A boundary value problem is posed for an integro-differential beam equation. An approximate solution is found using the Galerkin method and the Jacobi nonlinear iteration process. A theorem on the algorithm error is proved.

1. Introduction1.1. Statement of the Problem

We consider the equation
uıv(x)-(α+β∫0L(u′(ξ))2dξ)u′′(x)=f(x),0<x<L,
with the conditions
u(0)=u(L)=0,u′′(0)=u′′(L)=0.
Here α, β, and L are some positive constants, f(x) is a given function, and u(x) is the function we want to define.

1.2. Background of the Problem

Equation (1.1) is the stationary problem associated with
utt+EIρAuxxxx-(Hρ+E2ρL∫0Lux2dx)uxx=0
which was proposed by Woinowsky-Krieger [1] as a model for the deflection of an extensible beam with hinged ends. Here H, E, ρ, I, A, and L denote, respectively, the tension at rest, Young's elasticity modulus, density, cross-sectional moment of inertia, cross-section area and length of the beam. The nonlinear term in brackets is the correction to the classical Euler-Bernoulli equation
utt+EIρAuxxxx=0,
where tension changes induced by the vibration of the beam during deflection are not taken into account. This nonlinear term was for the first time proposed by Kirchhoff [2] who generalized D'Alembert's classical model. Therefore (1.3) is often called a Kirchhoff-type equation for a dynamic beam. Note that Arosio [3] calls the function of the integral ∫0Lux2dx the Kirchhoff correction (briefly, the K-correction) and makes a reasonable statement that the K-correction is inherent in a lot of physical phenomena.

The works dealing with the mathematical aspects of (1.3) and its generalization
utt+uxxxx-M(∫0Lux2dx)uxx=f(x,t,u),M(λ)≥const>0,
as well as some modifications of (1.3) and (1.5) belong to Ball [4, 5], Biler [6], Henriques de Brito [7], Dickey [8], B.-Z. Guo and W. Guo [9], Kouémou-Patcheu [10], Medeiros [11], Menezes et al. [12], Panizzi [13], Pereira [14], and to others. The subject of investigation concerned the questions of the existence and uniqueness of a solution [4, 5, 9–14], its asymptotic behavior [6–8, 10], stabilization and control problems [9], and so on.

As to the static Kirchhoff-type equation for a beam, Its more general form than (1.1), namely,
uıv-m(∫0Lu′2dx)u′′=f(x,u),m(λ)≥const>0,
was considered in Ma [15, 16], where the solvability under nonlinear boundary conditions is studied.

The topic of approximate solution of Kirchhoff equations, which the present paper is concerned with, was treated by Choo and Chung [17], Choo et al. [18], Clark et al. [19], and Geveci and Christie [20] for a dynamic beam, while Ma [16] and Tsai [21] studied the problem for the static case. Speaking more exactly, the finite difference and finite element Galerkin approximate solutions are investigated and the corresponding error estimates are derived in [17, 18]. Numerical analysis of solutions for a beam with moving boundary is carried out in [19]. The question of the stability and convergence of a semidiscrete and fully discrete Galerkin approximation is dealt with in [20]. To solve the problem with nonlinear boundary conditions, Ma [16] applies the difference method and the Gauss-Seidel iteration process. Finally, in [21] for the discretization of the problem, in particular the finite difference, finite element and spectral methods are used, while nonlinear systems of equations are solved by the Newton iteration and other methods.

In the present paper, a numerical algorithm is constructed and its total error estimated for (1.1). Formulas are given allowing us to calculate the upper bound of the error by using the initial data of the problem. The algorithm includes the Galerkin approximation reducing the problem to a system of cubic algebraic equations which are solved by means of the nonlinear Jacobi iteration process. We also use the Cardano formula due to which the current iteration approximation is expressed through the already found approximation in explicit form.

1.3. Assumptions

Let for each i=1,2,… there exists an integral
fi=2L∫0Lf(x)siniπxLdx,
and let the inequality
|fi|≤ωim,i=1,2,…
be fulfilled with ω and m being some known positive constants.

Assume that there exists a solution of problem (1.1)-(1.2) representable as a series
u(x)=∑i=1∞uisiniπxL,
whose coefficients satisfy the system of equations
(iπL)4ui+(iπL)2(α+βL2∑j=1∞(jπL)2uj2)ui=fi,i=1,2,….

2. The Algorithm2.1. Galerkin Method

An approximate solution of problem (1.1)-(1.2) will be sought for in the form of a finite series
un(x)=∑i=1nunisiniπxL,
where the coefficient uni is defined by the Galerkin method from the system
(iπL)4uni+(iπL)2(α+βL2∑j=1n(jπL)2unj2)uni=fi,i=1,2,…,n.

Here, incidentally, note that vast literature is available (e.g., see [22–25]) on the application of the Galerkin method to differential equations of second and fourth order.

2.2. Jacobi Iteration Process

To solve the nonlinear system (2.2) we use the Jacobi iteration process [26]
(iπL)4uni,k+1+(iπL)2[α+βL2((iπL)2uni,k+12+∑j=1j≠in(jπL)2unj,k2)]uni,k+1=fi,k=0,1,…,i=1,2,…,n,
where uni,k+l denotes the (k+l)th iteration approximation of uni, l=0,1.

For fixed i, (2.3) is a cubic equation with respect to (iπ/L)uni,k+1 (here uni,k+1 is taken with weight iπ/L just for convenience). Using the Cardano formula [27], we express (iπ/L)uni,k+1 through the kth iteration approximation
iπLuni,k+1=σi1,k-σi2,k,
where
σip,k=[(-1)psi+(si2+ri,k3)1/2]1/3,ri,k=13[2βL(α+(iπL)2)+∑j=1j≠in(jπL)2unj,k2],si=-1βiπfi,k=0,1,…,i=1,2,…,n.

The algorithm we have considered should be understood as the counting carried out by formula (2.4). Having uni,k, i=1,2,…,n, we construct the approximate solution of the problem
un,k(x)=∑i=1nuni,ksiniπxL.

2.3. Algorithm Error Definition

Let us compare the approximate solution (2.6) with the nth truncation of the exact solution (1.9)
pnu(x)=∑i=1nuisiniπxL.
This means that the algorithm error is defined as a difference
pnu(x)-un,k(x)
which we write as a sum
pnu(x)-un,k(x)=Δun(x)+Δun,k(x),
where Δun(x) is the Galerkin method error and Δun,k(x) the Jacobi process error which are equal, respectively, to
Δun(x)=pnu(x)-un(x),Δun,k(x)=un(x)-un,k(x).

3. The Algorithm Error

We set ourselves the task of estimating the L2(0,L)-norm of the algorithm error. For this we have to estimate the errors of the Galerkin method and the Jacobi process.

3.1. Galerkin Method Error

Let us expand Δun(x) into a series. Taking (2.10), (2.7), and (2.1) into account we write
Δun(x)=∑i=1nΔunisiniπxL,
where
Δuni=ui-uni,i=1,2,…,n.
By virtue of (3.1) we have
∥Δun(x)∥L2(0,L)=(L2∑i=1n(Δuni)2)1/2.
We will come back to (3.3) later, while now we denote
γln=(2-l)(iπL)4+12(iπL)2[α+βL2∑j=1n(jπL)2uj2+(-1)l+1(α+βL2∑j=1n(jπL)2unj2)],εn=12βL(iπL)2∑j=n+1∞(jπL)2uj2,∇n=14βL(iπL)2∑j=1n(jπL)2(uj+unj)Δunj,
and rewrite (1.10) and (2.2) in the form (γ1n+γ2n+εn)ui=fi and (γ1n-γ2n)uni=fi. Since by virtue of (3.4), (3.2), and (3.6) we have γ2n=∇n and therefore (γ1n+∇n+εn)ui=fi and (γ1n-∇n)uni=fi. Subtracting the last two equalities from each other and taking (3.2) into account, we obtain γ1nΔuni+∇n(ui+uni)+εnui=0 which we multiply by Δuni and sum over i=1,2,…,n. Using (3.4), (3.5), and the inequality ∑i=1n∇n(ui+uni)Δuni≥0 following from (3.6), we see that
∑i=1n(α+(iπL)2)(iπL)2(Δuni)2≤12βL∑i=1n(iπL)2|uiΔuni|∑i=n+1∞(iπL)2ui2.

By the Cauchy-Bunyakowsky-Schwarz inequality, we therefore have
(∑i=1n(α+(iπL)2)(iπL)2(Δuni)2)1/2≤12βL(∑i=1nui2)1/2∑i=n+1∞(iπL)2ui2.

Let us estimate the right-hand side of inequality (3.8). After multiplying (1.10) by (iπ/L)tui and summing the resulting relation over i=1,2,…,n in one case and over i=n+1,n+2,… in the other, we come to the formula common for both cases
∑i=vw(iπL)4+tui2+(α+βL2∑j=1∞(jπL)2uj2)∑i=vw(iπL)2+tui2=∑i=vw(iπL)tfiui,
where v=1, w=n or v=n+1, w=∞. Thus
(α+βL2∑j=1∞(jπL)2uj2)∑i=vw(iπL)2+tui2≤14∑i=vw(iπL)t-4fi2.

Let us put v=1, w=n, t=-2 in (3.10) and use the fact that ∑j=1∞(jπ/L)2uj2≥(π2/L2)∑j=1nuj2. We obtain
∑i=1nui2≤an,
where
an=Lπ2β[(α2+12Lπ2β∑i=1n(Liπ)6fi2)1/2-α].

Now assuming v=n+1, w=∞, t=0 in (3.10) and using in addition to this the inequality ∑j=1∞(jπ/L)2uj2≥∑j=n+1∞(jπ/L)uj2, we get
∑i=n+1∞(iπL)ui2≤bn,
where
bn=1βL[(α2+12βL∑i=n+1∞(Liπ)4fi2)1/2-α].

The use of (3.11) and (3.13) in (3.8) brings us to the inequality
(∑i=1n(α+(iπL)2)(iπL)2(Δuni)2)1/2≤12βL(an)1/2bn
which together with (3.3) gives
∥Δun(x)∥L2(0,L)≤12πβL2(L2anα+(π/L)2)1/2bn.

Let us substitute (3.12) and (3.14) into (3.16) and apply condition (1.8) and also the integral test for series convergence. As a result, if n>1, for the Galerkin method error we obtain the estimate
∥Δun(x)∥L2(0,L)≤c0[(1+c11n2m+3)1/2-1]{[1+c2(1+c3(1-1n2m+5))]1/2-1}1/2,
where the coefficients c0, c1, c2, and c3 do not depend on n and are defined by
c0=α2(Lπ)2(12β(1+(1/α)(π/L)2))1/2,cl=βLω22α2(1-2(l-2)(m+1))(Lπ)4,l=1,2,c3=12m+5.

3.2. Jacobi Process Error

Taking (2.10), (2.1), and (2.6) into account, we represent Δun,k(x) as a series
Δun,k(x)=∑i=1nΔuni,ksiniπxL,
where
Δuni,k=uni-uni,k,i=1,2,…,n.

Series (3.19) implies the formula
∥Δun,k(x)∥L2(0,L)=(L2∑i=1n(Δuni,k)2)1/2
to be used later.

Let us rewrite (2.4) in the form
iπLuni,k+1=φi(πLun1,k,2πLun2,k,…,nπLunn,k),
and introduce into consideration the Jacobian
J=(∂φi∂((jπ/L)unj,k))i,j=1n
(in this paper this is the second notion associated with the name of C. Jacobi, 1804–1851).

To establish the convergence condition for process (3.22) we have to estimate the norm of the matrix J. By virtue of (2.4), (2.9), and (3.22) there are zeros on the principal diagonal of this matrix,
∂φi∂((iπ/L)uni,k)=0.
As to the nondiagonal elements, i≠j, they are defined by the formula
∂φi∂((jπ/L)unj,k)=13ri,k2(si2+ri,k3)-1/2(σi1,k-2-σi2,k-2)jπLunj,k.
Using the relations
σi1,kσi2,k=ri,k,(si2+ri,k3)1/2=12(σi1,k3+σi2,k3),σi2,k3-σi1,k3=2si,
which follow from (2.5), we rewrite (3.25) as the equality
∂φi∂((jπ/L)unj,k)=-43si(σi1,k4+ri,k2+σi2,k4)-1jπLunj,k.
Apply to the latter equality the estimate σi1,k4+σi2,k4≥2ri,k2, which is obtained from the first relation in (3.26) and (2.5). Also use the fact that the maximal value of the function z(x)=x(a2+x2)-2, x≥0, is equal to (1/16)(3/a2)3/2. Thus we obtain the inequalities
|∂φi∂((jπ/L)unj,k)|≤49|si|ri,k2jπL|unj,k|≤4|fi|βiπ(2βL(α+(iπL)2)+(jπLunj,k)2)-2jπL|unj,k|≤38(32βL)1/2|fi|(iπ/L)(α+(iπ/L)2)3/2,i≠j,
which are fulfilled for the nondiagonal elements of the matrix J.

Let us use the vector and matrix norms equal, respectively, to ∑i=1n|vi| and max1≤j≤n∑i=1n|mij| for the vector v=(vi)i=1n and the matrix M=(mij)i,j=1n. Assume that for an arbitrary set of values unj,k, j=1,2,…,n, k=0,1,…, the elements of the matrix J satisfy the condition max1≤j≤n∑i=1n|∂φi/∂((jπ/L)unj,k)|≤q<1. For this, by virtue of (3.28), (3.24), and (1.8) it is sufficient that
38πLω(32βL)1/2∑i=1i≠jn1im+1(α+(iπ/L)2)3/2≤q<1,j=1,2,…,n.
Then, according to the map compression principle, the system of (2.2) has a unique solution uni, i=1,2,…,n, the iteration process (2.4) converges, limk→∞uni,k=uni, i=1,2,…,n, with the rate which in view of notation (3.20) is defined by the inequality ∑i=1ni|Δuni,k|≤(qk/(1-q))∑i=1ni|uni,1-uni,0|. From this and (3.21) we obtain the estimate for the Jacobi process error
∥Δun,k(x)∥L2(0,L)≤qk1-q(L2)1/2∑i=1ni|uni,1-uni,0|,k=0,1,….

To conclude this section, we would like to touch upon one auxiliary question. Let us see how condition (3.29) will change if we apply to it the integral test for the convergence of series and ignore i≠j under the summation sign. Besides, we restrict ourselves to the case where m is an integer number and apply the inequality α1/2+iπ/L≤[2(α+(iπ/L)2)]1/2. Then using the formula for the integral ∫dx/xm+1(a+bx)3, a,b>0, [28] instead of (3.29), we obtain
34ω(3βL)1/2{L2π2(α+(π/L)2)3/2+1α2(πLα)m-1(m+2)!×∑l=0m+2(-1)l1l!(m-l+2)!(m-l)[(1+Lαπ)m-l-(1+Lαπn)m-l]L2π2(α+(π/L)2)3/2}≤q<1.

3.3. Algorithm Error

Let us estimate error (2.8). By (2.9) we have
∥pnu(x)-un,k(x)∥L2(0,L)≤∥Δun(x)∥L2(0,L)+∥Δun,k(x)∥L2(0,L),
and therefore the application of (3.17) and (3.30) gives the inequality
∥pnu(x)-un,k(x)∥L2(0,L)≤c0[(1+c11n2m+3)1/2-1]{[1+c2(1+c3(1-1n2m+5))]1/2-1}1/2+qk1-q(L2)1/2∑i=1ni|uni,1-uni,0|.

The obtained result can be summarized as follows.

Theorem 3.1.

Let n>1 and q be some number from the interval (0,1). Assume that the conditions of Section 1.3 and restriction (3.29) (or (3.31) in the case of integer m) are fulfilled. Then the algorithm error is estimated by inequality (3.33), where the coefficients c0, c1, c2, and c3 are calculated by formulas (3.18).

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