNm−1tjT,j=1,…,M.(iii)
The computation of the vectorsu⟶jm=u⟶mtj=u1mtj,…,uNmtjT, j=1,…,M, can be done consecutively by the following steps:C1. The computation ofu⟶1m=u1mt1,…,uNmt1T.(i)
With the first given vectoru⟶0m=u⟶mt0=u˜0x1,…,u˜0xNT=X˜0, we calculate Ψ0∗,m as follows:(141)Ψ03,m=X˜1,Ψ01,m=Ψ02,m≡0.Hence,(142)Ψ0∗,m=−e−t0Ψ01,m+e−t0Ψ02,m+Ψ03,m=X˜1.(ii)
Findingu⟶1m=u1mt1,…,uNmt1T by(143)u⟶1m=u⟶0m+ΔtΨ0∗,m=X˜0+ΔtX˜1.C2. The computation ofu⟶2m=u1mt2,…,uNmt2T.(i)
CalculatingΨ1∗,m:(144)Ψ13,m=X˜1e−t1+e−t1Δtet0F⟶0m+et1F⟶1m2,Ψ11,m=Δtet0B0mu⟶0m+et1B1mu⟶1m2,Ψ12,m=Δt2et1∫0t1gt1−rCu⟶mrdr.Hence,(145)Ψ1∗,m=−e−t1Ψ11,m+e−t1Ψ12,m+Ψ13,m.(ii)
Findingu⟶2m=u1mt2,…,uNmt2T by(146)u⟶2m=u⟶1m+ΔtΨ1∗,m.C3. The computation ofu⟶j+1m=u1mtj+1,…,uNmtj+1T.Suppose thatu⟶1m, u⟶2m,…,u⟶jm have been calculated; then, we define u⟶j+1m=u1mtj+1,…,uNmtj+1T determined by recurrence as follows:(i)
CalculatingΨj∗,m:(147)Ψj3,m=X˜1e−tj+e−tjΔtet0F⟶0m+etjF⟶jm2+∑ν=1j−1etνF⟶νm,Ψj1,m=Δtet0B0mu⟶0m+etjBjmu⟶jm2+∑ν=1j−1etνBνmu⟶νm,Ψj2,m=Δt12etj∫0tjgtj−rCu⟶mrdr+∑ν=1j−1etν∫0tνgtν−rCu⟶mrdr,and(148)Ψj∗,m=−e−tjΨj1,m+e−tjΨj2,m+Ψj3,m.(ii)
Findingu⟶j+1m=u1mtj+1,…,uNmtj+1T by(149)u⟶j+1m=u⟶jm+ΔtΨj∗,m.When the process of the computations is reached toj=M−1, we obtain(150)u⟶jm=umx1,tj,…,umxN,tjT,1≤j≤M.C4. The error of two successively approximate iterations.(151)um−um−1M,N=max1≤j≤Mmax1≤i≤Numxi,tj−um−1xi,tj.The process of iteration will be stopped at themth step when the following estimate is satisfied:(152)um−um−1M,N<10−4.C5. The error between the approximate solution (at themth step) and the exact solution is defined by(153)EM,N=um−uexM,N=max1≤j≤Mmax1≤i≤Numxi,tj−uexxi,tj,in whichuexx,t=21+x−x2e−t is the exact solution of (114) satisfying datum (115).With datum (115) and the grid of N=50 and M=100, Figure 2 describes the surface of the finite-difference approximate solution of umx,t defined by 2-order iterative scheme (117) and (118), with respect to algorithm (135) and (136).Figure 2
The surface of the finite-difference approximate solution ofumx,t defined by 2-order iterative scheme (117) and (118), with respect to algorithm (135) and (136).Remark 2.
In scheme (117) and (118), as bmx,t=D3fx,t,um−1x,t=0, we have the following scheme (the single-iterative scheme):(154)u0≡0,u¨mt+u˙mt−Δumt+∫00gt−sΔumsds=Fmx,t,0<x<1,0<t<T,uxm0,t−um0,t=uxm1,t+um1,t=0,umx,0=u˜0x,u˙mx,0=u˜1x,where(155)Fmx,t=fx,t,um−1x,t.
At this time,(156)Bmt=C,∀t∈0,T,Bjm=Bmtj=C,is independent of m and j.
Similarly, applying (135)–(137) for (154) and (155), we obtain the difference equation of single-iterative scheme (154) and (155) as follows:(157)u⟶j+1m=u⟶jm+ΔtΨj∗,m,where(158)Ψj∗,m=−e−tjΨj1,m+e−tjΨj2,m+Ψj3,m,Ψj3,m=X˜1e−tj+e−tjΔtet0F⟶0m+etjF⟶jm2+∑ν=1j−1etνF⟶νm,Ψj1,m=Δtet0Cu⟶0m+etjCu⟶jm2+∑ν=1j−1etνCu⟶νm,Ψj2,m=Δt12etj∫0tjgtj−rCu⟶mrdr+∑ν=1j−1etν∫0tνgtν−rCu⟶mrdr,(159)F⟶jm=F⟶mtj=F1mtj,…,FNmtjT,Fimtj=Fmxi,tj=fxi,tj,uim−1tj.
Withdatum (115) and the grid of N=50 and M=100, Figure 3 describes the surface of the finite-difference approximate solution of umx,t defined by single-order iterative scheme (154) and (155), with respect to algorithm (157) and (158).
WithT=0.5, the computational results presented in Tables 1–5 get along with algorithms (135) and (136) and (157)–(159) of iterative schemes (117) and (118) and (154) and (155), respectively.
Table1 shows that the errors EM,N=um−uexM,N are decreased when the size of grid N,M is increased (smoother). It is shown that, with a certain grid N,M, the errors of the 2-order scheme are smaller than those of the single-order scheme line by line.
With the grid ofN=10 and M=20, the errors EM,N=um−uexM,N (Table 2) and DM,Nm=um−um−1M,N (Table 3) are also decreased when the number of iterations is increased. Comparing columns 2 and 3 of Table 3, the errors of the 2-order iterative scheme are still smaller than those of the single-iterative scheme on the same number of iterations.
Note that, with the grid ofN=10 and M=20, the above remarks on Tables 2 and 3 are also valid on Tables 4 and 5.
Finally, with the datum as in (115) and the grid of N=50 and M=100, we plot the curved surfaces of the finite-difference approximate solutions defined by algorithms (135) and (136) and (157)–(159), respectively, and the exact solution uex of problem (115).Figure 3
The surface of the finite-difference approximate solution ofumx,t defined by single-order iterative scheme (154) and (155), with respect to algorithm (157) and (158).Table 1
Errors of the approximate solution and the exact solution.
Single-iterative scheme2-order iterative schemeNMEM,N=um−uexM,NEM,N=um−uexM,N10200.06570216478078720.065702138866870420400.03432829993181640.034328241006146730600.02319605390516210.023195976323827440800.01740217967549800.0174020827329464501000.01459615530283780.0145960139057242Table 2
Errors of the approximate solution (at themth step) and the exact solution, with N=10 and M=20.
Number of iterationsSingle-iterative scheme2-order iterative schememEM,N=um−uexM,NEM,N=um−uexM,N10.94185108288965270.941851082889652720.19168225551605330.084687077398258930.07232902373928260.065704190687677440.06589093342180250.065702138866870450.06570500439442430.065702138866867760.06570216478078720.065702138866867570.06570213901661730.065702138866867580.06570213886744750.065702138866867590.06570213886686950.0657021388668675100.06570213886686770.0657021388668675110.06570213886686770.0657021388668675120.06570213886686770.0657021388668675130.06570213886686770.0657021388668675140.06570213886686770.0657021388668675150.06570213886686770.0657021388668675Table 3
Errors of two consecutive iterations, withN=10 and M=20.
Number of iterationsSingle-iterative scheme2-order iterative schememDM,Nm=um−um−1M,NDM,Nm=um−um−1M,N12.52.520.75781624584135130.885861665539614530.15018239722533670.035645324152086240.01299709346289569.425500365978223e(−06)55.099302560935826e(−04)8.992806499463768e(−14)61.061574497773776e(−05)2.220446049250313e(−16)71.316120679106803e(−07)081.043858777194373e(−09)095.554223747594733e(−12)0102.042810365310288e(−14)011001200130014001500Table 4
Errors of the approximate solution (at themth step) and the exact solution, with N=50 and M=100.
Number of iterationsSingle-iterative scheme2-order iterative schememEM,Nm=um−uexM,NEM,Nm=um−uexM,N10.94734840681237110.947348406812371120.19010341017014400.047870183148172930.02585753582596560.014600388562964040.01498159772680440.014596013905724250.01460511990315520.014596013905698460.01459615530283780.014596013905697570.01459601545430190.014596013905697580.01459601391824750.014596013905697890.01459601390577330.0145960139056978100.01459601390569330.0145960139056978110.01459601390569310.0145960139056978120.01459601390569310.0145960139056978130.01459601390569310.0145960139056978140.01459601390569310.0145960139056978150.01459601390569310.0145960139056978Table 5
Errors of two consecutive iterations, withN=50 and M=100.
Number of iterationsSingle-iterative scheme2-order iterative schememDM,Nm=um−um−1M,NDM,Nm=um−um−1M,N12.52.520.75724499664222710.900860682779292130.16757368242478800.042847212649330440.01791991917692931.897245848470064e(−05)59.579076481733839e(−04)7.116529587847253e(−13)62.975046644859702e(−05)1.088018564132653e(−14)76.028582577588537e(−07)3.996802888650564e(−15)88.580162136340164e(−09)8.881784197001252e(−16)99.013212398656378e(−11)2.220446049250313e(−16)107.265299473147024e(−13)0114.884981308350689e(−15)0122.220446049250313e(−16)0130014001500


## 6. Conclusion

In this article, an initial boundary value problem for a viscoelastic wave equation with nonlinear damping is investigated, and its main outcomes are summarized in two parts. In part 1, theoretically, the existence of a recurrent sequence via a high-order iterative scheme is proved, and the high-order convergence of this sequence to the unique weak solution of the proposed model is also claimed. In part 2, two specific cases of the high-order iterative scheme called the 2-order iterative scheme and the single-iterative scheme are considered, in which two numerical algorithms for finding the approximate solutions corresponding to these schemes are constructed by the finite-difference method and the techniques approximating double integrals. To close this part, a concrete example is numerically considered. And the studied results of the errors show that the convergent rate of the 2-order iterative scheme is faster than that of the single-order iterative scheme.



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