A Numerical Algorithm for a Kirchhoff-Type Nonlinear Static Beam

Abstract

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. Introduction

1.1. Statement of the Problem

We consider the equation 𝑢^𝚤v(𝑥)−(𝛼+𝛽^𝐿₀(𝑢^′(𝜉))²𝑑𝜉)𝑢^′′(𝑥)=𝑓(𝑥),0<𝑥<𝐿, (1.1) with the conditions 𝑢(0)=𝑢(𝐿)=0,𝑢^′′(0)=𝑢^′′(𝐿)=0. (1.2) 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 𝑢_𝑡𝑡+𝐸𝐼𝑢𝜌𝐴^{𝑥𝑥𝑥𝑥}𝐻−(𝜌+𝐸2𝜌𝐿^𝐿₀𝑢²_𝑥𝑑𝑥)𝑢_𝑥𝑥=0 (1.3) 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 𝑢_𝑡𝑡+𝐸𝐼𝑢𝜌𝐴^{𝑥𝑥𝑥𝑥}=0, (1.4) 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 ∫^𝐿₀𝑢²_𝑥𝑑𝑥 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 𝑢_𝑡𝑡+𝑢_{𝑥𝑥𝑥𝑥}−𝑀(^𝐿₀𝑢²_𝑥𝑑𝑥)𝑢_𝑥𝑥=𝑓(𝑥,𝑡,𝑢),𝑀(𝜆)≥const>0, (1.5) 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, 𝑢^𝚤v−𝑚(^𝐿₀𝑢^′2𝑑𝑥)𝑢^′′=𝑓(𝑥,𝑢),𝑚(𝜆)≥const>0, (1.6) 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 𝑓_𝑖=2𝐿^𝐿₀𝑓(𝑥)sin𝑖𝜋𝑥𝐿𝑑𝑥, (1.7) and let the inequality |𝑓_𝑖𝜔|≤𝑖^𝑚,𝑖=1,2,… (1.8) 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=1𝑢_𝑖sin𝑖𝜋𝑥𝐿, (1.9) whose coefficients satisfy the system of equations (𝑖𝜋𝐿)⁴𝑢_𝑖+(𝑖𝜋𝐿)²(𝛼+𝛽𝐿2^∞_𝑗=1(𝑗𝜋𝐿)²𝑢²_𝑗)𝑢_𝑖=𝑓_𝑖,𝑖=1,2,…. (1.10)

2. The Algorithm

2.1. Galerkin Method

An approximate solution of problem (1.1)-(1.2) will be sought for in the form of a finite series 𝑢_𝑛(𝑥)=^𝑛_𝑖=1𝑢_𝑛𝑖sin𝑖𝜋𝑥𝐿, (2.1) where the coefficient 𝑢_𝑛𝑖 is defined by the Galerkin method from the system (𝑖𝜋𝐿)⁴𝑢_𝑛𝑖+(𝑖𝜋𝐿)²(𝛼+𝛽𝐿2^𝑛_𝑗=1(𝑗𝜋𝐿)²𝑢²_𝑛𝑗)𝑢_𝑛𝑖=𝑓_𝑖,𝑖=1,2,…,𝑛. (2.2)

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)⁴𝑢_{𝑛𝑖,𝑘+1}+(𝑖𝜋𝐿)²[𝛼+𝛽𝐿2((𝑖𝜋𝐿)²𝑢²_{𝑛𝑖,𝑘+1}+^𝑛_{𝑗=1𝑗≠𝑖}(𝑗𝜋𝐿)²𝑢²_{𝑛𝑗,𝑘})]𝑢_{𝑛𝑖,𝑘+1}=𝑓_𝑖,𝑘=0,1,…,𝑖=1,2,…,𝑛, (2.3) where 𝑢_{𝑛𝑖,𝑘+𝑙} denotes the (𝑘+𝑙)th iteration approximation of 𝑢_𝑛𝑖, 𝑙=0,1.

For fixed 𝑖, (2.3) is a cubic equation with respect to (𝑖𝜋/𝐿)𝑢_{𝑛𝑖,𝑘+1} (here 𝑢_{𝑛𝑖,𝑘+1} is taken with weight 𝑖𝜋/𝐿 just for convenience). Using the Cardano formula [27], we express (𝑖𝜋/𝐿)𝑢_{𝑛𝑖,𝑘+1} through the 𝑘th iteration approximation 𝑖𝜋𝐿𝑢_{𝑛𝑖,𝑘+1}=𝜎_𝑖1,𝑘−𝜎_𝑖2,𝑘, (2.4) where 𝜎_{𝑖𝑝,𝑘}=[(−1)^𝑝𝑠_𝑖+(𝑠²_𝑖+𝑟³_𝑖,𝑘)^1/2]^1/3,𝑟_𝑖,𝑘=13[2𝛽𝐿(𝛼+(𝑖𝜋𝐿)²)+^𝑛_{𝑗=1𝑗≠𝑖}(𝑗𝜋𝐿)²𝑢²_{𝑛𝑗,𝑘}],𝑠_𝑖1=−𝑓𝛽𝑖𝜋^𝑖,𝑘=0,1,…,𝑖=1,2,…,𝑛. (2.5)

The algorithm we have considered should be understood as the counting carried out by formula (2.4). Having 𝑢_{𝑛𝑖,𝑘}, 𝑖=1,2,…,𝑛, we construct the approximate solution of the problem 𝑢_𝑛,𝑘(𝑥)=^𝑛_𝑖=1𝑢_{𝑛𝑖,𝑘}sin𝑖𝜋𝑥𝐿. (2.6)

2.3. Algorithm Error Definition

Let us compare the approximate solution (2.6) with the 𝑛th truncation of the exact solution (1.9) 𝑝_𝑛𝑢(𝑥)=^𝑛_𝑖=1𝑢_𝑖sin𝑖𝜋𝑥𝐿. (2.7) This means that the algorithm error is defined as a difference 𝑝_𝑛𝑢(𝑥)−𝑢_𝑛,𝑘(𝑥) (2.8) which we write as a sum 𝑝_𝑛𝑢(𝑥)−𝑢_𝑛,𝑘(𝑥)=Δ𝑢_𝑛(𝑥)+Δ𝑢_𝑛,𝑘(𝑥), (2.9) where Δ𝑢_𝑛(𝑥) is the Galerkin method error and Δ𝑢_𝑛,𝑘(𝑥) the Jacobi process error which are equal, respectively, to Δ𝑢_𝑛(𝑥)=𝑝_𝑛𝑢(𝑥)−𝑢_𝑛(𝑥),Δ𝑢_𝑛,𝑘(𝑥)=𝑢_𝑛(𝑥)−𝑢_𝑛,𝑘(𝑥). (2.10)

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 Δ𝑢_𝑛(𝑥) into a series. Taking (2.10), (2.7), and (2.1) into account we write Δ𝑢_𝑛(𝑥)=^𝑛_𝑖=1Δ𝑢_𝑛𝑖sin𝑖𝜋𝑥𝐿, (3.1) where Δ𝑢_𝑛𝑖=𝑢_𝑖−𝑢_𝑛𝑖,𝑖=1,2,…,𝑛. (3.2) By virtue of (3.1) we have ‖Δ𝑢_𝑛(𝑥)‖_{𝐿2(0,𝐿)}𝐿=(2^𝑛_𝑖=1(Δ𝑢_𝑛𝑖)²)^1/2. (3.3) We will come back to (3.3) later, while now we denote 𝛾_𝑙𝑛=(2−𝑙)(𝑖𝜋𝐿)⁴+12(𝑖𝜋𝐿)²[𝛼+𝛽𝐿2^𝑛_𝑗=1(𝑗𝜋𝐿)²𝑢²_𝑗+(−1)^𝑙+1(𝛼+𝛽𝐿2^𝑛_𝑗=1(𝑗𝜋𝐿)²𝑢²_𝑛𝑗𝜀)],_𝑛=12𝛽𝐿(𝑖𝜋𝐿)^2∞_𝑗=𝑛+1(𝑗𝜋𝐿)²𝑢²_𝑗∇_𝑛,=14𝛽𝐿(𝑖𝜋𝐿)^2𝑛_𝑗=1(𝑗𝜋𝐿)²(𝑢_𝑗+𝑢_𝑛𝑗)Δ𝑢_𝑛𝑗, (3.4)(3.5)(3.6) and rewrite (1.10) and (2.2) in the form (𝛾_1𝑛+𝛾_2𝑛+𝜀_𝑛)𝑢_𝑖=𝑓_𝑖 and (𝛾_1𝑛−𝛾_2𝑛)𝑢_𝑛𝑖=𝑓_𝑖. Since by virtue of (3.4), (3.2), and (3.6) we have 𝛾_2𝑛=∇_𝑛 and therefore (𝛾_1𝑛+∇_𝑛+𝜀_𝑛)𝑢_𝑖=𝑓_𝑖 and (𝛾_1𝑛−∇_𝑛)𝑢_𝑛𝑖=𝑓_𝑖. Subtracting the last two equalities from each other and taking (3.2) into account, we obtain 𝛾_1𝑛Δ𝑢_𝑛𝑖+∇_𝑛(𝑢_𝑖+𝑢_𝑛𝑖)+𝜀_𝑛𝑢_𝑖=0 which we multiply by Δ𝑢_𝑛𝑖 and sum over 𝑖=1,2,…,𝑛. Using (3.4), (3.5), and the inequality ∑^𝑛_𝑖=1∇_𝑛(𝑢_𝑖+𝑢_𝑛𝑖)Δ𝑢_𝑛𝑖≥0 following from (3.6), we see that ^𝑛_𝑖=1(𝛼+(𝑖𝜋𝐿)²)(𝑖𝜋𝐿)²(Δ𝑢_𝑛𝑖)²≤12𝛽𝐿^𝑛_𝑖=1(𝑖𝜋𝐿)²|𝑢_𝑖Δ𝑢_𝑛𝑖|^∞_𝑖=𝑛+1(𝑖𝜋𝐿)²𝑢²_𝑖. (3.7)

By the Cauchy-Bunyakowsky-Schwarz inequality, we therefore have (^𝑛_𝑖=1(𝛼+(𝑖𝜋𝐿)²)(𝑖𝜋𝐿)²(Δ𝑢_𝑛𝑖)²)^1/2≤12𝛽𝐿(^𝑛_𝑖=1𝑢²_𝑖)^∞1/2_𝑖=𝑛+1(𝑖𝜋𝐿)²𝑢²_𝑖. (3.8)

Let us estimate the right-hand side of inequality (3.8). After multiplying (1.10) by (𝑖𝜋/𝐿)^𝑡𝑢_𝑖 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 ^𝑤_𝑖=𝑣(𝑖𝜋𝐿)^4+𝑡𝑢²_𝑖+(𝛼+𝛽𝐿2^∞_𝑗=1(𝑗𝜋𝐿)²𝑢²_𝑗)^𝑤_𝑖=𝑣(𝑖𝜋𝐿)^2+𝑡𝑢²_𝑖=^𝑤_𝑖=𝑣(𝑖𝜋𝐿)^𝑡𝑓_𝑖𝑢_𝑖, (3.9) where 𝑣=1, 𝑤=𝑛 or 𝑣=𝑛+1, 𝑤=∞. Thus (𝛼+𝛽𝐿2^∞_𝑗=1(𝑗𝜋𝐿)²𝑢²_𝑗)^𝑤_𝑖=𝑣(𝑖𝜋𝐿)^2+𝑡𝑢²_𝑖≤14^𝑤_𝑖=𝑣(𝑖𝜋𝐿)^𝑡−4𝑓²_𝑖. (3.10)

Let us put 𝑣=1, 𝑤=𝑛, 𝑡=−2 in (3.10) and use the fact that ∑^∞_𝑗=1(𝑗𝜋/𝐿)²𝑢²_𝑗≥(𝜋²/𝐿²)∑^𝑛_𝑗=1𝑢²_𝑗. We obtain ^𝑛_𝑖=1𝑢²_𝑖≤𝑎_𝑛, (3.11) where 𝑎_𝑛=𝐿𝜋²𝛽[(𝛼²+1𝜋2𝐿²𝛽^𝑛_𝑖=1(𝐿)𝑖𝜋⁶𝑓²_𝑖)^1/2−𝛼]. (3.12)

Now assuming 𝑣=𝑛+1, 𝑤=∞, 𝑡=0 in (3.10) and using in addition to this the inequality ∑^∞_𝑗=1(𝑗𝜋/𝐿)²𝑢²_𝑗≥∑^∞_𝑗=𝑛+1(𝑗𝜋/𝐿)𝑢²_𝑗, we get ^∞_𝑖=𝑛+1(𝑖𝜋𝐿)𝑢²_𝑖≤𝑏_𝑛, (3.13) where 𝑏_𝑛=1𝛽𝐿[(𝛼²+12𝛽𝐿^∞_𝑖=𝑛+1(𝐿)𝑖𝜋⁴𝑓²_𝑖)^1/2−𝛼]. (3.14)

The use of (3.11) and (3.13) in (3.8) brings us to the inequality (^𝑛_𝑖=1(𝛼+(𝑖𝜋𝐿)²)(𝑖𝜋𝐿)²(Δ𝑢_𝑛𝑖)²)^1/2≤12𝛽𝐿(𝑎_𝑛)^1/2𝑏_𝑛 (3.15) which together with (3.3) gives ‖Δ𝑢_𝑛(𝑥)‖_{𝐿2(0,𝐿)}≤12𝜋𝛽𝐿²(𝐿2𝑎_𝑛𝛼+(𝜋/𝐿)²)^1/2𝑏_𝑛. (3.16)

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 ‖Δ𝑢_𝑛(𝑥)‖_{𝐿2(0,𝐿)}≤𝑐₀[(1+𝑐₁1𝑛^2𝑚+3)^1/2−1]{[1+𝑐₂(1+𝑐₃1(1−𝑛^2𝑚+5))]^1/2−1}^1/2, (3.17) 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,𝑐_𝑙=𝛽𝐿𝜔²2𝛼²(𝐿(1−2(𝑙−2)(𝑚+1))𝜋)⁴,𝑙=1,2,𝑐₃=1.2𝑚+5 (3.18)

3.2. Jacobi Process Error

Taking (2.10), (2.1), and (2.6) into account, we represent Δ𝑢_𝑛,𝑘(𝑥) as a series Δ𝑢_𝑛,𝑘(𝑥)=^𝑛_𝑖=1Δ𝑢_{𝑛𝑖,𝑘}sin𝑖𝜋𝑥𝐿, (3.19) where Δ𝑢_{𝑛𝑖,𝑘}=𝑢_𝑛𝑖−𝑢_{𝑛𝑖,𝑘},𝑖=1,2,…,𝑛. (3.20)

Series (3.19) implies the formula ‖Δ𝑢_𝑛,𝑘(𝑥)‖_{𝐿2(0,𝐿)}𝐿=(2^𝑛_𝑖=1(Δ𝑢_{𝑛𝑖,𝑘})²)^1/2 (3.21) to be used later.

Let us rewrite (2.4) in the form 𝑖𝜋𝐿𝑢_{𝑛𝑖,𝑘+1}=𝜑_𝑖(𝜋𝐿𝑢_𝑛1,𝑘,2𝜋𝐿𝑢_𝑛2,𝑘,…,𝑛𝜋𝐿𝑢_{𝑛𝑛,𝑘}), (3.22) and introduce into consideration the Jacobian 𝐽=(𝜕𝜑_𝑖𝜕((𝑗𝜋/𝐿)𝑢_{𝑛𝑗,𝑘}))^𝑛_𝑖,𝑗=1 (3.23) (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, 𝜕𝜑_𝑖𝜕((𝑖𝜋/𝐿)𝑢_{𝑛𝑖,𝑘})=0. (3.24) As to the nondiagonal elements, 𝑖≠𝑗, they are defined by the formula 𝜕𝜑_𝑖𝜕((𝑗𝜋/𝐿)𝑢_{𝑛𝑗,𝑘})=13𝑟²_𝑖,𝑘(𝑠²_𝑖+𝑟³_𝑖,𝑘)^−1/2(𝜎⁻²_𝑖1,𝑘−𝜎⁻²_𝑖2,𝑘)𝑗𝜋𝐿𝑢_{𝑛𝑗,𝑘}. (3.25) Using the relations 𝜎_𝑖1,𝑘𝜎_𝑖2,𝑘=𝑟_𝑖,𝑘,(𝑠²_𝑖+𝑟³_𝑖,𝑘)^1/2=12(𝜎³_𝑖1,𝑘+𝜎³_𝑖2,𝑘),𝜎³_𝑖2,𝑘−𝜎³_𝑖1,𝑘=2𝑠_𝑖, (3.26) which follow from (2.5), we rewrite (3.25) as the equality 𝜕𝜑_𝑖𝜕((𝑗𝜋/𝐿)𝑢_{𝑛𝑗,𝑘})4=−3𝑠_𝑖(𝜎⁴_𝑖1,𝑘+𝑟²_𝑖,𝑘+𝜎⁴_𝑖2,𝑘)⁻¹𝑗𝜋𝐿𝑢_{𝑛𝑗,𝑘}. (3.27) Apply to the latter equality the estimate 𝜎⁴_𝑖1,𝑘+𝜎⁴_𝑖2,𝑘≥2𝑟²_𝑖,𝑘, 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 |𝜕𝜑_𝑖𝜕((𝑗𝜋/𝐿)𝑢_{𝑛𝑗,𝑘})4|≤9|𝑠_𝑖|𝑟²_𝑖,𝑘𝑗𝜋𝐿|𝑢_{𝑛𝑗,𝑘}|≤4|𝑓_𝑖|(2𝛽𝑖𝜋𝛽𝐿(𝛼+(𝑖𝜋𝐿)²)+(𝑗𝜋𝐿𝑢_{𝑛𝑗,𝑘})²)⁻²𝑗𝜋𝐿|𝑢_{𝑛𝑗,𝑘}|≤38(32𝛽𝐿)^1/2|𝑓_𝑖|(𝑖𝜋/𝐿)(𝛼+(𝑖𝜋/𝐿)²)^3/2,𝑖≠𝑗, (3.28) which are fulfilled for the nondiagonal elements of the matrix 𝐽.

Let us use the vector and matrix norms equal, respectively, to ∑^𝑛_𝑖=1|𝑣_𝑖| and max_{1≤𝑗≤𝑛}∑^𝑛_𝑖=1|𝑚_𝑖𝑗| for the vector 𝑣=(𝑣_𝑖)^𝑛_𝑖=1 and the matrix 𝑀=(𝑚_𝑖𝑗)^𝑛_𝑖,𝑗=1. Assume that for an arbitrary set of values 𝑢_{𝑛𝑗,𝑘}, 𝑗=1,2,…,𝑛, 𝑘=0,1,…, the elements of the matrix 𝐽 satisfy the condition max_{1≤𝑗≤𝑛}∑^𝑛_𝑖=1|𝜕𝜑_𝑖/𝜕((𝑗𝜋/𝐿)𝑢_{𝑛𝑗,𝑘})|≤𝑞<1. For this, by virtue of (3.28), (3.24), and (1.8) it is sufficient that 338𝜋𝐿𝜔(2𝛽𝐿)^𝑛1/2_{𝑖=1𝑖≠𝑗}1𝑖^𝑚+1(𝛼+(𝑖𝜋/𝐿)²)^3/2≤𝑞<1,𝑗=1,2,…,𝑛. (3.29) Then, according to the map compression principle, the system of (2.2) has a unique solution 𝑢_𝑛𝑖, 𝑖=1,2,…,𝑛, the iteration process (2.4) converges, lim_𝑘→∞𝑢_{𝑛𝑖,𝑘}=𝑢_𝑛𝑖, 𝑖=1,2,…,𝑛, with the rate which in view of notation (3.20) is defined by the inequality ∑^𝑛_𝑖=1𝑖|Δ𝑢_{𝑛𝑖,𝑘}|≤(𝑞^𝑘/(1−𝑞))∑^𝑛_𝑖=1𝑖|𝑢_𝑛𝑖,1−𝑢_𝑛𝑖,0|. From this and (3.21) we obtain the estimate for the Jacobi process error ‖Δ𝑢_𝑛,𝑘(𝑥)‖_{𝐿2(0,𝐿)}≤𝑞^𝑘(𝐿1−𝑞2)^𝑛1/2_𝑖=1𝑖|𝑢_𝑛𝑖,1−𝑢_𝑛𝑖,0|,𝑘=0,1,…. (3.30)

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 𝛼^1/2+𝑖𝜋/𝐿≤[2(𝛼+(𝑖𝜋/𝐿)²)]^1/2. Then using the formula for the integral ∫𝑑𝑥/𝑥^𝑚+1(𝑎+𝑏𝑥)³, 𝑎,𝑏>0, [28] instead of (3.29), we obtain 34𝜔(3𝛽𝐿)^1/2{𝐿2𝜋√2(𝛼+(𝜋/𝐿)²)^3/2+1𝛼²(𝜋𝐿√𝛼)^𝑚−1×(𝑚+2)!_𝑚+2𝑙=0(−1)^𝑙1𝐿√𝑙!(𝑚−𝑙+2)!(𝑚−𝑙)[(1+𝛼𝜋)^𝑚−𝑙𝐿√−(1+𝛼)𝜋𝑛^𝑚−𝑙]}≤𝑞<1. (3.31)

3.3. Algorithm Error

Let us estimate error (2.8). By (2.9) we have ‖𝑝_𝑛𝑢(𝑥)−𝑢_𝑛,𝑘(𝑥)‖_{𝐿2(0,𝐿)}≤‖Δ𝑢_𝑛(𝑥)‖_{𝐿2(0,𝐿)}+‖Δ𝑢_𝑛,𝑘(𝑥)‖_{𝐿2(0,𝐿)}, (3.32) and therefore the application of (3.17) and (3.30) gives the inequality ‖𝑝_𝑛𝑢(𝑥)−𝑢_𝑛,𝑘(𝑥)‖_{𝐿2(0,𝐿)}≤𝑐₀[(1+𝑐₁1𝑛^2𝑚+3)^1/2−1]{[1+𝑐₂(1+𝑐₃1(1−𝑛^2𝑚+5))]^1/2−1}^1/2+𝑞^𝑘(𝐿1−𝑞2)^𝑛1/2_𝑖=1𝑖|𝑢_𝑛𝑖,1−𝑢_𝑛𝑖,0|. (3.33)

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).