Barycentric Interpolation Iterative Collocation Method For Free Boundary Value Problems

The free boundary problem is essentially a nonlinear problem,it is usually solved by an iterative method.The difficulty in solving the free boundary problem is that the partial boundary of the domain of the differential equation is unknown,the boundary is to be obtained as part of the solution.In order to make the free boundary problem have an approximate solution,over-constrained boundary conditions are generally given on free boundary.A part of the boundary conditions is used to solve the differential equation and the others is used to determine the free boundary position.Based on the meshless barycentric Lagrange Interpolation collocation method,two calculated formats of barycentric interpolation iterative collocation method for solving the free boundary problem of linear and nonlinear governing equations are proposed.Numerical examples are given to verify the effectiveness and accuracy of the proposed methods.First,when solving the free boundary problem of linear differential equations,after giving an initial hypothesis value of free boundary,the barycentric Lagrange interpolation collocation method is used to solve the boundary value problem of the differential equation,Then,the Newton method and the chord-cut method for determining the position of the free boundary are constructed by using any definite condition on the free boundary.The barycentric interpolation iterative collocation method for solving the linear free boundary problem is proposed.Second,the Newton nose-cone problem is a free boundary problem consisting of a highly nonlinear differential equation and three fixed solution conditions.For nonlinear differential equations,the nonlinear differential equation is linearized firstly by Newton's method,a linearized iterative scheme is constructed to approximate the nonlinear differential equation,and then the numerical solution of the differential equation is solved iteratively.The advantage of using linearized iterative method to solve nonlinear problems is that the numerical method for solving linear problems can be used to solve nonlinear problems without solving nonlinear algebraic equations.After processing the differential equation,assuming an initial function and the initial position of a free boundary,the high-precision barycentric Lagrange interpolation collocation method is used to solve the boundary value problem of linear differential equations.By using any one of defined conditions on the free boundary,the iterative scheme of the chord-cut method for determining the position of the free boundary is constructed.A high-precision linearized iterative collocation method for solving the nonlinear free boundary value is proposed.The computer program was compiled by MATLAB.The feasibility,calculation efficiency and accuracy of the free boundary problem solved by the barycentric interpolation iterative collocation method are analyzed and verified by numerical examples.The results of linear numerical examples show that the Newton method has fewer iterations than the chord-cut method.Newton method obtain high-precision solutions after 3⁴ iterations.The calculation of the chord-cut method is not affected by the boundary conditions and the governing equation itself.The numerical calculation results of the two iterative formats have extremely high calculation accuracy,and the error accuracy increases with the increase of the node,which can reach 10^-1110^-13.Chord-cut iteration collocation method as a complement to Newton iteration collocation,which makes the calculation method more universal applicability.Numerical analysis of two types of nodes,Chebyshev and Legendre,in nonlinear numerical examples.The numerical results show that with the increase of the number of computing nodes,the calculation accuracy is significantly improved,and the calculation error is at least 10^-10.Under the same number of compute nodes,the accuracy of the Legendre node is about 10^-2 higher than the accuracy of the Chebyshev node.Using the same initial position and the number of compute nodes,the type of nodes Chebyshev is more stable than the Legendre node.