NUMERICAL-SOLUTION OF A TIME-LIKE CAUCHY-PROBLEM FOR THE WAVE-EQUATION

Let D subset-of R(n) be a bounded domain with piecewise-smooth boundary, and q(x, t) a smooth function on D x [0, T]. Consider the time-like Cauchy problem u(tt) - DELTA(x)u + q(x, t)u = 0 in D x [0, T] u = g, u(n) = h on partial derivative D x [0, T]. Given g, h for which the equation has a solution, we show how to approximate u(x, t) by solving a well posed fourth-order elliptic partial differential equation (PDE). We use the method of quasi-reversibility to construct the approximating PDE. We derive error estimates and present numerical results.