Convergence and stability of the finite difference scheme for nonlinear parabolic systems with time delay (Q1385146)

The authors remark that little has been done in the domain decomposition method for nonlinear evolution equations with time delay. They consider the system: \[ u_t= Au_{xx}+ f(t, x, u, u_r),\quad (t, x)\in (0, T)\times (0,1), \] \[ u(0,x)= u(1, x)= 0,\quad t\in[0,T];\quad u(t, x)= \varphi(t,x),\quad (t,x)\in[- r,0]\times [0,1], \] where \(u= (u_1,\dots, u_J)^T\), \(f= (f_1,\dots, f_J)^T\), \(\varphi= (\varphi_1,\dots,\varphi_J)^T\), \(A\) is a constant matrix of order \(J\), \(u_r= u(t- r,x)\), \(r> 0\). The following topics are discussed: Existence of a generalized solution; the Picard-Schwarz iterative method for difference equations; the convergence of the difference solution and stability of the difference scheme. A numerical example illustrates the theoretical results.