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Theoretical analysis and numerical simulation for a hyperbolic equation with Dirichlet and acoustic boundary conditions. (English) Zbl 1415.35175
Summary: This paper is concerned with a theoretical and numerical study for the initial-boundary value problem for a linear hyperbolic equation with variable coefficient and acoustic boundary conditions. On the theoretical results, we prove the existence and uniqueness of global solutions, and the uniform stability of the total energy. Numerical simulations using the finite element method associated with the finite difference method are employed, for one-dimensional and two-dimensional cases, to validate the theoretical results. In addition, numerically the uniform decay rate for energy and the order of convergence of the approximate solution are also shown.

##### MSC:
 35L20 Initial-boundary value problems for second-order hyperbolic equations 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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##### References:
 [1] Bailly, C; Juve, D, Numerical solution of acoustic propagation problems using linearized Euler equations, AIAA J, 38, 22-29, (2000) [2] Beale, JT, Spectral properties of an acoustic boundary condition, Indiana Univ Math J, 25, 895-917, (1976) · Zbl 0325.35060 [3] Beale, JT; Rosencrans, SI, Acoustic boundary conditions, Bull Am Math Soc, 80, 1276-1278, (1974) · Zbl 0294.35045 [4] Ciarlet PG (2002) The finite element method for elliptic problems. SIAM, Philadelphia · Zbl 0999.65129 [5] Clark, H; Jutuca, LSG; Miranda, MM, On a mixed problem for a linear coupled system with variable coefficients, Eletron J Differ Equ, 1, 1-20, (1998) · Zbl 0886.35043 [6] Cousin, AT; Frota, CL; Larkin, NA, On a system of Klein-Gordon type equations with acoustic boundary conditions, J Math Anal Appl, 293, 293-309, (2004) · Zbl 1060.35118 [7] Frigeri, S, Attractors for semilinear damped wave equations with an acoustic boundary condition, J Evol Equ, 10, 29-58, (2010) · Zbl 1239.35025 [8] Frota, C; Medeiros, L; Vicente, A; etal., Wave equation in domains with non-locally reacting boundary, Differ Integral Equ, 24, 1001-1020, (2011) · Zbl 1249.35221 [9] Frota, CL; Goldstein, JA, Some nonlinear wave equations with acoustic boundary conditions, J Differ Equ, 164, 92-109, (2000) · Zbl 0979.35105 [10] Frota CL, Larkin NA (2005) Uniform stabilization for a hyperbolic equation with acoustic boundary conditions in simple connected domains. In: Cazenave T et al (eds) Contributions to nonlinear analysis. Progress in nonlinear differential equations and their applications, vol 66. Birkhäuser Basel, pp 297-312 [11] Graber, PJ, Uniform boundary stabilization of a wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping, J Evol Equ, 12, 141-164, (2012) · Zbl 1250.35134 [12] Jenkins, EW, Numerical solution of the acoustic wave equation using Raviart-Thomas elements, J Comput Appl Math, 206, 420-431, (2007) · Zbl 1148.65076 [13] Kobayashi, Y; Tanaka, N, An application of semigroups of locally Lipschitz operators to carrier equations with acoustic boundary conditions, J Math Anal Appl, 338, 852-872, (2008) · Zbl 1145.47044 [14] Límaco, J; Clark, HR; Frota, CL; Medeiros, LA, On an evolution equation with acoustic boundary conditions, Math Methods Appl Sci, 34, 2047-2059, (2011) · Zbl 1230.35058 [15] Medeiros, L; Miranda, M, On a boundary value problem for wave equations: existence, uniqueness-asymptotic behavior, Revista de Matemáticas Aplicadas, 17, 47-73, (1996) · Zbl 0859.35070 [16] Morse, PM; Ingard, KU; Beyer, R, Theoretical acoustics, J Appl Mech, 36, 382, (1969) [17] Mugnolo, D, Abstract wave equations with acoustic boundary conditions, Mathematische Nachrichten, 279, 299-318, (2006) · Zbl 1109.47035 [18] Rincon, M; Quintino, N, Numerical analysis and simulation for a nonlinear wave equation, J Comput Appl Math, 296, 247-264, (2016) · Zbl 1331.65142 [19] Silva, PB; Clark, H; Frota, C, On a nonlinear coupled system of thermoelastic type with acoustic boundary conditions, Comput Appl Math, 36, 397-414, (2017) · Zbl 1362.35046 [20] Vicente, A; Frota, C, Nonlinear wave equation with weak dissipative term in domains with non-locally reacting boundary, Wave Motion, 50, 162-169, (2013) · Zbl 1360.76311 [21] Vicente, A; Frota, C, On a mixed problem with a nonlinear acoustic boundary condition for a non-locally reacting boundaries, J Math Anal Appl, 407, 328-338, (2013) · Zbl 1310.35153 [22] Wheeler, MF, $$l_{∞ }$$ estimates of optimal orders for Galerkin methods for one-dimensional second order parabolic and hyperbolic equations, SIAM J Numer Anal, 10, 908-913, (1973) · Zbl 0266.65074
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