Title: Four Mutual Properties of Classical and Nonlocal Wave Equations

Abstract: We present nonlocal (NL) operators that enforce local boundary conditions (BCs).  The main advantage that our NL operators provide is the ability to enforce local BC through the use of a forcing function only on the local boundary, not in the interior of the domain.  The ability to incorporate such a widely accepted BC type into NL formulations is quite valuable.  

We provide a comparative study on classical and NL wave equations.  The NL operators employ local BCs, and this is why a comparison to the classical wave equation is relevant. We find out that the two equations are qualitatively identical in terms of the balance of linear momentum (BLM), conservation of energy, and the resonance and beating phenomena. For both equations, the BLM is satisfied for the Neumann and periodic BCs and fails for Dirichlet and antiperiodic BCs. 

We also reveal a close connection between classical and NL wave equations.  In d’Alembert’s formula on a bounded domain, the BC is encoded in the solution using the extension artifice known as the method of images. Whereas in our integral formulation, since the only degree of freedom is the kernel function, it is encoded in the kernel of the operator. This is a striking difference from the local formulation. What is even more striking is the following similarity: we discovered that the same combination of the function piece (even or odd) and extension type (antiperiodic or periodic) is used in structuring the kernel function.  We were able to discover such suitable kernel structures thanks to functional calculus.