A global domain/boundary integral equation method for the inverse wave source and backward wave problems

A g-analytic function theory, the Cauchy-Riemann equations of g-analytic function and the g-conformal invariant for the wave equation are derived in this paper. As a consequence, there exist two global spectral relations for the wave equation. By suitably choosing two different types of Trefftz test functions in the derived Green's second identity, we can recover unknown wave source function very well using the global domain/boundary integral equation method (BIEM), which is robust against large noise up to . Then, we develop a numerical algorithm based on the BIEM, which is effective for the backward wave problem, as well as for the long-term solution of the wave equation. Numerical examples are used to demonstrate the efficiency and accuracy of these methods.