| The second-order singular nonlinear two-point boundary value Lane-Emden equation plays a very important role in scientific fields such as physics and engineering.In mathematical physics and astrophysics,it is a frequently-used equation as mathematical modeling of some phenomena,such as stellar structure theory,isothermal dimensionless density distribution in gas spheres,theories of thermionic currents,etc.However,it is difficult to obtain accurate solutions to this problem.Therefore,it is necessary to study fast and high-precision numerical solutions of the second-order singular nonlinear two-point boundary value Lane-Emden equation.Meanwhile,the Lane-Emden equation is regarded as a typical problem for testing new methods for solving nonlinear differential equations due to its nonlinearity and singularity at the origin.In this paper,the modified Legendre wavelet method is proposed for the first time to solve singular nonlinear Lane-Emden equations with different boundary conditions.To solve the objective equation,the singularity of the equation needs to be eliminated,and the nonlinear term of the Lane-Emden equation is processed by Newton’s iteration method.According to the specific form of the obtained linear equation and the different boundary conditions of the equation,the scheme of solution space is defined as the reproducing kernel space.We find a set of Legendre wavelet basis functions in square integrable space,construct mappings from square integrable space to reproducing kernel space,and obtain the modified Legendre multiscale orthogonal basis in reproducing kernel space.Based on the collocation method and the least square method,the modified multiwavelets are applied to get the approximate solution of the linear equation in definite reproducing kernel space.At the same time,stability of the method is demonstrated in this paper.Finally,the feasibility and effectiveness of the algorithm in the paper are verified by numerical examples. |
PDF Full Text Request