Generalized quasilinearization method for semilinear hyperbolic problems (Q1811374)

The paper deals with the initial-boundary value problem (IBVP) for hyperbolic differential equations of the form \[ u_{tt} + Lu = f(x,t,u) \quad \text{ in } \Omega \times (0,T], \] \[ u = 0 \quad \text{ on } \partial \Omega \times (0,T] \] \[ u = g, \;u_t = h \quad \text{ on } \Omega \times \{ t = 0 \}, \] where \(L\) is a second order elliptic partial differential operator in divergence form. The authors extend the method of generalized quasilinearization to the IBVP (1)--(3). Two monotone sequences are obtained which converge quadratically to the unique solution of the problem under consideration.