High-order iterative schemes for a nonlinear Kirchhoff-Carrier wave equation associated with the mixed homogeneous conditions

In this paper we consider the following nonlinear wave equation (1){utt−μ(t,‖u(t)‖2,‖ux(t)‖2)uxx=f(x,t,u),00, h1≥0 are given constants. In Eq. (1)1, the nonlinear term μ(t,‖u(t)‖2,‖ux(t)‖2) depends on the integrals ‖u(t)‖2=∫01|u(x,t)|2dx, ‖ux(t)‖2=∫01|ux(x,t)|2dx. In this paper we associate with Eq. (1)1 a recurrent sequence {um} defined by (2)∂2um∂t2−μ(t,‖um(t)‖2,‖umx(t)‖2)umxx=∑i=0N1i!∂if∂ui(x,t,um−1)(um−um−1)i,0