Analytical method and its convergence analysis based on homotopy analysis for the integral form of doubly singular boundary value problems

In this paper, we consider the nonlinear doubly singular boundary value problems $(p(x)y^{\mathrm{\prime }}(x){)}^{\mathrm{\prime }}+q(x)f(x,y(x))=0,0<x<1$ with Dirichlet/Neumann boundary conditions at $x=0$ and Robin type boundary conditions at $x=1$. Due to the presence of singularity at $x=0$ as well as discontinuity of $q(x)$ at $x=0$, these problems pose difficulties in obtaining their solutions. In this paper, a new formulation of the singular boundary value problems is presented. To overcome the singular behavior at the origin, with the help of Green’s function theory the problem is transformed into an equivalent Fredholm integral equation. Then the optimal homotopy analysis method is applied to solve integral form of problem. The optimal control-convergence parameter involved in the components of the series solution is obtained by minimizing the squared residual error equation. For speed up the calculations, the discrete averaged residual error is used to obtain optimal value of the adjustable parameter $c_0$ to control the convergence of solution. The proposed method $\mathbf{\text{(a)}}$ avoids solving a sequence of transcendental equations for the undetermined coefficients $\mathbf{\text{(b)}}$ it is a general method $\mathbf{\text{(c)}}$ contains a parameter $c_0$ to control the convergence of solution. Convergence analysis and error estimate of the proposed method are discussed. Accuracy, applicability and generality of the present method is examined by solving five singular problems.