Numerical solution for third order singularly perturbed turning point problems with integral boundary condition

In this paper, a third-order singularly perturbed differential equation with integral boundary condition (IBC) is considered. The problem is reduced into system of differential equation, one compromises initial value problem and another one is second order singularly perturbed differential equation with integral boundary condition. Due to the presence of turning point at r=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=0,$$\end{document} the problem exhibit boundary layer at r=-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=-1$$\end{document} and r=1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=1.$$\end{document} To tackle this type of problem, a thorough study is required to obtain a priori estimations on the solution and its derivatives of the considered problem. We present a numerical technique adopting an upwind finite difference scheme on a dense piece-wise uniform mesh at the boundary layers. The proposed method is almost first-order convergent. Some numerical examples are provided to validate the theoretical findings.