Fundamental solutions of the linear equations governing mechanical and electromagnetic oscillations are kinematically represented by delay time along ray trajectories. The fundamental solutions can contain components which are not physically justified, if their ray trajectories are partly located outside the actual medium in accordance with Fermat's principle. To exclude all non-physical components and consider only the physically feasible fundamental solution, ray trajectories and delay time must satisfy the generalized Fermat's principle, as introduced by Hadamard in 1910. We introduce a rigorous dynamic description of this feasible fundamental solution satisfying the generalized Fermat's principle and being physically justifiable. The description is based on an integral condition of absolute absorption at the boundary of an effective medium. This condition selects a subset of the physically feasible fundamental solutions. We prove that, in homogeneous domains, the feasible fundamental solution is the sum of Green's function for unbounded medium and an operator Neumann series describing cascade diffraction at the boundary. In inhomogeneous domains we represent the feasible fundamental solution by an equation with a volume integral operator. The integral kernel contains the feasible fundamental solution for a homogeneous domain. We introduce feasible surface and volume integral operators that eliminate the unfeasible wavefields in the geometrical shadow zones.
