Analytic solution of the wave equation in complex structures with defects, waveguides, sources

We discuss a derivation of the analytic solution of the wave equations in complex structures perturbed by local defects, long waveguides, and various sources. After obtaining the exact analytic form of the solution, the numerical implementation becomes more or less straightforward. The corresponding real-time simulations will be demonstrated. Another important point is that the analytic solutions do not have disadvantages associated with the noise of reflections from the artificial boundaries of the model and other drawbacks inherent in purely numerical simulations. The solution is based on integral and algebraic transforms, including the active use of special functions. Even for linear waves that propagate in inhomogeneous structures, the solution is very complex. This fact probably makes the process of obtaining exact analytic solutions for nonlinear waves practically hopeless.