Numerical approximations for a class of nonlinear higher order singular boundary value problem by using homotopy perturbation and variational iteration method

In this work, we focus on the non-linear fourth order class of singular boundary value problem which contains the parameter $\lambda$. We convert this non-linear differential equation into third order non-linear differential equation. The third order problem is singular, non self adjoint, and nonlinear. Moreover, depending upon $\lambda$, it admits multiple solutions. Hence, it is too difficult to capture these solutions by any discrete method such as finite difference and so forth. Here we develop an iterative technique with the help of homotopy perturbation method and variational iteration method in a suitable way. Convergence of this series solution is studied in a novel way. We compute these solutions numerically. For small positive values of $\lambda$, singular BVP has two solutions while solutions can not be found for large positive values of $\lambda$. Furthermore, we also find dual solutions for $\lambda <0$.