Three-Layer Semidiscrete Scheme For Generalized Kirchhoff Equation Jemal Rogava I. Vekua Institute of Applied Mathematics 2 University St., 0186, Tbilisi Georgia jrogava@viam.sci.tsu.ge Mikheil Tsiklauri I. Vekua Institute of Applied Mathematics 2 University St., 0186, Tbilisi Georgia mtsiklauri@gmail.com Abstract: In the present work Cauchy problem for abstract generalization of Kirchhoff equation is considered. For approximate solution of this problem symmetric three-layer semi-discrete scheme is constructed. In this scheme, value of the gradient in nonlinear term is taken in the middle point. Stability of the offered scheme is proved and error of approximate solution is estimated. Key–Words: Nonlinear Kirchhoff wave equation, three-layer semi-discrete scheme. 1 Introduction In the work there is considered nonlinear abstract hyperbolic equation with self-adjoint positively de- fined operator corresponding to classic beam Kirch- hoff equation (obviously it also comprises spatial multi-dimensional case). We search the approximate solution of Cauchy problem stated for this equation using symmetric three-layer semi-discrete scheme. In this scheme, value of the gradient in nonlinear term is taken in the middle point. It makes possible to find approximate solution at each time step by inverting of linear operator. Investigation of stability and convergence of the considered scheme is based on two following facts: (a) approximate solution and difference analog of the first order derivative are uniformly bounded; (b) For the solution of the corresponding linear discrete prob- lem is valid a priori estimate, where the right-hand side term contains square root from the inversion of the main operator. Fact (b) makes it possible to weaken nonlinear term in the way that after taking into account fact (a), it will be possible to use Gronwell’s lemma for perturbation. As well as we know, issues of approximate so- lution for abstract generalization of Kirchhoff equa- tion are less studied, as distinct from classic string and beam equations. 2 Statement of the problem and semi-discrete scheme Let us consider the Cauchy problem for abstract hy- perbolic equation in the Hilbert space H : d 2 u(t) dt 2 + A 2 u (t)+ a    A 1/2 u    2  Au (t) = f (t) , t ∈ [0,T ] , (1) u (0) = ϕ 0 , du (0) dt = ϕ 1 . (2) where A is a self-adjoint (A does not depend on t), positively defined (generally unbounded) opera- tor with the definition domain D (A), which is every- where dense in H , i.e. D (A)= H, A = A ∗ and (Au, u) ≥ ν ‖u‖ 2 , ∀u ∈ D (A) , ν = const > 0, where by ‖·‖ and (·, ·) are defined correspondingly the norm and scalar product in H ; a    A 1/2 u    2  = λ +   A 1/2 u    2 ,λ> 0; ϕ 0 and ϕ 1 are given vectors from H ; u (t) is a continuous, twice continuously dif- ferentiable, searched function with values in H and f (t) is given continuous function with values in H. As in the linear case (see [1], T. 1.5 p. 301) u (t) vector function with values in H , defined on the inter- val [0,T ] is called a solution of the problem (1)-(2) if it satisfies the following conditions: (a) u (t) is twice continuously differentiable in the interval [0,T ]; (b) u (t) ∈ D ( A 2 ) for any t from [0,T ] and the func- tion A 2 u (t) is continuous; (c) u (t) satisfies equation (1) on the [0,T ] interval and the initial condition (2). Here continuity and differentiability is meant by met- ric H . Existence and uniqueness of the solution of the problem (1)-(2) is shown in [2]. Proceedings of the 2nd WSEAS Int. Conf. on FINITE DIFFERENCES, FINITE ELEMENTS, FINITE VOLUMES, BOUNDARY ELEMENTS ISSN: 1790-2769 193 ISBN: 978-960-474-089-5