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Analysis and numerical approximation of an integro-differential equation modelling non-local effects in linear elasticity

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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