Numerical Solutions For Systems Of Singular Time-varying Nonlinear Equations

In mathematical modeling of scientific and engineering problems,more and more scientists and engineers pay attention to systems of time-varying nonlinear equations which are usually applied to robotics,motion control of manipulator,Chaotic system and multi-dimensional spectrum estimation.Therefore,solving time-varying nonlinear equations plays a crucial role in science and engineering.Like time-invariant nonlinear equations,solving time-varying nonlinear equations also encounters singular problems.Based on Zeroing neural network(ZNN)which is a model for solving time-varying equations,combining the method of solving singular problems in the numerical solution of time invariant equations and differential equations,we explore the effective numeri-cal solution of singular time-varying nonlinear equations.Firstly,for singular time-varying nonlinear equations,we propose two precondi-tioned ZNN models.Considering the stiffness problem,we apply the implicit method of differential equations for the iterations.A Jacobian-free method is applied to solving its nonlinear equations subproblems in real-time.The numerical results show that the preconditioned implicit ZNN model with Jacobian-free method can effectively solve singular time-varying nonlinear equations.Secondly,using the property of the right term which is easy to differentiate in the ZNN model,we propose the implicit RK method and Rosenbrock-type method which are with singular matrix coefficients.In order to reduce the complexity and operation time of the algorithm,an simplified Newton method is applied to solving the nonlinear equations subproblems,and the nonsingular condition of nonlinear equations in sub-problems has also been analyzed.The numerical results show that both the simplified Newton implicit RK method and the Rosenbrock-type method which are with singular coefficient matrix can solve singular time-varying nonlinear equations effectively.Finally,through theoretical analysis,the orders of global error and function's error of ZNN model and preconditioned ZNN model are derived.