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Analysis and Numerical Approximation of an Integro-differential Equation Modeling Non-local Effects in Linear Elasticity

Emmrich, Etienne; Weckner, Olaf

Inst. Mathematik

Long-range interactions for linearly elastic media resulting in nonlinear dispersion relations are modeled 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 ∞(ℝ) as well as jump relations are proved. Moreover, the construction of the micromodulus function from the dispersion relation is studied. 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.
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  • This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.