and note that \(S(x)=x .\) Similarly, we have \(s_{n+1}=S(x+\Delta x)\) and \(s_{n-1}=S(x-\Delta x)\), where \(\Delta x=1 / N .\) Then, with \(\Delta t=2 / N,(5.5.4)\) transforms into<\p> \[\begin{ali...and note that \(S(x)=x .\) Similarly, we have \(s_{n+1}=S(x+\Delta x)\) and \(s_{n-1}=S(x-\Delta x)\), where \(\Delta x=1 / N .\) Then, with \(\Delta t=2 / N,(5.5.4)\) transforms into<\p> \[\begin{align} \nonumber P(x, t&+\Delta t)-P(x, t)=S(x+\Delta x)(1-S(x+\Delta x)) P(x+\Delta x, t) \\[4pt] &-2 S(x)(1-S(x)) P(x, t)+S(x-\Delta x)(1-S(x-\Delta x)) P(x-\Delta x, t) \end{align} \nonumber \] To simplify further, we use the well-known central-difference approximation to the second-derivative of a…
