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Numerical methods for the nonlocal wave equation of the peridynamics. (English) Zbl 1436.65195

Summary: In this paper we will consider the peridynamic equation of motion which is described by a second order in time partial integro-differential equation. This equation has recently received great attention in several fields of Engineering because seems to provide an effective approach to modeling mechanical systems avoiding spatial discontinuous derivatives and body singularities. In particular, we will consider the linear model of peridynamics in a one-dimensional spatial domain. Here we will review some numerical techniques to solve this equation and propose some new computational methods of higher order in space; moreover we will see how to apply the methods studied for the linear model to the nonlinear one. Also a spectral method for the spatial discretization of the linear problem will be discussed. Several numerical tests will be given in order to validate our results.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65D30 Numerical integration
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
35R09 Integro-partial differential equations
45K05 Integro-partial differential equations
74B10 Linear elasticity with initial stresses
35Q74 PDEs in connection with mechanics of deformable solids
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References:

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