A Wavelet-Galerkin Method for a 1D Elastic Continuum with Long-Range Interactions

An elastic continuum model with long-range forces is addressed in this study. The model stems from a physically-based approach to non-local mechanics where non-adjacent volume elements exchange mutual central forces that depend on the relative displacement and on the product between the interacting volume elements; further, they are taken as proportional to a material dependent and distance-decaying function. Smooth-decay functions lead to integro-differential equations while hypersingular, fractional-decay functions lead to a fractional differential equation of Marchaud type. In both cases the governing equations are solved by the Galerkin method with different sets of basis functions, among which also discrete wavelets are used. Numerical applications confirm the accuracy of the Galerkin solution as compared to finite difference solutions