A nonlinear wave equation with a nonlinear integral equation involving the boundary value

Date

2004-09-03

Authors

Nguyen, Thanh Long
Bui, Tien Dung

Journal Title

Journal ISSN

Volume Title

Publisher

Texas State University-San Marcos, Department of Mathematics

Abstract

We consider the initial-boundary value problem for the nonlinear wave equation utt - uxx + ƒ(u, ut) = 0, x ∈ Ω = (0, 1), 0 < t < T, ux(0, t) = P(t), u(1, t) = 0, u(x, 0) = u0(x), ut(x, 0) = u1(x), where u0, u1, ƒ are given functions, the unknown function u(x, t) and the un-known boundary value P(t) satisfy the nonlinear integral equation P(t) = g(t) + H(u(0, t)) - ∫t0 K(t - s, u(0, s))ds, where g, K, H are given functions. We prove the existence and uniqueness of weak solutions to this problem, and discuss the stability of the solution with respect to the functions g, H and K. For the proof, we use the Galerkin method.

Description

Keywords

Galerkin method, Integrodifferential equations, Schauder fixed point theorem, Weak solutions, Stability of the solutions

Citation

Nguyen, T. L., & Bui, T. D. (2004). A nonlinear wave equation with a nonlinear integral equation involving the boundary value. <i>Electronic Journal of Differential Equations, 2004</i>(103), pp. 1-21.

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Attribution 4.0 International

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