Skip to main content

Advertisement

SpringerLink
Log in

Theoretical analysis and numerical simulation for a hyperbolic equation with Dirichlet and acoustic boundary conditions

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Bailly C, Juve D (2000) Numerical solution of acoustic propagation problems using linearized euler equations. AIAA J 38(1):22–29

    Article  Google Scholar 

  • Beale JT (1976) Spectral properties of an acoustic boundary condition. Indiana Univ Math J 25(9):895–917

    Article  MathSciNet  MATH  Google Scholar 

  • Beale JT, Rosencrans SI (1974) Acoustic boundary conditions. Bull Am Math Soc 80(6):1276–1278

    Article  MathSciNet  MATH  Google Scholar 

  • Ciarlet PG (2002) The finite element method for elliptic problems. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  • Clark H, Jutuca LSG, Miranda MM (1998) On a mixed problem for a linear coupled system with variable coefficients. Eletron J Differ Equ 1(04):1–20

    MATH  Google Scholar 

  • Cousin AT, Frota CL, Larkin NA (2004) On a system of Klein–Gordon type equations with acoustic boundary conditions. J Math Anal Appl 293(1):293–309

    Article  MathSciNet  MATH  Google Scholar 

  • Frigeri S (2010) Attractors for semilinear damped wave equations with an acoustic boundary condition. J Evol Equ 10(1):29–58

    Article  MathSciNet  MATH  Google Scholar 

  • Frota C, Medeiros L, Vicente A et al (2011) Wave equation in domains with non-locally reacting boundary. Differ Integral Equ 24(11/12):1001–1020

    MathSciNet  MATH  Google Scholar 

  • Frota CL, Goldstein JA (2000) Some nonlinear wave equations with acoustic boundary conditions. J Differ Equ 164(1):92–109

    Article  MathSciNet  MATH  Google Scholar 

  • 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

  • Graber PJ (2012) Uniform boundary stabilization of a wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping. J Evol Equ 12(1):141–164

    Article  MathSciNet  MATH  Google Scholar 

  • Jenkins EW (2007) Numerical solution of the acoustic wave equation using Raviart–Thomas elements. J Comput Appl Math 206(1):420–431

    Article  MathSciNet  MATH  Google Scholar 

  • Kobayashi Y, Tanaka N (2008) An application of semigroups of locally Lipschitz operators to carrier equations with acoustic boundary conditions. J Math Anal Appl 338(2):852–872

    Article  MathSciNet  MATH  Google Scholar 

  • Límaco J, Clark HR, Frota CL, Medeiros LA (2011) On an evolution equation with acoustic boundary conditions. Math Methods Appl Sci 34(16):2047–2059

    Article  MathSciNet  MATH  Google Scholar 

  • Medeiros L, Miranda M (1996) On a boundary value problem for wave equations: existence, uniqueness-asymptotic behavior. Revista de Matemáticas Aplicadas 17:47–73

    MathSciNet  MATH  Google Scholar 

  • Morse PM, Ingard KU, Beyer R (1969) Theoretical acoustics. J Appl Mech 36:382

    Article  Google Scholar 

  • Mugnolo D (2006) Abstract wave equations with acoustic boundary conditions. Mathematische Nachrichten 279(3):299–318

    Article  MathSciNet  MATH  Google Scholar 

  • Rincon M, Quintino N (2016) Numerical analysis and simulation for a nonlinear wave equation. J Comput Appl Math 296:247–264

    Article  MathSciNet  MATH  Google Scholar 

  • Silva PB, Clark H, Frota C (2017) On a nonlinear coupled system of thermoelastic type with acoustic boundary conditions. Comput Appl Math 36(1):397–414

    Article  MathSciNet  MATH  Google Scholar 

  • Vicente A, Frota C (2013a) Nonlinear wave equation with weak dissipative term in domains with non-locally reacting boundary. Wave Motion 50(2):162–169

    Article  MathSciNet  MATH  Google Scholar 

  • Vicente A, Frota C (2013b) On a mixed problem with a nonlinear acoustic boundary condition for a non-locally reacting boundaries. J Math Anal Appl 407(2):328–338

    Article  MathSciNet  MATH  Google Scholar 

  • Wheeler MF (1973) \(l_{\infty }\) estimates of optimal orders for galerkin methods for one-dimensional second order parabolic and hyperbolic equations. SIAM J Numer Anal 10(5):908–913

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

A. A. Alcantara was partially supported by CAPES-Brazil. M. A. Rincon was partially supported by CNPq-Brazil.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Federal University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil

    Adriano A. Alcântara & Mauro A. Rincon

  2. Department of Mathematics, Federal University of Piauí, Parnaíba, PI, Brazil

    Haroldo R. Clark

Authors
  1. Adriano A. Alcântara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Haroldo R. Clark
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mauro A. Rincon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adriano A. Alcântara.

Additional information

Communicated by Luz de Teresa.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alcântara, A.A., Clark, H.R. & Rincon, M.A. Theoretical analysis and numerical simulation for a hyperbolic equation with Dirichlet and acoustic boundary conditions. Comp. Appl. Math. 37, 4772–4792 (2018). https://doi.org/10.1007/s40314-018-0601-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40314-018-0601-y

Keywords

Mathematics Subject Classification

Search

Navigation