High-Order Iterative Scheme for a Viscoelastic Wave Equation and Numerical Results

Abstract

In this paper, we consider a Robin problem for a viscoelastic wave equation. First, by the high-order iterative method coupled with the Galerkin method, the existence of a recurrent sequence via an N -order iterative scheme is established, and then the N -order convergent rate of the obtained sequence to the unique weak solution of the proposed model is also proved. Next, with N = 2 , a numerical algorithm given by the finite-difference method is constructed to approximate the solution via the 2-order iterative scheme. Moreover, the same algorithm for the single-iterative scheme generated by the 2-order iterative scheme is also considered. Finally, comparison with errors of the numerical solutions obtained by the single-iterative scheme and the 2-order iterative scheme shows that the convergent rate of the 2-order iterative scheme is faster than that of the single-iterative scheme.

High-order iterative scheme to the Robin problem for a nonlinear wave equation with viscoelastic term

“…Under some suitable conditions, we establish a high-order iterative scheme and then prove that the scheme converges to the weak solution of the original problem along with the error estimate. This result extends the result in [9].…”

Continuous dependence and general decay of solutions for a wave equation with a nonlinear memory term

“…(1.1). The results obtained here may be considered as relative generalizations of those in [28] - [31], [34].…”

“…The viscoelastic problem of the form (1.1) has been studied by many authors, for example, we refer to [3], [17], [18], [22]- [24], [26] - [30], [32], [34]. By using di¤erent methods together with various techniques in functional analysis, several results concerning the existence/global existence and the properties of solutions of viscoelastic problems such as blow-up, decay, stability have been established.…”

“…(1.2) and Eq. (1.3) such as general decay or blow up in …nite time, we can see in [17], [18], [30], [34].…”

“…1 was studied. Recently, Quynh et al [34] has considered Eq. (1.4), in which, an N -order recurrent sequence has been established and its convergence to the unique solution of (1.4) satisfying an estimation of convergent rate in N order has been proved.…”