Analytical Solutions of Peridynamic Equations. Part I: Transient Heat Diffusion

In this paper, we construct formal analytical solutions for peridynamic models of transient diffusion using the separation of variables technique. We show that the infinite series nonlocal solutions can be obtained directly from corresponding classical solutions by inserting “peridynamic (nonlocal) factors” in the time-exponential part of the solution. We find analytical expressions for the nonlocal factor. In 2D rectangular domains, these can be written in terms of Bessel functions. The nonlocal factor depends on the horizon size and converges to value one as the horizon size goes to zero, recovering the classical form of the solution for the corresponding partial-differential equations. We also show that, as time goes to infinity, the nonlocal solution converges to the classical one, for a fixed horizon. We consider examples of transient diffusion problems with Dirichlet and Neumann boundary conditions. Their analytical solutions are compared with the corresponding classical solutions. While most of the analytical solutions we present here are formal, for a number of cases, we are able to prove uniform convergence of the series solutions. This is the first contribution that presents analytical (formal) solutions to peridynamic transient diffusion problems in 1D or 2D finite domains by separation of variables, with arbitrary boundary conditions, and shows their connections to the corresponding solutions to the classical/local problem. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.