The article deals with the following nonlinear integro-differential problem
\[ \begin{cases} \dot\frac{\partial U}{\partial t} - \dot\frac{\partial }{\partial x} \bigg\{\bigg(1 + \dot{\int_0^t} \bigg[\bigg(\dot\frac{\partial U}{\partial x}\bigg)^2 + \bigg(\dot\frac{\partial V}{\partial x}\bigg)^2\bigg] \, d\tau \dot\frac{\partial U}{\partial x}\bigg\} = f_1(x,t), \\ \dot\frac{\partial V}{\partial t} - \dot\frac{\partial }{\partial x} \bigg\{\bigg(1 + \dot{\int_0^{t}} \bigg[\bigg(\dot\frac{\partial U}{\partial x}\bigg)^2 + \bigg(\dot\frac{\partial V}{\partial x}\bigg)^2\bigg] \, d\tau \dot\frac{\partial V}{\partial x}\bigg\} = f_2(x,t), \\ U(0,t) = U(1,t) = V(0,t) = V(1,t) = 0, \\ U(x,0) = U_0(x), \quad V(x,0) = V_0(x) \end{cases}\tag{1} \]
in the cylinder \([0,1] \times [0,T]\). It is assumed that this system has a sufficiently smooth solution \(U = U(x,t)\), \(V = V(x,t)\) and considered the finite difference scheme that is a natural approximation to (1) with the uniform net \((x_i,t_j) = (ih,j\tau)\), \(0 \leq i \leq M\), \(0 \leq j \leq N\). It is proved that solutions \(u^j = (u_1^j,\dots,u_{M-1}^j)\), \(v^j = (v_1^j,\dots,v_{M-1}^j)\), \(j = 1,2,\dots,N\), of the difference scheme tend to \(U^j = (U_1^j,\dots,U_{M-1}^j)\), \(V^j = (V_1^j,\dots,V_{M-1}^j)\), \(j = 1,2,\dots,N\) of (1) as \(\tau, h \to 0\) and estimates of type \(\|u^j - U^j\|, \|v^j - V^j\| = O(\tau + h)\). Some numerical illustration of this result is presented.