Author(s) :
Etienne Emmrich
,
Olaf Weckner
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 02-2005
MSC 2000
- 74H20 Existence of solutions
-
74H25 Uniqueness of solutions
-
74H30 Regularity of solutions
-
74H55 Stability
-
74H15 Numerical approximation of solutions
-
74B99 None of the above, but in this section
-
45K05 Integro-partial differential equations
-
34G10 Linear equations
-
47G20 Integro-differential operators
-
65R20 Integral equations
Abstract :
Long-range interactions
for linearly elastic media resulting in
nonlinear dispersion relations are modelled by an
initial-value problem for an integro-differential equation (IDE) that
incorporates non-local effects. Interpreting this IDE as an
evolutionary equation of second order, well-posedness in
$L^{\infty}(\rz)$ as well as jump
relations are proved. A numerical approximation based upon quadrature
is suggested and carried out for two examples, one
involving jump discontinuities in the initial data corresponding to a
Riemann-like problem.
Keywords :
Long-range interactions,peridynamic theory,nonlinear dispersion relations,integro-differential equation, existence and uniqueness,jump discontinuity,numerical approximation