Oscillation analysis of numerical solution in the \(\theta\)-methods for equation \(x\prime (t) + ax(t) + a_{1}x([t - 1]) = 0\)
This paper is concerned with the preservation of the oscillatory behavior of the solutions of functional differential equations (1) \( x'(t) + a x(t) + a_1 x([t-1])=0\), \(t>0\), where \( a, a_1 >0\) are constants and \([\cdot]\) denotes the greatest integer functions, when integrated numerically by the \(\theta\)-methods. Since for any given set of starting values \( x(-1), x(0)\) there exist a unique solution of (1) which is oscillatory iff \( 4 a_1 (1-e^{-a})-a>0\), some conditions on the \(\theta\)--methods are given for the preservation of the oscillatory behavior of the solutions of (1) for sufficiently small step size. Further it is shown that the zeros of the linear interpolation of the numerical solutions converge to the zeros of the exact solution with first order accuracy when \( \theta \neq 1/2\) and second order accuracy when \( \theta =1/2\). Finally the results of some numerical experiments are presented to show the oscillatory behavior preservation according to the values of the step size.
scientific article; zbMATH DE number 5147034