AN ITERATION METHOD OF SOLUTION OF A NONLINEAR EQUATION FOR A STATIC BEAM. 2 University Str., Tbilisi 0186, Georgia 2 Georgian Technical University

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1 AN ITERATION METHOD OF SOLUTION OF A NONLINEAR EQUATION FOR A STATIC BEAM T. Davitashvili 1, J. Peradze 1,, Z. Tsiklauri 1 Ivane Javakhishvili Tbilisi State University University Str., Tbilisi 186, Georgia Georgian Technical University 77 Kostava Str., Tbilisi 174, Georgia (Received: ; accepted: Abstract The paper deals with the boundary value problem for the nonlinear integrodifferential equation u m ( l u dx u = f(x, m(z α >, z <, modeling the static state of the Kirchhoff beam. The problem is reduced to an integral equation which is solved using the iteration method. The convergence of the iteration process is established and the error estimate is obtained. Key words and phrases: Kirchhoff type equation, Green s function, iteration method, error estimate. AMS subject classification: 34B7, 65L1, 65L7, 65R. 1 Statement of the problem Let us consider the nonlinear beam equation u (x m with the conditions u (x dx u (x = f(x, < x < l, (1 u( = u(l =, u ( = u (l =. ( Here u = u(x is the displacement function of the length l of the beam subjected to a force given by the function f(x, the function m(z, m(z α >, z <, (3 describes the type of relationship between stress and strain. Namely, if the function m(z is linear, this means that the latter relation is consistent with Hooke s linear law, while otherwise case we deal with material nonlinearities.

2 AMIM Vol.19 No.1, 14 T. Davitashvili, J. Peradze,... + Equation (1 is the stationary problem associated with the equation u tt + u xxxx m m(z const >, u x dx u xx = f(x, t, which for the case m(z = m + m 1 z, m, m 1 >, and f(x, t =, was proposed by Woinowsky Krieger [11] as a model of deflection of an extensible dynamic beam with hinged ends. The nonlinear term l u x dx was for the first time used by Kirchhoff [] who generalized D Alembert s classical linear model. Therefore (1 is frequently called a Kirchhoff type equation for a static beam. The topic of solvability of equations of (1 type is studied in [3] [5] and [1], while the problem of construction of numerical algorithms and estimation of their accuracy is investigated in [1], [4], [6], [8] and [9]. In the present paper, in order to obtain an approximate solution of problem (1, (, an approach different from the ones applied in the above references is used. It consists in reducing problem (1, ( by means of Green s function to a nonlinear integral equation, to solve which we use an iteration method. A condition for convergence of the method is established and its accuracy is estimated. The Green s function method with a further iteration procedure has been applied by us previously also to a nonlinear problem for the axially symmetric Timoshenko plate [7]. Assumptions Suppose that f(x L (, l and the function m(z satisfies, in addition to requirement (3, the Lipschitz condition on an interval L = const > and m(z m(z 1 L z z 1, (4 z 1, z c, c = l ( α + l l f (x dx. (5 1

3 + An Iteration Method of Solution... AMIM Vol.19 No.1, 14 3 The Method Applying the Green s function of the problem u (x au (x = f(x, < x < l, u( = u(l =, u ( = u (l =, a = const >, and performing some transformations, from problem (1, ( we come as a result to the equivalent nonlinear integral equation = 1 τ τ sinh( τ l φ(x = 1 l u(x = l G(x, ξf(ξ dξ + 1 φ(x, (6 τ G(x, ξ = { sinh( τ (x l sinh( τ ξ, < ξ x < l, ( (l x sinh( τ (ξ l sinh( τ x, < x ξ < l, τ = m x u (x dx, ξf(ξ dξ + x l x (l ξf(ξ dξ. Equation (6 is solved by the method of ordinary iterations. After choosing a function u (x, x l, which together with its second derivative vanishes for x = and x = l, we find subsequent approximations by the formula u k+1 (x = l G k (x, ξf(ξ dξ + 1 τ k φ(x, < x < l, (7 k =, 1,..., 1 G k (x, ξ = τ k τk sinh( τ k l { sinh( τ k (x l sinh( τ k ξ, < ξ x < l, sinh( τ k (ξ l sinh( τ k x, < x ξ < l, τ k = m k =, 1,..., (8 u k (x dx, (9 11

4 AMIM Vol.19 No.1, 14 T. Davitashvili, J. Peradze,... + and u k (x is the k-th approximation of the solution of equation (6. At the (k + 1-th iteration step, having the k-th approximation u k (x, to find u k+1 (x we first calculate the parameter τ k by (9 and then, taking (8 into account, find the function G k (x, ξ, which is used together with τ k in formula (7. The numerical integration technique can be used for calculating definite integrals in the general case. 4 Estimation of the Method Error Our aim is to estimate the error of the method, by which we understand the difference between the approximate and the exact solution δu k (x = u k (x u(x, k =, 1,.... (1 For this, it is advisable to use not formula (7, but the system of equalities u k+1 (x m u k (x dx u k+1 (x = f(x, < x < l, (11 u k ( = u k (l =, u k ( = u k (l =, (1 k =, 1,..., from which formula (7 follows. If we subtract the respective equalities in (1 and ( from (11 and (1, we get ( ( δu k (x 1 m u k 1 +m (x dx ( + m u k 1 m (x dx ( l ( l u (x dx δu k (x+ (u u (x dx k (x+u (x =, (13 δu k ( = δu k (l =, δu k ( = δu k (l =, (14 k = 1,,.... System (13 and conditions (14 are the starting point of estimation of the method error. We shall have to derive preliminarily several a priori estimates. Let us denote the norms in W 1 (, l W (, l as u(x p = ( d p u 1 dx p (x dx, p =, 1,, u(x = u(x. 1

5 + An Iteration Method of Solution... AMIM Vol.19 No.1, 14 The symbol (, is understood as a scalar product in L (, l. Lemma 1. The inequalities are valid for u(x W (, l. l u(x u(x 1 Proof. Using the equality u(x = u(x ( x 1 dξ ( x l u(x (15 x u (ξ dξ we obtain 1 u 1 (ξ dξ x u(x 1, which implies the left inequality of (15. Applying the latter and taking into account that u(x 1 = u(xu (x l ( u(x, u (x = = ( u(x, u (x u(x u(x, we complete the proof. Lemma. For the solution of problem (1, ( the inequality u(x 1 c 1, (16 ( c 1 = max z, < l z l + m(z f(x, (17 is true. Proof. We multiply equation (1 by u(x and integrate the resulting equality with respect to x from to l. Using (, we get the relation By (15 we obtain u(x + m ( u(x 1 u(x 1 = ( f(x, u(x. ( l + m( u(x 1 u(x 1 l f(x u(x 1, which means the fulfillment of (16. Note that when we deal with the problem of finding c 1 by (17, by virtue of (3 c 1 can be defined by c 1 = max z, < l z( + α f(x. l As a result we get c 1 = l ( + α 1 f(x, but, speaking in general, this l solution is not the best way to cope with the situation. 13

6 AMIM Vol.19 No.1, 14 T. Davitashvili, J. Peradze,... + Lemma 3. Approximations of the iteration method (7 satisfy the inequality u k (x 1 c, k = 1,,..., (18 c = l ( l + α 1 f(x. (19 Proof. We multiply equation (11 by u k (x and integrate the resulting relation with respect to x from to l. Taking (1 into account, we get u k (x + m ( u k 1 (x 1 uk (x 1 = ( f(x, u k (x, Applying (3 and (15 we have k = 1,,.... ( l + α u k (x 1 l f(x u k (x 1, which implies (18. Let us compare the parameters c, c 1 and c. By (5, (17, (3 and (19 5 Convergence of the Method c 1 c = c. ( Multiplying (13 by δu k (x, integrating the resulting equality with respect to x from to l and using (14, we come to the relation δu k (x + 1 ( ( m ( u k 1(x ( 1 + m u(x 1 δu k (x 1+ ( + m ( u k 1 (x ( 1 ( m u(x 1 u k (x + u (x, δu k (x =. Using (3, (4, (5 together with (16, (18 and ( we obtain δu k (x + α δu k (x ( L u k p (x + u (x, δu k p (x p= 1 L 1 p= ( u k p (x 1 + u(x 1 δu k p (x 1. 14

7 + An Iteration Method of Solution... AMIM Vol.19 No.1, 14 By (15, (16 and (18 we get δu k (x 1 1 ( 1 δuk 1 1 L l +α ( (x 1 uk p (x 1 + u(x 1 p= q δu k 1 (x 1, q = 1 L ( l + α 1(c1 + c. Taking (1, (15, (17 and (19 into consideration, we come to the following result. Theorem. Let the assumptions (3 and formulated in Section be fulfilled and besides q = 1 ( ( 1 L l + α max z + l ( 1 f(x l + α < 1, ( < l z l + m(z f(x. Then approximations of the iteration method (7 converge to the exact solution of problem (1, ( and for the error the following estimate u k (x u(x p ( l 1 pq k u (x u(x 1, k = 1,,..., p =, 1, is true. In conclusion, note that since relations (11, (1 do not necessarily lead to (7, the theorem will also be valid in the case of another extension of (11, (1, for instance, when the current iteration approximation in (11, (1 is defined by finite difference, finite element or other methods. References 1. Bernardi C., Copetti M. I. M. Finite element discretization of a thermoelastic beam. Archive ouverte HAL-UPMC, 9/5/13, 3 pp.. Kirchhoff G. Vorlesungen über mathematische physik. I. Mechanik. Teubner, Leipzig, Ma T. F. Existence results for a model of nonlinear beam on elastic bearings. Appl. Math. Lett. 13 (, no. 5,

8 AMIM Vol.19 No.1, 14 T. Davitashvili, J. Peradze, Ma T. F. Existence results and numerical solutions for a beam equation with nonlinear boundary conditions. nd International Workshop on Numerical Linear Algebra, Numerical Methods for Partial Differential Equations and Optimization (Curitiba, 1. Appl. Numer. Math. 47 (3, no., Ma T. F. Positive solutions for a nonlocal fourth order equation of Kirchhoff type. Discrete Contin. Dyn. Syst. 7, Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, suppl., Peradze J. A numerical algorithm for a Kirchhoff-type nonlinear static beam. J. Appl. Math. 9, Art. ID 81869, 1 pp. 7. Peradze J. On an iteration method of finding a solution of a nonlinear equilibrium problem for the Timoshenko plate. ZAMM Z. Angew. Math. Mech. 91 (11, No. 1, Temimi H., Ansari A. R., Siddiqui A. M. An approximate solution for the static beam problem and nonlinear integro-differential equations. Comput. Math. Appl. 6 (11, no. 8, Tsai S. Y. Numerical computation for nonlinear beam problems. M.S. thesis, National Sun Yat-Sen University, Kaohsiung, Taiwan, Wang F., An Y. Existence and multiplicity of solutions for a fourthorder elliptic equation. Bound. Value Probl. 1, 1:6, 9 pp. 11. Woinowsky-Krieger S. The effect of an axial force on the vibration of hinged bars. J. Appl. Mech. 17 (195,