Among his topics are solid continuum mechanics, the finite element method, the wave equation
for solids, the simulation of strong ground motion, elasto-plasticity and fracture mechanics, the analysis of faulting, the simulation of faulting with the boundary element method, multi-agent simulation for evacuation process analysis, high performance computing application, and meta-modeling theory for the construction of numerical analysis models.
The absorbing boundary conditions are constructed from paraxial approximations of the wave equation
(Clayton and Enquist, 1977).
Indeed, the enforcement of the wave equation
for the electric field in conjunction with the divergence-free condition, may enter in conflict with boundary conditions in a very wide range of circumstances.
As a further step, the first author proved the generalized Hyers-Ulam stability of the wave equation
without source (see [16,17]).
Hirota, "Exact N-soliton solutions of the wave equation
of long waves in shallow-water and in nonlinear lattices," Journal of Mathematical Physics, vol.
Extending the wave equation
from outside the event horizon into the inside by rotating -[pi] through the lower half of the complex plane, the thermal radiation spectra can be derived.
The wave equation
for [u.sup.vsub.[parallel]] describes the propagation of longitudinal displacements, while the wave equation
for [u.sup.v.sub.[perpendicular to]] describes the propagation of transverse displacements in the spacetime continuum.
Therefore, it requires tremendous amount of floating point calculations and run times to carry out forward modeling of the wave equation
during an iteration.
The main idea is to use the material momentum equation where the forces are explicitly shown  for deriving the wave equation
and the Clausius-Duhem inequality for determining the governing evolution equations for internal variables.
We denote the Maxwell's curl equations formulation as DGM-TM-ME (or simply ME) and the wave equation
formulations as DGM-TM-WE and DGM-TM-WH (simply WE or WH), for the electric and magnetic wave equations respectively.
Adomian Decomposition method was successfully applied to nonlinear differential delay equations ,a non-linear dynamic systems, the heat equation [8,9], the wave equation
, coupled non-linear Partial differential equations [11,12], linear and non-linear integro-differential equations .
Chapters four through six are heavy with mathematics, discussing wave reduction with data processing, integral solutions to the wave equation
with boundary and initial value conditions--using a variety of mathematical tools including Green's functions, Kirchoff integral formula, and the eikonal equation--and decomposition and continuation of seismic wave field using Fourier integrals.