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In this study, the homotopy analysis method (HAM) is used to solve the generalized Duffing equation. Both the frequencies and periodic solutions of the nonlinear Duffing equation can be explicitly and analytically formulated. Accuracy and validity of the proposed techniques are then verified by comparing the numerical results obtained based on the HAM and numerical integration method. Numerical simulations are extended for even very strong nonlinearities and very good correlations which achieved between the results. Besides, the optimal HAM approach is introduced to accelerate the convergence of solutions.
The Laplace transform is a very useful tool for the solution of problems involving an impulsive excitation, usually represented by the Dirac delta, but it does not work in nonlinear problems. In contrast with this, the parametric representation of the Dirac delta presented here works both in linear and nonlinear problems. Furthermore, the parametric representation converts the differential equation of a problem with an impulsive excitation into two equations: the first equation referring to the impulse instant (absent in the conventional solution) and the second equation referring to post-impulse time. The impulse instant equation contains fewer terms than the original equation and the impulse is represented by a constant, just as in the Laplace transform, the post-impulse equation is homogeneous. Thus, the solution of the parametric equations is considerably simpler than the solution of the original equation. The parametric solution, involving the equations of both the dependent and independent variables in terms of the parameter, is readily reconverted into the usual equation in terms of the dependent and independent variables only. This parametric representation may be taught at an earlier stage because the principle on which it is based is easily visualized geometrically and because it is only necessary to have a knowledge of elementary calculus to understand it and use it.
This paper mainly concerns oblique derivative problems
for nonlinear nondivergent elliptic equations of second order with measurable
coefficients in a multiply connected domain. Under certain condition, we derive
a priori estimates of solutions. By using these estimates and the fixed-point
theorem, we prove the existence of solutions.