In this paper, we consider the following nonlinear wave equation: (1) u tt − B(∥ u x ∥ 2) u xx = f( x, t, u, u x , u t ,∥ u x ∥ 2), x∈(0,1),0< t< T, (2) u x (0, t)− h 0 u(0, t)= u(1, t)=0, (3) u(x,0)= u ̃ 0(x) , u t(x,0)= u ̃ 1(x) , where B,f, u ̃ 0, u ̃ 1 are given functions. In Eq. (1), the nonlinear terms B(∥ 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 N+1 ( R +), B⩾ b 0>0, B 1∈ C N ( R +), B 1⩾0, f∈ C N+1 ([0,1]× R +× R 3× R +) and f 1∈ C N ([0,1]× R +× R 3× R +) we obtain from the following equation: u tt −[ B(∥ u x ∥ 2)+ εB 1(∥ 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 N+1 in ε, for ε sufficiently small.