On implicit algorithms of continuation method with applications to dynamic systems

In this work, we show that numerical solution of the Cauchy problem for a system of ODEs of the second order resolved with respect to the higher derivative can be obtained by constructing the simple and effective implicit step-by-step integration procedures without involving laborious iterative processes like Newton–Raphson. The problem is initially transformed to a new argument, an integral curve length. Such transformation involves one equation that relates the initial parameter of problem and integral curve length. on the linear acceleration method, we demonstrate a procedure of constructing an implicit algorithm, which uses simple iterations to numerically solve the transformed Cauchy problem. The definitions of computational properties of iterational process are formulated and proven. Explicit estimates of integration step providing the convergence of simple iterations are given. Effectiveness of the proposed method is demonstrated upon three problems solved with and without preliminary parameterisation. The problem of celestial mechanics “Pleiades” is considered as a test one. The second example deals with modelling nonlinear dynamic of elastic cantilever flexible beam, which is rolled in initial static state into a ring by the bending moment. In third example, we give a solution for deployment of mechanical system of three flexible beams under prescribed control laws.