Finite amplitude steady-state wave groups with multiple near resonances in deep water

In this paper, finite amplitude steady-state wave groups with multiple nearly resonant interactions in deep water are investigated theoretically. The nonlinear water wave equations are solved by the homotopy analysis method (HAM), which imposes no constraint on either the number or the amplitude of the wave components, to resolve the small-divisor problems caused by near resonances. A new kind of auxiliary linear operator in the framework of the HAM is proposed to transform the small divisors associated with the non-trivial nearly resonant components to singularities associated with the exactly resonant ones. Primary components, exactly resonant components together with nearly resonant components are considered as the initial non-trivial components, since all of them are homogeneous solutions to the auxiliary linear operator. For wave groups with weak nonlinearity, the energy transfer between nearby nearly resonant components is remarkable. As the nonlinearity increases, the number of steady-state wave groups increases as more components join the near resonance. This indicates that the probability of existence of steady-state resonant waves increases with the nonlinearity of wave groups. The frequency band broadens and spectral asymmetry becomes more and more pronounced. The amplitude of each component may either increase or decrease with the nonlinearity of wave groups, while the amplitude of the whole wave group increases continuously and finite amplitude wave groups are obtained. This work shows the wide existence of steady-state waves when multiple nearly resonant interactions are considered.

Approximate analytical solution of the nonlinear system of differential equations having asymptotically stable equilibrium

The present paper is concerned with the purely analytic solutions of the highly nonlinear systems of differential equations possessing an asymptotically stable equilibrium. A methodology combined with the homotopy analysis method is proposed. The methodology involves proper introduction of an auxiliary linear operator and an auxiliary function during the implementation of the homotopy method so that it can yield uniformly valid solutions, not affected from the existing parameters or initial conditions. The technique is applied to the systems particularly appearing in mathematical biology. The obtained explicit analytical expressions for the solution generate results that compare excellently with the numerically computed ones.

Dynamical analysis of the avian–human influenza epidemic model using multistage analytical method

In this paper, analytical result of avian–human influenza epidemic model has been investigated by applying homotopy analysis method (HAM) and by expanding it to hybrid numeric-analytic method which is known as multistage HAM (MSHAM). HAM is an algorithm which gives us the approximate solution of the problem in an arrangement of time interims and by modifying it to multistage one. Some advantages such as flexibility of picking the auxiliary linear operator and the auxiliary parameter are emerged, that leads us to achieve some excellent results in this work. Furthermore, in this analytical work, obtained results are compared and reported with numerical ones which were obtained previously from methods such as the Runge–Kutta (RK4) method.

On the Optimal Auxiliary Linear Operator for the Spectral Homotopy Analysis Method Solution of Nonlinear Ordinary Differential Equations

The purpose of this study is to identify the auxiliary linear operator that gives the best convergence and accuracy in the implementation of the spectral homotopy analysis method (SHAM) in the solution of nonlinear ordinary differential equations. The auxiliary linear operator is an essential element of the homotopy analysis method (HAM) algorithm that strongly influences the convergence of the method. In this work we introduce new procedures of defining the auxiliary linear operators and compare solutions generated using the new linear operators with solutions obtained using well-known linear operators. The applicability and validity of the proposed linear operators is tested on four highly nonlinear ordinary differential equations with fluid mechanics applications that have recently been reported in the literature. The results from the study reveal that the new linear operators give better results than the previously used linear operators. The identification of the optimal linear operator will direct future research on further applications of HAM-based methods in solving complicated nonlinear differential equations.