Research Of Nonlinear Equation And Optimal Control Solution Based On RHPM

With the rapid development of science and technology,nonlinear equations and optimal control problems play important roles in a wide range of fields.However,in the current research,the method of solving the nonlinear equation still has limitations,and there are various problems for the algorithm of solving the optimal control problem.Homotopy Perturbation Method(HPM)is not only an effective tool for solving various linear and non-linear problems,but also plays a vital role in solving optimal control problems.The Rational Homotopy Perturbation method(RHPM)is based on the HPM,which expands the HPM in the form of two power series quotients.Compared with the HPM,the RHPM has the advantages of fewer iterations,faster convergence speed and higher accuracy.In this paper,based on the convergence of the RHPM,we apply the RHPM for solving nonlinear equations and optimal control problems.At the same time,we improve the solution form of RHPM,and propose a new form of equivalent solution.Then we apply the Equivalent Rational Homotopy Perturbation Method(ERHPM)to solve some nonlinear equations.Finally,the validity of the above two methods is verified by some examples.The main contents of this paper are as follows:Firstly,the current research status of HPM,nonlinear equation and optimal control problem are briefly summarized.At the same time,the homotopy method and the HPM are simply introduced,and the convergence of HPM is proved.The optimal control model studied in this paper is also given.Secondly,we introduce the idea and convergence proof of the RHPM in detail,and give the new form of solution of the ERHPM,and the application of the RHPM theory in solving nonlinear equations and optimal control problems.Finally,the ERHPM and the RHPM are used to solve Riccati differential equation,convection-radiation combined cooling set system,Abel integral equation and Van del Por oscillator equation.On the other hand,the RHPM is used to solve the optimal control problems of three first-order linear systems and a second-order linear system.The above examples show that RHPM has the advantages of less iterations,better convergence and more accurate approximate solutions in solving nonlinear equations and optimal control problems.The ERHPM further improves the accuracy of the approximate solutions which from the RHPM.