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

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Introduction
In this paper, we consider the following initial-boundary problem: u(x, 0) � u 0 (x), where R > 1 and ζ ≥ 0 are given constants and μ(t), α(t), f, g, and u 0 are given functions satisfying conditions specified later, with u � u(x, 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][2][3][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][6][7][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: where w(r, t) is the velocity along the z-axis, ] 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][14][15][16][17][18][19] and the references therein. In [15], Dai and Huang studied the solvability and the well-posedness of solutions for the nonlinear pseudoparabolic equation with the nonlocal moment boundary conditions β α u(x, t)dx � β α xu(x, t)dx � 0, 0 ≤ t ≤ T. In [17], Sun et al. considered the Dirichlet problem for the nonlinear pseudoparabolic equation with a memory term as follows: on zΩ ×(0, T), where Ω is a bounded domain of R n (n ≥ 1) with smooth boundary zΩ, p > 2, T ∈ (0, ∞], u 0 ∈ H 1 (Ω) and g: R + ⟶ R + is a positive nonincreasing function. e 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.
is paper consists of two main parts. In Part 1, by using the N-order iterative method, Faedo-Galerkin method, and compact method, we prove existence and uniqueness of a weak solution of problems (1)-(3) (see eorem 2). We begin with the establishment of the N-order nonlinear approximate sequence u (m) in case of f ∈ C N ([0, 1] × [0, T * ] × R) via the N-order iterative scheme associated with problems (1)-(3) as follows: and next, we prove that u (m) converges to the unique solution u of problems (1)-(3) at a rate of order N(N ≥ 2); it means that ‖u (m) − u‖ X ≤ C‖u (m− 1) − u‖ N X , 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: , 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 u (m) via a singleiterative 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][21][22][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 approximate u (m) in the 2-order iterative scheme (8). In order to do this, we shall use a simple finitedifference scheme which is a standard model given in [24]. We first use the uniform partition x i � ih, h � (1/N 0 ), i � 0, 1, . . ., N 0 , and the forward difference formulas (see [24], pages 36 and 43) to approximate the k th derivatives, k � 1, 2, in spatial variable, as follows: is is also a technique used in [20,[25][26][27][28][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: where A(t), B (m) (t) are functional matrices depending on a time variable t and C ∈ M N 0 (M N 0 is the set of real N 0 -size matrices). Next, we make discretizations in time variables t j � jΔt, Δt � (T/M), and j � 0, 1, . . . , M and approximate the integral t 0 g(t − s)C u →(m) (s)ds by the trapezoidal formula (see [24], page 56), and we remark that this technique was also used in [26,27,30]. en, we obtain the following algorithm to determine the finite-difference approximate solutions of u (m) given by the 2-order iterative scheme formula (71) where . Similarly, we have constructed the algorithm to find the finite-difference approximate solutions of u (m) 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 initialboundary value problem for the pseudoparabolic equation with delay where Q � (0, l) × (0, T], r represents the delay parameter, and f(x, t, u(x, t), u(x, 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. e 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 Mathematical Problems in Engineering presented. e 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 where e 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: with nonlocal integral boundary conditions and initial condition where f, φ, μ i , and i � 1, 2, 3, 4 are given functions and η, c 1 , and c 2 > 0 are given constants. ey decomposed problem (15) into two locally one-dimensional problems from layer t � t n to layer t � t n+1 as follows: and next, they changed equation (18) to the following onedimensional difference schemes: . ey proved the difference equation (19) approximating the differential equation (18) with the truncation error O(h 2 + τ). Moreover, if c 1 + c 2 < 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: where F: R n ⟶ R n is a regular map. e backward Euler scheme applied to the above equation generates the iterations and the nonlinear term F(u (k+1) ) is approximated by Consequently, the following difference equation is established: 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 wellknown 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. ey 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. e 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.

Existence and Uniqueness
roughout this paper, we set Ω � (1, R) and use L 2 � L 2 (Ω) to denote the Lebesgue space with the inner product defined by . Moreover, we also introduce three weighted scalar products and then, L 2 , H 1 , and H 2 are the Hilbert spaces with respect to the abovementioned scalar products. We denote e symmetric bilinear form a(·, ·) is defined by with ζ ≥ 0 being a given constant and ‖v‖ a � ����� � a(v, v). en, we have the following lemmas.
Lemma 2. e symmetric bilinear form a(·, ·) is continuous on V × V and coercive on V, i.e., there exist two positive constants C 0 and C 1 such that e notation ‖ · ‖ X is the norm in the Banach space X, and X ′ is the dual space of X. We denote by L p (0, T; X), For a fixed constant T * > 0, we make the following assumptions: and u satisfies the following variational equation: Mathematical Problems in Engineering Now, we construct the recurrent sequence u (m) defined by u (0) ≡ 0, and suppose that en, u (m) is found by the fact that u (m) ∈ B T (M), m ≥ 1, and u (m) satisfies where Using the standard Faedo-Galerkin method, which is introduced by Lions in [44], we can prove the following theorem.
By using eorem 1 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 then W 1 (T) is a Banach space with respect to the norm (see [44]) for all m ≥ 1, where C is a suitable constant. On the other hand, the following estimate is fulfilled: where C T and 0 < k T < 1 are the constants depending only on T.
Proof. We shall prove that u (m) is a Cauchy sequence in where F (m) (x, t) is defined by (34). (38), after integrating in t, we have Next, we have to estimate the integrals on the right-hand side of (40). We Using the inequality )ds], with β * � min μ * , α * , the integrals J 1 and J 2 are estimated as follows: Using Taylor's expansion of the functions f( where erefore, we have Mathematical Problems in Engineering Since the above inequality, the integral J 3 can be estimated by By (39) and (42), it follows from (47) that where c 3 (M) � 4c 2 Using Gronwall's Lemma, we deduce from (48) that where Choosing T > 0 small enough such that k T � M(μ T ) (1/(N− 1)) < 1, it follows from (49) that, for all m and p, e above inequality ensures that u (m) is a Cauchy sequence in W 1 (T). en, there exists u ∈ W 1 (T) such that u (m) ⟶ u strongly in W 1 (T). (51) Note that u (m) ∈ B T (M), then there exists a subsequence u (m j ) of u (m) such that We note that Hence, erefore, we deduce from (51) and (54) that Letting m � m j ⟶ ∞ in (33) and (34) and using (51, 52, and 55), it implies that there exists u ∈ B T (M) satisfying (31). e 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). eorem 2 is proved completely.  t)). en, the assumption for f is weakened as follows: We note more that the scheme obtained here is called the single-iterative scheme.

Numerical Scheme
In this section, we first construct a difference scheme to approximate the solution u of problems (1)-(3) via approximating u (m) in the 2-order iterative scheme (8). It is implied from eorems 1 and 2 that, u (m) is definite by problems (33) and (34) with the nonlinear term F (m) (x, t) given on right-hand side of (33) as follows: Putting (58) (60) Replacing the derivatives in spatial variable x of (59) by the following approximations (see [24], pages 36 and 43), with We get from (59) that
(C1) e computation of u →(m) and the vectors ) T by solving the following system (ii) Finding the vector u →(m) 2 � (u (m) 1 (t 2 ), . . . , u (m) N (t 2 )) T by solving the following system  t j+1 ), . . . , u (m) N (t j+1 )) T by recurrence as follows: (i) Calculating the matrices (82) ) T by solving the following system When the process of computation is reached to (C4) e error of two consecutive steps of the iteration, at the m th step and at the (m − 1) th step, is defined as follows: e process of the iteration will be stopped at the m th step when the following estimate is satisfied: (C5) e error of the approximate solution (at the m th step) and the exact solution is defined by where u ex (x, 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 2order iterative scheme is faster than that of the singleiterative scheme (which is schemes (32)- (34) with t)), as in Remark 1).
In case t)) � 0, and then the approximate scheme (8) has the form en, the matrix is independent of m. In this case, scheme (71) leads to the following approximate scheme, which is also called a singleiterative scheme where 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.
According to the numerical results in Table 1, we can see that two iterative schemes used here are effective. Indeed, (i) e third column of Table 1 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) e fourth column of Table 1 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) e errors in the fourth column are less than these of the third column with the same grid (N, M), respectively.
To compare the convergent speed of the single-iterative scheme and of 2-order iterative scheme, we establish the errors as in Tables 2-5. For more details, it is as follows.
First, with M and N fixed, we put the following errors.           Tables 3 and 5.
According to the numerical results in Tables 2-5, we have the following: (i) e values of the error E (m) M,N 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) e values of the third column are less than these of the second column, line by line, in Tables 2 and 4. is 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 D (m) M,N 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).

Conclusion
is paper has proved the solvability of problems (1)-(3) for a nonlinear pseudoparabolic align with Robin-Dirichlet conditions by establishing an approximate sequence u (m) based on a high-order iterative scheme or a single-iterative scheme. e proposed schemes are tested on an example in which a standard finite-difference scheme is used suitably. e 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.

Data Availability
Research data used in this study are available from the references.