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.
Highlights
In this paper, we consider the following initial boundary value problem for a viscoelastic wave equation with nonlinear damping: utt −uxx + λutq− t 2ut + g(t −s)uxx(x, s)ds (1)f(x, t, u), 0 < x < 1, 0 < t < T, ux(0, t) − h0u(0, t) ux(1, t) + h1u(1, t) 0, (2)u(x, 0) u0(x), ut(x, 0) u1(x), (3)where λ > 0, q ≥ 2, h0 ≥ 0, and h1 ≥ 0 are constants, with h0 + h1 > 0, and f, g, u0, and u1 are given functions.Equation (1) arises naturally within frameworks of mathematical models in engineering and physical sciences
∈ R2+: s ≤ t}, by the high-order iterative method coupled with the Galerkin method, the existence of a recurrent sequence um associated with equation (7) and defined by u0 ≡ 0, z2um zt2
An initial boundary value problem for a viscoelastic wave equation with nonlinear damping is investigated, and its main outcomes are summarized in two parts
Summary
We consider the following initial boundary value problem for a viscoelastic wave equation with nonlinear damping: utt −. In the presence of the strong damping − Δut and the linear damping ut(m 2), Li and He [8] proved the global existence of solutions and established a general decay rate estimate for problem (10) They showed the finite-time blow-up results of solutions with both negative initial energy and positive initial energy. We will construct the algorithm to find the finite-difference approximate solutions of u(m) given by the single-iterative scheme (formulas (157)–(159) below) and present a numerical example to compare convergent rates of two schemes.
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