Numerical and theoretical treatment for solving linear and nonlinear delay differential equations using variational iteration method (Q375183)

This paper is concerned with the approximate solution of some initial value problems (IVP) for delay differential equations by using the so-called variational iteration method (VIM) proposed by \textit{J.-H. He} [Int. J. Non-Linear Mech. 34, No. 4, 699--708 (1999; Zbl 1342.34005)]. The IVPs under consideration have the form: NEWLINE\[NEWLINE Lu(t) \equiv u^{(n)}(t) = f(t, u(t), u( \alpha (t))),\, t \in [0,T],\tag{1}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE u^{(k)}(0) = u_0^{k}\, (k=0,1, \dots n-1),\,\,u(t)= \phi(t), \,t \leq 0,NEWLINE\]NEWLINE where \( \alpha (t) \leq t\) is a given delay function.NEWLINENEWLINE For the application of the VIM the authors assume that the equation (1) can be written in the form \( L u + R u + N(u) = 0\), where \(L\) and \(R\) are linear bounded operators and \(N(u)\) is a nonlinear Lipschitz continuous operator. In this setting, the VIM proposes a sequence of functions \( ( u_p(x) )_{p \geq 0} \) with \( u_0\) defined by the initial conditions and \( u_{p+1}\) computed recursively from \( u_{p+1} = u_p + \int_0^t \lambda (\tau) [ L u_p + R \widetilde{u}_p + N \widetilde{u}_p ] d \tau \) where \( \widetilde{u}_p\) is a restricted variation, i.e., \( \delta ( \widetilde{u}_p) = 0 \) and the Lagrange multiplier \( \lambda \) is determined so that \( \delta u_{p+1} = 0\).NEWLINENEWLINE In Section 3, under suitable assumptions, some results about the convergence of the sequence \( ( u_p )\) to a solution \(u\) of the IVP (1) are given. Also, the existence of a unique solution is proved. In Section 4, a number of examples is presented to show the convergence of the iterative sequence generated by the VIM.