| The boundary value problem of ordinary differential equations has important applications in space science and engineering technology.A large number of mathematical models in the fields of engineering,astronomy,mechanics,economics,etc.,are often described by ordinary differential boundary value problems.Except for a few special types,the exact solution of the boundary value problem of ordinary differential equations is difficult to express in analytical form.It is especially important to find an approximate solution to obtain its numerical solution.At present,the commonly used numerical solutions include test-shot method,finite difference method and finite element method.The finite difference method and the finite element method need to discretize the differential equation into a large system of equations.When the differential equation is a nonlinear equation,it needs to face great difficulties.In this paper,for a class of ordinary differential equation boundary value problems which can be transformed into initial value problems with a single unknown parameter,according to the boundary value condition,the condition required to set the initial value method is the unknown parameter as the initial condition of the differential equation,which is transformed into a single unknown,conversion to initial value problem of ordinary differential equations with single unknown parameters.Then use some numerical solution of the initial value problem to perform the operation under a certain step size.Thus,a parameter expression of an approximation of the function value at the node can be obtained,and a parameter expression of another boundary point is recursively obtained.Then,using the definite condition of the original boundary value problem,the unary nonlinear equation satisfied by the previously set parameters is established,and then the approximate value of the parameter is solved by the corresponding iterative method,that is,the approximation of the initial condition of the solution satisfying the boundary condition.Finally,the above iterative result is directly used as the initial condition,and the numerical solution of the boundary value problem is given againby the initial value solution.Since the function involved in the one-dimensional nonlinear equation established by the method is complicated,in order to solve the equation effectively,it is necessary to adopt a high-order convergence iterative method and try to avoid the calculation of the function derivative.To this end,an improved Stephenson method is designed.It only needs to calculate the function value three times for each iteration,and the fourth-order convergence effect can be achieved without calculating the derivative function,thereby greatly reducing the calculation amount.In this paper,by transforming the boundary value problem of some ordinary differential equations into the initial value problem of differential equations with single unknown parameter,it provides effective support for solving the problem of solving nonlinear equations.The iterative method enriches the method of rooting of nonlinear equations,and has high value and significance both in theory and in application.In this paper,a new numerical method for solving nonlinear ordinary differential equations with certain boundary conditions is proposed by combining the numerical solution of initial value problem and the iterative method of unary nonlinear equation.The calculation format and convergence of the method are given.And comparative analysis was carried out by numerical examples. |
