Approximation algorithm for differential-difference equations and simulation of electrodynamic processes (Q2703334)

The authors deal with the differential-difference equation of neutral type NEWLINE\[NEWLINE{d\over dt}[x(t)-A(t)x(t-\tau)]= f(t,x(t),x(t-\tau)), \quad t\in[0,T],NEWLINE\]NEWLINE with initial condition \(x(t)=\phi(t)\), \(t\in[-\tau,0]\). Let us define functions \(z_{j}(t)\), \(j=0,\ldots,m\), as solutions to the Cauchy problem NEWLINE\[NEWLINE{d\over dt}[z_0(t)- A(t)z_{m}(t)]= f(t,z_0(t),z_{m}(t)), \quad {d\over dt}z_{j}(t)={m\over\tau} (z_{j-1}(t)-z_{j}(t)), \quad j=1,\ldots,m,NEWLINE\]NEWLINE NEWLINE\[NEWLINEz(0)=\phi \biggl(-{j\tau\over m}\biggr), \quad j=1, \ldots,m.NEWLINE\]NEWLINE The authors prove that if function \(f(t,u,v)\) is continuous and Lipschitz continuous on \((u,v)\), \(A(t)\) is continuous and \(\max_{t\in[0,T]}|A(t)|<1\), then \(|x(t-j{\tau\over m})- z_{j}(t)|\to 0\), \(j=0,\ldots,m\), \(t\in [0,T]\), as \(m\to\infty\). An application of this algorithm for simulations in electrodynamics is considered.