The Variational Iteration Method For Delay Differential Equations

Variational iteration method is a very effective approach,which can give approximate or exact solutions to both linear and nonlinear problems.Moreover, this technique reduces the volume of calculations because it has no need of discretization of the variables, linearization or small perturbations. So far, it has been applied to wave equations, oscillation equations, fractional differential equations, some delay differential equations, and so on. In this paper, variational iteration method is applied to initial value problems of neutral differential equations and parabolic delay differential equations. The main contents and results are listed as follows:In Chapter1, the background of the variational iteration method is introduced, and we overviewe the development and applications of this method in recent years. Then the main contents of this paper is proposed.In Chapter2, the basic concepts, preparation theorem, operational properties in variations are introduced. What’s more, the main concepts and ideas in the variational iteration method, such as general Lagrange multiplier, stationary conditions and restricted variation is also explained.In Chapter3, the variational iteration method is applied to obtain approximately analytical solutions of initial value problem of neutral differential equations. First, we construct the iteration scheme, substitute the initial value, and obtain an iterative sequence, then we prove that the iterative sequence converges to the exact solution of the problem. The numerical results show that our theoretical analyses are right and the variational iteration method is a powerful method for solving this problem.In Chapter4, two kinds of variational iteration method are applied to initial value problem of parabolic delay differential equations. The first one directly uses variational iteration method to the problem. The second one should discrete the original problem on space first, and then apply the variational iteration method to the new problem. The illustrate examples reveal that these methods are effective and convenient.