Stability analysis of \(\theta\)-methods for neutral functional- differential equations
This paper deals with the numerical stability of the neutral functional differential equation \(y'(t) = ay(t) + by(qt) + cy'(pt)\), \(t > 0\). It is proved that \(\theta\)-methods are convergent if \(| c| < 1\). Experiments suggest they are divergent if \(| \theta|\) is large. The problem is transferred to a neutral equation with constant time lags. Using the later equation as a test model, it is shown that the linear \(\theta\)-method is \(\Lambda\)-stable if \(\text{Re }a < 0\) and \(| a| > | b|\) if and only if \(\theta \geq 1/2\) and the one-leg \(\theta\)-method is \(\Lambda\)-stable if \(\theta = 1\). It is also shown that inappropriate stepsize causes spurious solutions in the marginal case when \(\text{Re }a < 0\) and \(| a| = | b|\).
neutral functional differential equation
scientific article; zbMATH DE number 770169