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We propose a numerical method to solve an inverse source problem of computingthe initial condition of hyperbolic equations from the measurements of Cauchydata. This problem arises in thermo- and photo- acoustic tomography in abounded cavity, in which the reflection of the wave makes the widely-usedapproaches, such as the time reversal method, not applicable. In order to solvethis inverse source problem, we approximate the solution to the hyperbolicequation by its Fourier series with respect to a special orthogonal basis of$L^2$. Then, we derive a coupled system of elliptic equations for thecorresponding Fourier coefficients. We solve it by the quasi-reversibilitymethod. The desired initial condition follows. We rigorously prove theconvergence of the quasi-reversibility method as the noise level tends to 0.Some numerical examples are provided. In addition, we numerically prove thatthe use of the special basic above is significant.

The quasi-reversibility method to numerically solve an inverse source problem for hyperbolic equations