Numerical Study For Linear Singular Two-point Boundary Value Problems Based On Bernstein Polynomials

The singular two-point boundary value problem arises in a variety of differential applied mathematics and physics such as gas dynamics, nuclear physics, chemical reaction, studies of atomic structures and atomic calculations and in the study of positive radial solutions of nonlinear elliptic equations, physiological studies, in the study of steady state oxygen diffusion in a spherical cell and the distribution of heat sources in the human head. Accurate and efficient numerical methods are often necessary for the solution of singular two-point boundary value problems (SBVPs) for ordinary differential equations. Therefore, the problem has attracted much attention and has been studied by many authors.In chapter one of the paper, the development of the singular two-point boundary value problem is introduced.The discrete variable numerical solution of the singular two-point boundary value problems has been considered by many authors and the references given in these papers.In chapter two,the numerical solution was suggested in the case of linear singular two point boundary value problems. First, in comparison with finite difference methods, the spline solution has its own advantages. For example, once the solution has been computed, the information required for spline interpolation between mesh points is available.This is particularly significant when the solution of the boundary value problem is required at various locations in the interval [0,1].In this paper, we study the cubic spline method,the quartic B-spline method,the second order spline finite difference method,the Parametric spline method for regular linear singular two point boundary value problems.Secondly, we discuss same numerical methods for (regular) linear singular two point boundary value problems via Chebyshev economization.Attempts by many researchers for the removal of singularity are based on using the series expansion procedures in the neighborhood (0,δ) of singularity and then solve the regular boundary value problem in the interval (δ,1) using any numerical method.In chapter three, we summarize three numerical solutions of nonlinear singular two point boundary value problems, such as:The Adomian decomposition method(ADM), The improved Adomian decomposition method (IADM), Taylor series method(TSM).In chapter four, we develop a formula based on Bernstein polynomials in compression for a class of singular two-point boundary value problems. This method does not require non-singularity of the equation, and if the exact solution is a polynomial solution, we can get the exact solution by using this method. Some examples have been included and comparison of the numerical results made with other methods. The reliability and efficiency of the proposed method are demonstrated by several numerical examples.In the last chapter, the paper draws the conclusion and makes some prospects of the further work.