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homotopy approaches are able to estimate the frequency

and period of the oscillations in a better way. Moreover,

unlike the solution of He, no singularity appears in the

displacement solution. Thus it can be concluded that the

homotopy method presented here can safely handle the

situations of high nonlinearity occurring in the Duffing

problems.

5. Concluding Remarks

In this paper the nonlinear problem of Duffing equation

has been considered with the homotopy analysis tech-

nique. As compared to the perturbation methods, the

homotopy treatment employed here does not require any

small parameter, yielding explicit formulas for the

quantities of physical desire, such as the displacement,

frequency and the period of the oscillations.

Four homotopy approaches have been pursued in this

analysis. First, the homotopy method of He in [15] has

been reinvestigated further calculating the second-order

contribution that was missed in [15]. Taking the

advantage of free selection of the linear operator and the

initial approximation to the solution, two more approa-

ches have been proposed here. The homotopy methods

with these choices are proven to generate analytic

approximations that are more accurate than the result

presented in [15]. Moreover, the approximate solutions

obtained by these techniques are valid not only for small

parameters but also very large parameters that are in-

volved in the degree of the nonlinearity. The purely

explicit analytical formulas obtained for the frequency

and period agree excellently with the exact values even

for infinitely large parameters, the discrepancy of the

approximate period with respect to the exact one is as

low as 0.26%. In addition to this, the displacement

function is uniformly valid and does not exhibit any

singularities as compared to the solution in [15].

The homotopy technique proposed here can be safely

adopted for sets of fully coupled, highly nonlinear

equations governing other physical problems in science

and engineering. Analytical solutions obtained here also

provide a good scientific base for the validation of the

numerically computed values using different schemes in

the literature.

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