In this paper we consider the following nonlinear wave equation (1) u tt−B t,‖u x‖ 2 u xx=f x,t,u,u x,u t,‖u x‖ 2 , x∈Ω=(0,1), 0<t<T, (2) u x(0,t)−h 0u(0,t)=g 0(t), u(1,t)=g 1(t), (3) u(x,0)= u ̃ 0(x), u t(x,0)= u ̃ 1(x), where B,f,g 0,g 1, u ̃ 0, u ̃ 1 are given functions. In Eq. (1), the nonlinear terms B( t,‖ u x ‖ 2), f( x, t, u, u x , u t ,‖ u x ‖ 2) depending on an integral ‖u x‖ 2=∫ 0 1|u x(x,t)| 2 dx . In this paper we associate with problem (1)–(3) a linear recursive scheme for which the existence of a local and unique solution is proved by using standard compactness argument. In case of B∈ C 3( R + 2), B⩾ b 0>0, B 1∈ C 2( R + 2), B 1⩾0, f∈ C 3([0,1]× R +× R 3× R +) and f 1∈ C 2([0,1]× R +× R 3× R +) we obtain from the equation u tt −[ B( t,‖ u x ‖ 2)+ εB 1( t,‖ u x ‖ 2)] u xx = f( x, t, u, u x , u t ,‖ u x ‖ 2)+ εf 1( x, t, u, u x , u t ,‖ u x ‖ 2) associated to (2), (3) a weak solution u ε ( x, t) having an asymptotic expansion of order 3 in ε, for ε sufficiently small.