An elastic continuum model with long-range forces is addressed in this study within the context of\udapproximate analytical methods. Such a model stems from a mechanically-based approach to non-local\udtheory where long-range central forces are introduced between non-adjacent volume elements. Specifically,\udlong-range forces depend on the relative displacement, on the volume product between interacting\udelements and they are proportional to a proper, material-dependent, distance-decaying function.\udSmooth-decay functions lead to integro-differential governing equations whereas hypersingular, fractional-\uddecay functions lead to a fractional differential governing equation of Marchaud type. In this paper\udthe Galerkin and the Rayleigh–Ritz method are used to build approximate solutions to the integro-differential\udand the fractional differential governing equations. Numerical applications show the accuracy of\udthe proposed approximate solutions as compared to the finite difference approximation and to the fractional\udfinite difference approximation.
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