Analysis of the nonlocal wave propagation problem with volume constraints

2020;
: pp. 334–344
https://doi.org/10.23939/mmc2020.02.334
Received: June 12, 2020
Revised: July 24, 2020
Accepted: July 27, 2020
1
Cadi Ayyad University
2
Institut Polytechnique UniLaSalle
3
Cadi Ayyad University
4
Sultan Moulay Slimane University

In the current paper, we develop a nonlocal propagation model, which describes the diffusion wave process.  The main motivation of this work is to apply the nonlocal vector calculus, introduced and developed by Du et al. [1] to such hyperbolic problem.  Moreover, based on some density arguments, some a priori estimates and using the Galerkin approach, we prove existence and uniqueness of a weak solution to the nonlocal wave equation widely adopted in various applications.

  1. Du Q., Gunzburger M., Lehoucq R. B., Zhou K.  A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws.  Mathematical Models and Methods in Applied Sciences. 23 (03), 493–540 (2013).
  2. Gilboa G., Osher S.  Nonlocal operators with applications to image processing.  Multiscale Model. Simul. 7 (3), 1005–1028 (2008).
  3. Guan Q., Gunzburger M.  Analysis and approximation of a nonlocal obstacle problem.  J. Comput. Appl. Math. 313,  102–118 (2017).
  4. Gunzburger M., Lehoucq R. B.  A nonlocal vector calculus with application to nonlocal boundary value problems.  Multiscale Model. Simul. 8 (5), 1581–1598 (2010).
  5. Kindermann S., Osher S., Jones P. W.  Deblurring and denoising of images by nonlocal functionals.  Multiscale Model. Simul. 4 (4), 1091–1115 (2005).
  6. Alberti G., Bellettini G.  A nonlocal anisotropic model for phase transitions. I. The optimal profile problem.  Mathematische Annalen. 310, 527–560 (1998).
  7. Bates P. W., Chmaj A.  An integrodifferential model for phase transitions: stationary solutions in higher space dimensions.  J. Stat. Phys. 95 (5/6), 1119–1139 (1999).
  8. Rosasco L., Belkin M., De Vito E.  On learning with integral operators.  J. Mach. Learn. Res. 11, 905–934 (2010).
  9. Du Q., Gunzburger M., Lehoucq R. B., Zhou K.  Analysis and approximation of nonlocal diffusion problems with volume constraints.  SIAM Rev. 54 (4), 667–696 (2012).
  10. Zeidler E.  II/A: Linear Monotone Operators.  Nonlinear Functional Analysis and Its Applications. Springer-Verlag, New York (1990).
Mathematical Modeling and Computing, Vol. 7, No. 2, pp. 334–344 (2020)