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.
Highlights
The homotopy analysis method (HAM) is an analytical solution method which allows one to approximate the solution to a variety of problems, such as nonlinear ordinary differential equations, partial differential equations, integral equations [1,2,3]
We demonstrate that the approach is efficient for an appropriate selection of auxiliary operator and convergence control parameter
We have considered the application of the optimal homotopy analysis method (OHAM) to the solution of implicitly defined ordinary differential equations
Summary
The homotopy analysis method (HAM) is an analytical solution method which allows one to approximate the solution to a variety of problems, such as nonlinear ordinary differential equations, partial differential equations, integral equations [1,2,3]. The HAM gives one great freedom in selecting the form of the solutions via representation of solutions [1, 16], since one has control over the type of basis functions employed in such a representation This is tied closely to the choice of auxiliary operator used in the construction of the homotopy. A review of the open literature reveals that the problem of implicitly defined differential equations has not been well-studied in the context of the HAM or for other related analytic methods.
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