An iterative method to compute dual solutions on a class of nonlinear
higher order SBVPs arising in MBE
Abstract
In this work, we focus on the following non-linear fourth order SBVP
\begin{eqnarray} \nonumber
\frac{1}{r}\left[ r
\left\lbrace
\frac{1}{r} \left(r
\phi’ \right)^{’}
\right\rbrace^{’}\right]^{’}=\frac{\phi’
\phi’‘}{r}+\lambda,
\end{eqnarray} where $\lambda$ is a
parameter. We convert this non-linear differential equation into third
order non-linear differential equation, which is given by
\begin{eqnarray} \nonumber
\frac{1}{r}\left[ r
\left\lbrace
\frac{1}{r} \left(r y
\right)^{’}
\right\rbrace^{’}\right]^{’}=\frac{y
y’}{r}+\lambda. \end{eqnarray} The
problem is singular, non self adjoint, 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 etc. Here we propose an iterative technique by
using homotopy perturbation method (HPM) with the help of variational
iteration method (VIM) in a suitable way. We compute these solutions
numerically. Convergence of this series solution is studied in a novel
way. 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 $\lambda <0$.