# A High-Order Iterative Scheme for a Nonlinear Pseudoparabolic Equation and Numerical Results

Authors: Nguyen Huu Nhan; Tran Trinh Manh Dung; Le Thi Mai Thanh; Le Thi Phuong Ngoc; Nguyen Thanh Long
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

In this paper, by applying the Faedo-Galerkin approximation method and using basic concepts of nonlinear analysis, we study the initial-boundary value problem for a nonlinear pseudoparabolic equation with Robin–Dirichlet conditions. It consists of two main parts. Part 1 is devoted to proof of the unique existence of a weak solution by establishing an approximate sequenceum based on a N-order iterative scheme in case of f∈CN0,1×0,T∗×ℝN≥2, or a single-iterative scheme in case of f∈C1Ω¯×0,T∗×ℝ. In Part 2, we begin with the construction of a difference scheme to approximate um of the N-order iterative scheme, with N=2. Next, we present numerical results in detail to show that the convergence rate of the 2-order iterative scheme is faster than that of the single-iterative scheme.


---

## Body

## 1. Introduction

In this paper, we consider the following initial-boundary problem:(1)ut−μt+αt∂∂tuxx+1xux+∫0tgt−suxxs+1xuxsds=fx,t,u,1<x<R,0<t<T,(2)ux1,t−ζu1,t=uR,t=0,(3)ux,0=u˜0x,where R>1 and ζ≥0 are given constants and μt, αt, f, g, and u˜0 are given functions satisfying conditions specified later, with u=ux,t being the unknown function.Equation (1) is a form of the Sobolev-type differential equations; it is also called pseudoparabolic equation after Showalter’s works [1–4] in the seventies. Since then, numerous interesting results for linear/nonlinear pseudoparabolic equations have been obtained. It is also well known that the Sobolev-type differential equations or the pseudoparabolic equations appear in the study of various problems of hydrodynamics, thermodynamics, and filtration theory, see [5–8] and the references therein. In the absence of the memory term in (1), i.e., g=0, the nonlinear pseudoparabolic problem of the types (1)–(3) is arisen in the investigations about second-grade or third-grade fluid flows, see [9, 10, 11] and references therein. In [9], a mathematical model describing the unsteady flow of second-grade fluid in a circular cylinder is considered as follows:(4)∂w∂t=ν+α∂∂twrr+1rwr−Nw,0<r<a,t>0,wa,t=W,t>0,wr,0=0,0≤r<a,where wr,t is the velocity along the z-axis, ν is the kinematic viscosity, α is the material parameter, and N is the imposed magnetic field. In the boundary and initial conditions, W is the constant velocity at r=a and a is the radius of the cylinder. In the presence of the memory term in (1), i.e., g≠0, the problems of the types (1)–(3) are also studied in the theory of viscoelasticity, see [12]. Besides, it is well known that pseudoparabolic equations with nonlocal boundary conditions/nonlocal terms have been studied and many interesting results have been obtained such as stability, global existence, and finite time blow-up, for example, we refer to [13–19] and the references therein. In [15], Dai and Huang studied the solvability and the well-posedness of solutions for the nonlinear pseudoparabolic equation(5)ut+ax,tuxtx=Fx,t,u,ux,uxx,α<x<β,0<t<T,with the nonlocal moment boundary conditions ∫αβux,tdx=∫αβxux,tdx=0,0≤t≤T.In [17], Sun et al. considered the Dirichlet problem for the nonlinear pseudoparabolic equation with a memory term as follows:(6)ut−Δu−Δut+∫0tgt−τΔuτdτ=up−2u,inΩ×0,T,u=0,on∂Ω×0,T,u0=u0,inΩ,where Ω is a bounded domain of ℝnn≥1 with smooth boundary ∂Ω, p>2, T∈0,∞, u0∈H1Ω and g:ℝ+⟶ℝ+ is a positive nonincreasing function. The authors used the concavity method and the improved potential well method to obtain the global existence and the finite time blow-up phenomena of solutions.This paper consists of two main parts. In Part 1, by using theN-order iterative method, Faedo-Galerkin method, and compact method, we prove existence and uniqueness of a weak solution of problems (1)–(3) (see Theorem 2). We begin with the establishment of the N-order nonlinear approximate sequence um in case of f∈CN0,1×0,T∗×ℝ via the N-order iterative scheme associated with problems (1)–(3) as follows:(7)u0≡0,utm−μt+αt∂∂tuxxm+1xuxm−∫0tgt−suxxms+1xuxmsds=∑i=0N−11i!D3ifx,t,um−1x,tumx,t−um−1x,ti,uxm1,t−ζum1,t=umR,t=0,um0=u˜0,and next, we prove that um converges to the unique solution u of problems (1)–(3) at a rate of order NN≥2; it means that um−uX≤Cum−1−uXN, for some C>0, where X is a suitable space. Scheme (7) is called the high-order iterative scheme or the N-order iterative scheme. Specially, when N=2, the 2-order iterative scheme is given as follows:(8)u0≡0,utmt−μtLumt−αtLutmt+∫0tgt−sLumsds=fx,t,um−1x,t+D3fx,t,um−1x,tumx,t−um−1x,t,uxm1,t−ζum1,t=umR,t=0,um0=u˜0,where Lu=uxx+1/xux. In case of f∈C1Ω¯×0,T∗×ℝ, it is clear to see that the local existence and uniqueness of problems (1)–(3) also can be established by using the linear approximate sequence um via a single-iterative scheme (see Remark 1). We note more that the abovementioned high-order iterative scheme is also used to obtain the existence of solutions in the previous papers [20–23]. In [23], Truong et al. studied the initial-boundary problem for a nonlinear wave equation of Kirchhoff-Carrier type. Here, by Galerkin method and compactness method, the existence and the convergence at N-order rate of a recurrent sequence associated with the proposed problem were proved. Furthermore, when N=3, the 3-order iterative scheme was established and solved numerically.In this paper, the numerical results are also given in Part 2. First, this part is devoted to the construction of the difference scheme to approximateum in the 2-order iterative scheme (8). In order to do this, we shall use a simple finite-difference scheme which is a standard model given in [24]. We first use the uniform partition xi=ih, h=1/N0, i=0,1,…, N0, and the forward difference formulas (see [24], pages 36 and 43) to approximate the kth derivatives, k=1,2, in spatial variable, as follows:(9)uxmxi,t≃uimt−ui−1mth,uxmx0,t≃u1mt−u0mth,uxxmxi,t≃ui−1mt−2uimt+ui+1mth2,Luimt=uxxmxi,t+1xiuxmxi,t≡aiui−1mt+biuimt+γui+1mt,ai=1h2−1xih,bi=−2h2+1xih,γ=1h2,2≤i≤N0.This is also a technique used in [20, 25–29]. After replacing (9)1,2,3 in problem (8), we obtain the first-order integro-differential equation with a vector-function variable in the form as follows:(10)A¯tdu⟶mdtt+Bmtu⟶mt+∫0tgt−sCu⟶msds=F⟶mt,where A¯t, Bmt are functional matrices depending on a time variable t and C∈MN0 (MN0 is the set of real N0-size matrices). Next, we make discretizations in time variables tj=jΔt, Δt=T/M, and j=0,1,…,M and approximate the integral ∫0tgt−sCu⟶msds by the trapezoidal formula (see [24], page 56), and we remark that this technique was also used in [26, 27, 30]. Then, we obtain the following algorithm to determine the finite-difference approximate solutions of um given by the 2-order iterative scheme formula (71)(11)A¯0u⟶1m=A¯0−ΔtB0mu⟶0m+ΔtF⟶mt0,A¯1u⟶2m=A¯1−ΔtB1m−12Δt2g0Cu⟶1m−12Δt2g1Cu⟶0m+ΔtF⟶mt1,A¯ju⟶j+1m=A¯j−ΔtBjm−12Δt2g0Cu⟶jm−12Δt2g1Cu⟶0m−Δt2∑ν=1j−1gj−νCu⟶νm+ΔtF⟶mtj,2≤j≤M−1,where A¯j=A¯tj and Bjm=Bmtj. Similarly, we have constructed the algorithm to find the finite-difference approximate solutions of um given by the single-order iterative scheme (formula (92)).It is well known that the finite-difference method to solve nonlinear elliptic/parabolic/pseudoparabolic equations and the consistency, accuracy, efficiency, stability, convergence, and the other properties of difference schemes are mentioned in many works [25–27, 29, 31–45].In [31], Amirali et al. considered the following initial-boundary value problem for the pseudoparabolic equation with delay(12)Lu≔∂u∂t−∂∂xax,t∂2u∂t∂x−∂∂xbx,t∂u∂x=fx,t,ux,t,ux,t−r,x,t∈Q,u0,t=ul,t=0,t∈0,T,ux,t=φx,t,x,t∈Q¯×−r,0,where Q=0,l×0,T, r represents the delay parameter, ax,t≥α>0, bx,t≤b∗, φx,t, and fx,t,ux,t,ux,t−r are given sufficiently smooth functions satisfying certain regularity conditions. Here, the finite-difference technique was applied to the numerical solution of problem (12). By the method of integral identities with use of the piecewise linear basis functions in space and interpolating quadrature rules with weight and remainder term in integral forms, two-level difference scheme was constructed for singular perturbation cases without delay. The finite-difference discretization was shown to be absolutely stable and convergent of order two in space and of order one in time. Based on the method of energy estimates, the error analysis for the approximate solution was presented. The error estimates were obtained in the discrete norm. Some numerical results confirming the expected behavior of the method were shown.In [32], Beshtokov studied the following nonlocal boundary value problem for a third-order pseudoparabolic equation with variable coefficients(13)ut=Lu+fx,t,0<x<l,0<t≤T,L0u0,t=0,0≤t≤T,ul,t=β∫0lxmux,tdx−μt,0≤t≤T,ux,0=u0x,0≤x≤l,where(14)Lu=1xmxmkx,tuxx+1xmηx,tuxxt+rx,tux−qx,tu,L0u=kux+ηuxt.The existence and uniqueness of the solution of problems (13) and (14) were proved by the Riemann function method. For its solution, in the differential and finite-difference settings, the author derived a priori estimates that implied the stability of the solution with respect to the initial data and the right-hand side on a layer as well as the convergence of the solution of the difference problem to the solution of the differential problem.In [26], Jachimavičienė and Sapagovas studied the following two-dimensional pseudoparabolic equation:(15)ut=Δu+ηΔut+fx,y,t,0<x,y<1,0<t<T,with nonlocal integral boundary conditions(16)u0,y,t=γ1∫01ux,y,tdx+μ1y,t,u1,y,t=γ2∫01ux,y,tdx+μ2y,t,ux,0,t=μ3x,t,ux,1,t=μ4x,t,and initial condition(17)ux,y,0=φx,y,where f, φ, μi, and i=1,2,3,4 are given functions and η, γ1, and γ2>0 are given constants. They decomposed problem (15) into two locally one-dimensional problems from layer t=tn to layer t=tn+1 as follows:(18)12∂u∂t=∂2u∂x2+η∂∂t∂2u∂x2+12f,tn≤t≤tn+1/2,12∂u∂t=∂2u∂y2+η∂∂t∂2u∂y2+12f,tn+1/2≤t≤tn+1,and next, they changed equation (18) to the following one-dimensional difference schemes:(19)uijn+1/2−uijnτ=Λ1uijn+1/2+ηΛ1uijn+1/2−Λ1uijnτ/2+12fijn+1/2,uijn+1−uijn+1/2τ=Λ2uijn+1+ηΛ2uijn+1−Λ2uijn+1/2τ/2+12fijn+1,where Λ1uijn=ui−1jn−2uijn+ui+1jn/h2 and Λ2uijn+1=uij−1n+1−2uijn+1+uij+1n+1/h2. They proved the difference equation (19) approximating the differential equation (18) with the truncation error Oh2+τ. Moreover, if γ1+γ2<2, then the difference schemes (19) are stable for all values of h and τ.In [34], Brachet and Chehab considered the following nonlinear parabolic equation:(20)ut+Fu=0,where F:ℝn⟶ℝn is a regular map. The backward Euler scheme applied to the above equation generates the iterations(21)uk+1−uk+ΔtFuk+1=0,and the nonlinear term Fuk+1 is approximated by(22)Fuk+1≈Fuk+F′ukuk+1−uk.Consequently, the following difference equation is established:(23)uk+1=uk−ΔtId+ΔtF′uk−1Fuk.Stability results in the linear and the nonlinear case and numerical simulations of 2D incompressible Navier-Stokes equations for illustrating the robustness of the method were also presented here. It is clear that the approximation given by (22) is similar to the approximation of the nonlinear term on the left hand side of the 2-order iterative scheme (8).In [41], the authors undeveloped two new B-spline collocation algorithms based on cubic trigonometric B-spline functions to find approximate solutions of a nonlinear parabolic partial differential equations with Dirichlet and Neumann boundary conditions. Some well-known nonlinear parabolic problems were also solved here to check the applicability, accuracy, and efficiency of the proposed algorithms.In [37], departing from a generalized Burgers–Huxley partial differential equation, the authors provided a Mickens-type, nonlinear, finite-difference discretization of the model. They proved that the method proposed also preserves many of the relevant characteristics of these solutions, such as the positivity, the boundedness, and the spatial and temporal monotonicity, and then, in [42], the authors established the property of convergence for a finite-difference discretization of a diffusive partial differential equation with generalized Burgers convective law and generalized Hodgkin–Huxley reaction. The authors proved that the method introduced in [37] was convergent with linear order in time and quadratic order in space. Some numerical experiments were provided in order to support the analytical results.In this paper, at the end of Part 2, an illustrated example and the numerical results are detailed to show that the convergence rate of the 2-order iterative scheme is faster than that of the single-iterative scheme.


## 2. Existence and Uniqueness

Throughout this paper, we setΩ=1,R and use L2=L2Ω to denote the Lebesgue space with the inner product defined by u,v=∫1Ruxvxdx, and L2-norm of a function u∈L2 is denoted by u=u,u. We use Hm=HmΩ to denote the Sobolev spaces with the norm uHm=∑i=0mDiu21/2.Moreover, we also introduce three weighted scalar products(24)u,v=∫1Rxuxvxdx,u,v∈L2,u,v1=u,v+ux,vx,u,v∈H1,u,v2=u,v+ux,vx+uxx,vxx,u,v∈H2,and then, L2, H1, and H2 are the Hilbert spaces with respect to the abovementioned scalar products. We denote u0=u,u, u∈L2; u1=u,u1, u∈H1; u2=u,u2, u∈H2.Put(25)V=v∈H1:vR=0.The symmetric bilinear forma⋅,⋅ is defined by(26)au,w=ux,wx+ζu1w1,for allu,w∈V,with ζ≥0 being a given constant and va=av,v. Then, we have the following lemmas.Lemma 1.
The imbeddingsV⟼C0Ω¯ are compact and(27)ivC0Ω¯≤R−1vx0≤R−1va,for allv∈V,iiv0≤2RR−12vx0,for allv∈V,iiivx0≤va≤1+ζR−1vx0,for allv∈V.Lemma 2.
The symmetric bilinear forma⋅,⋅ is continuous on V×V and coercive on V, i.e., there exist two positive constants C0 and C1 such that(28)iau,v≤C1ux0vx0,iiav,v≥C0vx02,for all u, v∈V. Moreover, C1=1+ζR−1 and C0=1.The notation⋅X is the norm in the Banach space X, and X′ is the dual space of X. We denote by Lp0,T;X, 1≤p≤∞, for the Banach space of functions u:0,T⟶X measurable, such that(29)uLp0,T;X=∫0TutXpdt1/p<∞,for1≤p<∞,uL∞0,T;X=esssup0<t<TutX,for p=∞.Denoteutx=ux,t, u˙t=u′t=utt=∂u/∂tt, uxt=∂u/∂xt, and uxx=∂2u/∂x2t. With f∈CN0,1×0,T∗×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≤N, D0f=D0,0,0f=f.For a fixed constantT∗>0, we make the following assumptions:(30)H1u˜0∈V∩H2,u˜0x1−ζu˜01=0;H2g∈L20,T∗;H3α∈H10,T∗,αt≥α∗>0,for allt∈0,T∗;H4μ∈H10,T∗,μt≥μ∗>0,for allt∈0,T∗;H5f∈C1Ω¯×0,T∗×ℝsatisfies the conditionsD3if,D2D3jf∈C0Ω¯×0,T∗×ℝ,1≤i≤N,1≤j≤N−1.Definition 1.
The weak solution of problems (1)–(3) is a function u∈C0,T;V∩H2 such that u′∈L∞0,T;V∩H2 and u satisfies the following variational equation:(31)u′t,w+αtau′t,w+μtaut,w=∫0tgt−saus,wds+fut,w,for allw∈V,a.e.,t∈0,T,u0=u˜0,where fux,t=fx,t,ux,t. For each T∈0,T∗, we define WT=v∈C0,T;V∩H2:v′∈L∞0,T;V∩H2, and then WT is a Banach space with the norm vWT=maxvC0,T;V∩H2,v′L∞0,T;V∩H2. For M>0, we put BTM=v∈WT:vWT≤M.
Now, we construct the recurrent sequenceum defined by u0≡0, and suppose that(32)um−1∈BTM.
Then,um is found by the fact that um∈BTM, m≥1, and um satisfies(33)u˙mt,w+αtau˙mt,w+μtaumt,w=∫0tgt−saums,wds+ℱmt,w,for allw∈V,a.e.,t∈0,T,um0=u˜0,where(34)ℱmx,t=∑i=0N−11i!D3ifx,t,um−1x,tumx,t−um−1x,ti.
Using the standard Faedo-Galerkin method, which is introduced by Lions in [44], we can prove the following theorem.Theorem 1.
Assume thatu˜0, g, αt,μt, and f satisfy the conditions H1−H5, respectively, then there exist the constants M>0 and T>0 such that problems (33), (34) admit um∈BTM.By using Theorem1 and the compact imbedding theorems, we shall prove the existence and uniqueness of weak local solution in time to problems (1)–(3).First, we consider the space(35)W1T=v∈C0,T;V:v′∈L20,T;V,then W1T is a Banach space with respect to the norm (see [44]) vW1T=vC0,T;V+v′L20,T;V.Theorem 2.
LetH1−H5 hold. Then, there exist constants M>0 and T>0 such that problems (1)–(3) have a unique weak solution u∈BTM and the recurrent sequence um, defined by (32)–(34), converges at a rate of order N to the solution u strongly in the space W1T in the sense(36)um−uW1T≤Cum−1−uW1TN,for all m≥1, where C is a suitable constant. On the other hand, the following estimate is fulfilled:(37)um−uW1T≤CTkTNm,for allm∈ℕ,where CT and 0<kT<1 are the constants depending only on T.Proof.
We shall prove thatum is a Cauchy sequence in W1T.
First, we putvm=um+1−um. Then, vm satisfies the variational problem,(38)v˙mt,w+αtav˙mt,w+μtavmt,w=∫0tgt−savms,wds+ℱm+1t−ℱmt,w,∀w∈V,vm0=0,where ℱmx,t is defined by (34).
Takingw=v˙mt in (38), after integrating in t, we have(39)Zmt=2∫0tμ′svmsa2ds+2∫0tdτ∫0τgτ−savms,v˙mτds+2∫0tℱm+1s−ℱms,v˙msds=J1+J2+J3,where(40)Zmt=μtvmta2+2∫0tv˙ms02+αsv˙msa2ds.
Next, we have to estimate the integrals on the right-hand side of (40).
We putKMf=fC0Ω¯M+∑i=1ND3ifC0Ω¯M+∑j=1N−1D2D3jfC0Ω¯M, where(41)fC0Ω¯M=supfx,t,y:x,t,y∈Ω¯M,Ω¯M=1,R×0,T∗×−R−1M,R−1M.
Using the inequalityZmt≥β∗vmta2+2∫0tv˙ms02+v˙msa2ds, with β∗=minμ∗,α∗, the integrals J1 and J2 are estimated as follows:(42)J1=2∫0tμ′svmsa2ds≤1β∗μ′L∞0,T∗∫0tZmsds,J2=2∫0tdτ∫0tgτ−savms,v˙mτds≤14Zmt+2T∗gL20,T∗2β∗∫0tZmsds.
Using Taylor’s expansion of the functionsfx,t,um=fx,t,um−1+vm−1 around the point um−1 up to order N, we obtain(43)fx,t,um−fx,t,um−1=∑i=1N−11i!D3ifx,t,um−1vm−1i+1N!D3Nfx,t,u¯mvm−1N,where u¯m=um−1x,t+θvm−1x,t, 0<θ<1.
Hence,(44)ℱm+1t−ℱmt=∑i=1N−11i!D3ifx,t,umvmi+1N!D3Nfx,t,u¯mvm−1N.
Note that(45)vmx,ti≤R−1vmtai≤R−1i2Mi−1Zmtβ∗,vm−1x,tN≤R−1vm−1taN≤R−1Nvm−1W1TN.
Therefore, we have(46)ℱm+1t−ℱmt0≤R2−12KMf∑i=1N−11i!R−1i2Mi−1Zmtβ∗+1N!R2−12KMfR−1Nvm−1W1TN≡γ¯1MZmt+γ¯2Mvm−1W1TN,where γ¯1M=R2−1/2KMf∑i=1N−11/i!R−1i2Mi−1/β∗, γ¯2M=1/N!R2−1/2KMfR−1N. Since the above inequality, the integral J3 can be estimated by(47)J3=2∫0tℱm+1s−ℱms,v˙msds≤2∫0tℱm+1s−ℱms02ds+12∫0tv˙ms02ds≤4Tγ¯22Mvm−1W1T2N+4γ¯12M∫0tZmsds+14Zmt.
By (39) and (42), it follows from (47) that(48)Zmt≤8Tγ¯22Mvm−1W1T2N+2γ¯3M∫0tZmsds,where γ¯3M=4γ¯12M+μ′L∞0,T∗+2T∗gL20,T∗2/β∗.
Using Gronwall’s Lemma, we deduce from (48) that(49)vmW1T≤μTvm−1W1TN,∀m∈ℕ,where μT=1+1/222/β∗γ¯2MTexpTγ¯3M.
ChoosingT>0 small enough such that kT=MμT1/N−1<1, it follows from (49) that, for all m and p,(50)um−um+pW1T≤1μT1/N−11−kTkTNm.
The above inequality ensures thatum is a Cauchy sequence in W1T. Then, there exists u∈W1T such that(51)um⟶ustrongly inW1T.
Note thatum∈BTM, then there exists a subsequence umj of um such that(52)umj⟶u,inL∞0,T;V∩H2weakly∗,u˙mj⟶u′,inL∞0,T;V∩H2weakly∗,u∈BTM.
We note that(53)ℱmx,t−fum−1x,t≤KMf∑i=1N−11i!R−1um−um−1W1Ti,fum−1x,t−fux,t≤KMfR−1um−1−uW1T.
Hence,(54)ℱm−fuC0QT¯≤ℱm−fum−1C0QT¯+fum−1−fuC0QT¯≤KMf∑i=1N−11i!R−1ium−um−1W1Ti+KMfR−1um−1−uW1T.
Therefore, we deduce from (51) and (54) that(55)ℱm⟶fustrongly inC0QT¯.
Lettingm=mj⟶∞ in (33) and (34) and using (51, 52, and 55), it implies that there exists u∈BTM satisfying (31). The proof of the existence is completed. Next, it is not difficult to prove the uniqueness of a solution of (31). Afterward, by passing to the limit in (50) as p⟶∞ for fixed m, we get (37). Theorem 2 is proved completely.Remark 1.
The local existence and uniqueness of problems (1)–(3) can be established by using the linear approximate sequence um corresponding to schemes (32)–(34) with ℱmx,t=fx,t,um−1x,t. Then, the assumption for f is weakened as follows: f∈C1Ω¯×0,T∗×ℝ.
We note more that the scheme obtained here is called the single-iterative scheme.


## 3. Numerical Scheme

In this section, we first construct a difference scheme to approximate the solutionu of problems (1)–(3) via approximating um in the 2-order iterative scheme (8). It is implied from Theorems 1 and 2 that, um is definite by problems (33) and (34) with the nonlinear term ℱmx,t given on right-hand side of (33) as follows:(56)ℱmx,t=fx,t,um−1x,t+bmx,tumx,t−um−1x,t,with(57)bmx,t=D3fx,t,um−1x,t.Putting(58)uimt=umxi,t,xi=1+ih,i=0,1,…,N+1,h=R−1N+1.Rewriting (8) at the node x=xi,(59)u˙imt−αtLu˙imt−μtLuimt+∫0tgt−sLuimsds−bimtuimt=Fimt,0≤i≤N,0<t<T,uxmx0,t−ζu0mt=uN+1mt=0,uim0=umxi,0=u˜0xi=u˜0i,0≤i≤N+1,in which(60)bimt=bmxi,t=D3fxi,t,uim−1t,Fimt=Fmxi,t=fxi,t,uim−1t−bimtuim−1t.Replacing the derivatives in spatial variablex of (59) by the following approximations (see [24], pages 36 and 43),(61)uxmxi,t≃uimt−ui−1mth,uxmx0,t≃u1mt−u0mth,uxxmxi,t≃ui−1mt−2uimt+ui+1mth2,Luimt=uxxmxi,t+1xiuxmxi,t≃ui−1mt−2uimt+ui+1mth2+1xiuimt−ui−1mth=1h2−1xihui−1mt+−2h2+1xihuimt+1h2ui+1mt≡aiui−1mt+biuimt+γui+1mt,with(62)ai=1h2−1xih,bi=−2h2+1xih,γ=1h2,2≤i≤N.We get from (59) that(63)u˙imt−αtaiu˙i−1mt+biu˙imt+γu˙i+1mt−μtaiui−1mt+biuimt+γui+1mt+∫0tgt−saiui−1ms+biuims+γui+1msds−bimtuimt=Fimt,1≤i≤N,u1t−u0th−ζu0t=uN+1t=0,ui0=uxi,0=u˜0xi=u˜0i,0≤i≤N.Using the boundary conditions (63)2 with u0mt=u1mt/1+hζ,uN+1t=0 and after eliminating the unknown functions u0t and uN+1t in the first align i=1 and the last align i=N, respectively, then system (63) is rewritten as follows:(64)u˙1mt−αta˜1u˙1mt+γu˙2mt−μta˜1u1mt+γu2mt+∫0tgt−sa˜1u1ms+γu2msds−b1mtu1mt=F1mt,i=1,u˙imt−αtaiu˙i−1mt+biu˙imt+γu˙i+1mt−μtaiui−1mt+biuimt+γui+1mt+∫0tgt−saiui−1ms+biuims+γui+1msds−bimtuimt=Fimt,2≤i≤N−1,u˙Nmt−αtaNu˙N−1mt+bNu˙Nmt−μtaNuN−1mt+buNmt+∫0tgt−saNuN−1ms+buNmsds−bNmtuNmt=FNmt,i=N,where a˜1=a1/1+hζ+b1.Using (64) to rewrite (63) into a vector align as follows:(65)A¯tdu⟶mdtt+Bmtu⟶mt+∫0tgt−sCu⟶msds=F⟶mt,u⟶m0=u1m0,…,uNm0T=u˜0x1,…,u˜0xNT,where(66)u⟶mt=u1mt,…,uNmtT=umx1,t,…,umxN,tT,F⟶mt=F1mt,…,FNmtT,and A¯t,Bmt,C∈MN (MN is the set of real N-order matrices) are defined by(67)C=a˜1γa2b2γa3b3γ⋱⋱⋱⋱⋱⋱aN−1bN−1γaNbN,B˜mt=b1mtb2mt⋱⋱bNmt,A¯t=I−αtC,Bmt=−B˜mt−μtC.Approximating the derivativesdu⟶m/dttj by the differences in time variable and the following partition,(68)du⟶mdttj≃u⟶j+1m−u⟶jmΔt,u⟶jm=u⟶mtj,tj=jΔt,j=0,…,M,Δt=TM,A¯j=A¯tj,Bjm=Bmtj,then the align (65) was rewritten as follows:(69)A¯ju⟶j+1m−u⟶jmΔt+Bjmu⟶jm+∫0tjgtj−sCu⟶msds=F⟶mtj.Note that, the integral∫0tjgtj−sCu⟶msds can be approximated by the trapezoidal formula (see [24], page 56) as follows:(70)∫0tjgtj−sCu⟶msds≃Δtg1Cu⟶0m+g0Cu⟶1m2,j=1,ΔtgjCu⟶0m+g0Cu⟶jm2+∑ν=1j−1gj−νCu⟶νm,2≤j≤M,where gj=gtj, 0≤j≤M.Hence, align (69) can be rewritten as follows:(71)A¯0u⟶1m=A¯0−ΔtB0mu⟶0m+ΔtF⟶mt0≡b¯1m,A¯1u⟶2m=A¯1−ΔtB1m−12Δt2g0Cu⟶1m−12Δt2g1Cu⟶0m+ΔtF⟶mt1≡b¯2m,A¯ju⟶j+1m=A¯j−ΔtBjm−12Δt2g0Cu⟶jm−12Δt2g1Cu⟶0m−Δt2∑ν=1j−1gj−νCu⟶νm+ΔtF⟶mtj≡b¯j+1m,2≤j≤M−1,in which(72)u⟶mt=u1mt,…,uNmtT=umx1,t,…,umxN,tT,u⟶jm=u⟶mtj=umx1,tj,…,umxN,tjT,u⟶0m=u⟶m0=u˜0x1,…,u˜0xNT,F⟶mtj=F1mtj,…,FNmtjT,Fimtj=Fmxi,tj=fxi,tj,uim−1tj−bimtjuim−1tj,bimtj=bmxi,tj=∂f∂uxi,tj,uim−1tj.Description of scheme (71):(A)
LetM,N to be fixed constants. At the first iteration with m=0, we set up the given vector(73)u⟶j0=u⟶0tj=u10tj,…,uN0tjT=u0x1,tj,…,u0xN,tjT≡0,j=1,…,M.(B)
At them−1th iteration, suppose that we have(74)u⟶jm−1=u⟶m−1tj=u1m−1tj,…,uNm−1tjT,j=1,…,M.(C)
Then, the vectorsu⟶jm=u⟶mtj=u1mtj,…,uNmtjT,j=1,…,M can be computed consecutively by the following steps.(C1)
The computation ofu⟶1m=u1mt1,…,uNmt1T.(i)
With the first given vectoru⟶0m=u˜0x1,…,u˜0xNT, we calculate the matrices(75)A¯0=A¯t0=I−αt0C,Bmt0=−B˜mt0−μt0C,and the vectors(76)F⟶mt0=F1mt0,…,FNmt0T,b¯1m=A¯0−ΔtB0mu⟶0m+ΔtF⟶mt0.(ii)
Finding the vectoru⟶1m=u1mt1,…,uNmt1T by solving the following system(77)A¯0u⟶1m=b¯1m.(C2)
The computation ofu⟶2m=u1mt2,…,uNmt2T.(i)
Calculating the matrices(78)A¯1=A¯t1=I−αt1C,Bmt1=−B˜mt1−μt1C,and the vectors(79)F⟶mt1=F1mt1,…,FNmt1T,b¯2m=A¯1−ΔtB1m−12Δt2g0Cu⟶1m−12Δt2g1Cu⟶0m+ΔtF⟶mt1.(ii)
Finding the vectoru⟶2m=u1mt2,…,uNmt2T by solving the following system(80)A¯1u⟶2m=b¯2m.(C3)
The computation ofu⟶j+1m=u1mtj+1,…,uNmtj+1T. Suppose that the vectors u⟶1m, u⟶2m,…, u⟶jm are calculated; we determine the vector u⟶j+1m=u1mtj+1,…,uNmtj+1T by recurrence as follows:(i)
Calculating the matrices(81)A¯j=A¯tj=I−αtjC,Bmtj=−B˜mtj−μtjC,and the vectors(82)F⟶mtj=F1mtj,…,FNmtjT,b¯j+1m=A¯j−ΔtBjm−12Δt2g0Cu⟶jm−12Δt2g1Cu⟶0m−Δt2∑ν=1j−1gj−νCu⟶νm+ΔtF⟶mtj.(ii)
Finding the vectoru⟶j+1m=u1mtj+1,…,uNmtj+1T by solving the following system(83)A¯ju⟶j+1m=b¯j+1m.When the process of computation is reached toj=M−1, we get(84)u⟶jm=umx1,tj,…,umxN,tjT,1≤j≤M.(C4)
The error of two consecutive steps of the iteration, at themth step and at the m−1th step, is defined as follows:(85)um−um−1M,N=max1≤j≤Mmax1≤i≤Numxi,tj−um−1xi,tj.The process of the iteration will be stopped at themth step when the following estimate is satisfied:(86)um−um−1M,N<10−4.(C5)
The error of the approximate solution (at themth step) and the exact solution is defined by(87)EM,N=um−uexM,N=max1≤j≤Mmax1≤i≤Numxi,tj−uexxi,tj,whereuexx,t is the exact solution. Next, we present an illustrated example and the corresponding numerical results in order to show that the convergence rate of the 2-order iterative scheme is faster than that of the single-iterative scheme (which is schemes (32)–(34) with ℱmx,t=fx,t,um−1x,t, as in Remark 1).For example, we consider problems (1)–(3) with(88)fx,t,u=ρup−2u+Gx,t,μt=μ∗+e−μ¯t,μ∗>0,μ¯>0,αt=α∗+e−α¯t,α∗>0,α¯>0,gt=gmaxe−g¯t,gmax>0,0<g¯≠1,Gx,t=1+ρx2−52x+1p−2e−p−2tx2−52x+1e−t−4−52xα∗−μ∗+e−α¯t−e−μ¯t+gmaxg¯−11−e−g¯−1te−t,u˜0x=−x2+52x−1.Then,uexx,t=−x2+5/2x−1e−t is the exact solution of problems (1)–(3) corresponding to the constants ζ=1,R=2,p=4, ρ=10−2, α∗=μ∗=α¯=μ¯=1, and g¯=2 and the given functions fx,t,u, μt, αt, and gt as in (88).In caseℱmx,t=fx,t,um−1x,t, it means that bmx,t=D3fx,t,um−1x,t=0, and then the approximate scheme (8) has the form(89)u0≡0,u˙mt−μtLum−αtLu˙mt+∫0tgt−sLumsds=Fmx,t,1<x<R,0<t<T,uxm1,t−h1um1,t=umR,t=0,umx,0=u˜0x,where(90)Fmx,t=fx,t,um−1x,t.Then, the matrix(91)Bjm=Bmtj=−μtjC≡Btj≡Bjis independent of m. In this case, scheme (71) leads to the following approximate scheme, which is also called a single-iterative scheme(92)A¯0u⟶1m=A¯0−ΔtB0u⟶0m+ΔtF⟶mt0≡b¯1m,A¯1u⟶2m=A¯1−ΔtB1−12Δt2g0Cu⟶1m−12Δt2g1Cu⟶0m+ΔtF⟶mt1≡b¯2m,A¯ju⟶j+1m=A¯j−ΔtBj−12Δt2g0Cu⟶jm−12Δt2g1Cu⟶0m−Δt2∑ν=1j−1gj−νCu⟶νm+ΔtF⟶mt1≡b¯j+1m,2≤j≤M−1,where F⟶mtj is determined by(93)F⟶mtj=F1mtj,…,FNmtjT,Fimtj=Fmxi,tj=fxi,tj,uim−1tj.With the datum as in (88) and the error as in (86), the corresponding numerical results given by the single-iterative (92) and (93) and the 2-order iterative scheme are presented in Tables 1–5 and Figures 1–3.Table 1
Errors of the approximate solution and the exact solution.
Single-iterative scheme2-order iterative schemeNMEM,N=um−uexM,NEM,N=um−uexM,N10200.0035749012028080.00357489618204620400.0016756752669700.00167567044279130600.0010926047546700.00109259989879840808.098476439145630×10−48.098428422048665×10−4501006.434079705471274×10−46.434032161508618×10−4Table 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,Nm=um−uexM,NEM,Nm=um−uexM,N10.0036093354834680.00360933548346820.0035749012028080.00357489618204630.0035748961821220.00357489618168340.0035748961816840.00357489618168350.0035748961816840.00357489618168360.0035748961816840.00357489618168370.0035748961816840.00357489618168380.0035748961816840.00357489618168390.0035748961816840.003574896181683100.0035748961816840.003574896181683Table 3
Errors of two consecutive steps in each iteration, withN=10 and M=20.
Number of iterationsSingle-iterative scheme2-order iterative schememDM,Nm=um−um−1M,NDM,Nm=um−um−1M,N10.5619834710743800.56198347107438023.700981639570555×10−53.701583173776335×10−536.797891999310579×10−96.646627692674656×10−1348.078537838684952×10−133.330669073875470×10−1657.771561172376096×10−160.000000000000000060.00000000000000000.0000000000000000Table 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,N16.765921811121567×10−46.765921811121567×10−426.434079705471274×10−46.434032161508618×10−436.434032163748493×10−46.434032158795233×10−446.434032159262637×10−46.434032158795233×10−456.434032159262637×10−46.434032158795233×10−466.434032159262637×10−46.434032158795233×10−476.434032159262637×10−46.434032158795233×10−486.434032159262637×10−46.434032158795233×10−496.434032159262637×10−46.434032158795233×10−4106.434032159262637×10−46.434032158795233×10−4Table 5
Errors of two consecutive steps in each iteration, withN=50 and M=100.
Number of iterationsSingle-iterative scheme2-order iterative schememDM,Nm=um−um−1M,NDM,Nm=um−um−1M,N10.5624759707804690.56247597078046923.613601710050762×10−53.614203006274685×10−536.811192498901164×10−96.195877144676842×10−1348.621436897726653×10−131.337818744673314×10−1451.412758798835512×10−140.000000000000000060.00000000000000000.0000000000000000Figure 1
The surface of the finite-difference approximate solution ofumx,t defined by the 2-order iterative scheme (8), with respect to equation (71) and the grid of N=50 and M=100 as mentioned above.Figure 2
The surface of the finite-difference approximate solution ofumx,t defined by the single-iterative scheme (89), with respect to equation (92) and the grid of N=50 and M=100 as mentioned above.Figure 3
The exact solution of problems (1)–(3) given by uexx,t=−x2+5/2x−1e−t, with the input datum as in (88).According to the numerical results in Table1, we can see that two iterative schemes used here are effective. Indeed,(i)
The third column of Table1 shows that the errors of the approximate solution and the exact solution given by the single-iterative schemes (92) and (93) are decreased when N and M are increased.(ii)
The fourth column of Table1 shows that the errors of the approximate solution and the exact solution given by the 2-order scheme (71) are also decreased when N and M are increased.(iii)
The errors in the fourth column are less than these of the third column with the same gridN,M, respectively.To compare the convergent speed of the single-iterative scheme and of 2-order iterative scheme, we establish the errors as in Tables2–5. For more details, it is as follows.First, withM and N fixed, we put the following errors.(i)
The error of the approximate solution (at themth step) and the exact solution:(94)EM,Nm=um−uexM,N=max1≤j≤Mmax1≤i≤Numxi,tj−uexxi,tj.(ii)
The error of two consecutive steps of the iteration, at themth step and at the m−1th step:(95)DM,Nm=um−um−1M,N=max1≤j≤Mmax1≤i≤Numxi,tj−um−1xi,tj.Next, we compute the errors as in Tables2–5. The errors EM,Nm of two iterative schemes according to the iterative steps are given in Tables 2 and 4, where the grids are considered with N=10, M=20 and N=50, M=100 respectively. Similarly, the errors DM,Nm of two iterative schemes are given in Tables 3 and 5.According to the numerical results in Tables2–5, we have the following:(i)
The values of the errorEM,Nm given in the columns 2 and 3 of Table 4 are decreased when the iterative steps are increased from 1 to 10. It is reasonable by the fact that both schemes are convergent.(ii)
The values of the third column are less than these of the second column, line by line, in Tables2 and 4. This shows that the convergent speed of 2-order iterative scheme is faster than that of the single-iterative scheme. It is similar to the errors DM,Nm of two schemes given in Tables 3 and 5.Finally, we have drawn the approximated solutions and the exact solution of problems (1)–(3) with the datum as in (88).


## 4. Conclusion

This paper has proved the solvability of problems (1)–(3) for a nonlinear pseudoparabolic align with Robin-Dirichlet conditions by establishing an approximate sequence um based on a high-order iterative scheme or a single-iterative scheme. The proposed schemes are tested on an example in which a standard finite-difference scheme is used suitably. The numerical results obtained here show that the convergence rate of the 2-order iterative scheme is faster than that of the single-iterative scheme. Because of the efficient convergence rate, the present high-order iterative scheme offers a good alternative to find a solution of nonlinear problems for partial differential aligns.



---

*Source: 8886184-2021-03-20.xml*