On the Exact Solution of Nonlocal Euler–Bernoulli Beam Equations via a Direct Approach for Volterra-Fredholm Integro-Differential Equations

First, we develop a direct operator method for solving boundary value problems class of nth order linear Volterra–Fredholm integro-differential equations convolution type. The proposed technique is based on the assumption that Volterra bijective and its inverse known in closed form. Existence uniqueness criteria are established exact solution derived. We then apply this to construct form fourth equilibrium bending Euler–Bernoulli beams context Eringen’s nonlocal theory elasticity (two phase integral model) under transverse distributed load simply supported conditions. An easy use algorithm obtaining symbolic algebra system also given.