Optimal homotopy analysis and control of error for implicitly defined fully nonlinear differential equations

Abstract

Implicitly defined fully nonlinear differential equations can admit solutions which have only finitely many derivatives, making their solution via analytical or numerical techniques challenging. We apply the optimal homotopy analysis method (OHAM) to the solution of implicitly defined ordinary differential equations, obtaining solutions with low error after few iterations or even one iteration of the method. This is particularly true in cases where an auxiliary nonlinear operator was employed (in contrast to the commonly used choice of an auxiliary linear operator), highlighting the need for further study on using auxiliary nonlinear operators in the HAM. Through various examples, we demonstrate that the approach is efficient for an appropriate selection of auxiliary operator and convergence control parameter.