Artificial boundary conditions for Euler-Bernoulli beam equation

In a semi-discretized Euler-Bernoulli beam equation, the non-nearest neighboring interaction and large span of temporal scales for wave propagations pose challenges to the effectiveness and stability for artificial boundary treatments. With the discrete equation regarded as an atomic lattice with a three-atom potential, two accurate artificial boundary conditions are first derived here. Reflection coefficient and numerical tests illustrate the capability of the proposed methods. In particular, the time history treatment gives an exact boundary condition, yet with sensitivity to numerical implementations. The ALEX (almost EXact) boundary condition is numerically more effective.