Homotopy Pade Technique Research Articles

Size dependent analysis of tapered FG micro-bridge based on a 3D beam theory

In the current study, an analytical solution based on the modified couple stress theory for a nonlinear model describing the couple 3D motion of a functionally graded tapered micro-bridge is presented. The small scale effects and the nonlinearity arising from the mid-plane stretching are taken into consideration. Governing equations of motions are derived utilizing the modified couple stress theory and applying Hamilton principle. Dynamic and static analyses to determine the effects of lateral distributed forces and mid-plane stretching are investigated. To this aim, analytical Homotopy-pade technique is employed to capture the nonlinear natural frequencies in high amplitude vibrations of tapered micro-bridges with different types of geometries and material compositions. The obtained results of frequencies propose that there is a good agreement between the present analytical results and the numerical ones in opposed to well-known multiple-scale method. Furthermore, comparing the results in 2D and 3D analyses shows that in 2D analysis, the stiffness and natural frequency of the micro-beam is underestimated and it is found that increasing the tapered ratio has different impacts on natural frequencies for micro-beams with different slender ratios.

Application of the optimal homotopy asymptotic method for solving a strongly nonlinear oscillatory system

In this paper, the optimal homotopy asymptotic method (OHAM) and the traditional homotopy analysis method (HAM) are used to obtain analytical solution for a strongly nonlinear oscillation. Moreover, the homotopy-pade technique is employed to accelerate the convergence of solution series of traditional HAM. Results show that the second-order approximation by the OHAM is quick convergence and more accurate than the high-order of approximation by the homotopy-pade technique.

Homotopy analysis and Homotopy Pad$acute{e}$ methods for two-dimensional coupled Burgers' equations

In this paper, analytic solutions of two-dimensional coupled Burgers' equations are obtained by the Homotopy analysis and the Ho- motopy Pade methods. The obtained approximation by using Homotopy method contains an auxiliary parameter which is a simple way to con- trol and adjust the convergence region and rate of solution series. The approximation solutions by (m, m) Homotopy Pade technique is often independent of auxiliary parameter and this technique accelerate the convergence of the related series.

A Series Solution of the Cauchy Problem for Turing Reaction-diffusion Model

In this paper, the series pattern solution of the Cauchy problem for Turing reaction-diffusion model is obtained by using the homotopy analysis method (HAM). Turing reaction-diffusion model is nonlinear reaction-diffusion system which usually has power-law nonlinearities or may be rewritten in the form of power-law nonlinearities. Using the HAM, it is possible to find the exact solution or an approximate solution of the problem. This technique provides a series of functions which converges rapidly to the exact solution of the problem. The efficiency of the approach will be shown by applying the procedure on two problems. Furthermore, the so-called homotopy-Pade technique (HPT) is applied to enlarge the convergence region and rate of solution series given by the HAM.

A uniformly valid series solution to the unsteady stagnation-point flow towards an impulsively stretching surface

The paper studies the problem of the unsteady two-dimensional stagnation-point flow of an incompressible viscous fluid over a flat deformable sheet. The flow is started impulsively from rest and the sheet is suddenly stretched in its own plane with a velocity proportional to the distance from the stagnation point. An analytical series solution is obtained by means of the homotopy analysis method (HAM). Also, the homotopy-Pade technique is employed. An explicit formula for the local friction coefficient is provided. The present formula, different from the perturbation solution, is accurate and uniformly valid for all dimensionless time in the whole spatial region and for all possible values of physical parameter λ, defined as the ratio of the potential flow velocity to the sheet sudden stretching velocity. Numerical tests are done to verify the present formula for its validity and accuracy.

Series solution of non-similarity natural convection boundary-layer flows over permeable vertical surface

The non-similarity solution for natural convection from a permeable isothermal vertical wall is considered. The governing boundary-layer equations for non-similarity flow and temperature fields are solved using the homotopy analysis method. The homotopy-Pade technique is applied to accelerate the convergence of the homotopy-series solution. The influence of physical parameters on the non-similarity flows is investigated in detail. Different from the previous analytic results, the homotopy-series solutions are convergent and valid for all physical parameters in the whole domain 0 ⩽ x < ∞ and 0 ⩽ y < ∞.

Accurate approximate analytical solutions for multi-degree-of-freedom coupled van der Pol-Duffing oscillators by homotopy analysis method

In this paper, the homotopy analysis method (HAM) is presented to establish the accurate approximate analytical solutions for multi-degree-of-freedom (MDOF) nonlinear coupled oscillators. The periodic solutions for the three-degree-of-freedom (3DOF) coupled van der Pol-Duffing oscillators are applied to illustrate the validity and great potential of this method. For given physical parameters of nonlinear systems and with different initial conditions, the frequency ω, displacements x1(t),x2(t) and x3(t) can be explicitly obtained. In addition, comparisons are conducted between the results obtained by the HAM and the numerical integration (i.e. Runge–Kutta) method. It is shown that the analytical solutions of the HAM are in excellent agreement with respect to the numerical integration solutions, even if time t progresses to a certain large domain in the time history responses. Finally, the homotopy Pade technique is used to accelerate the convergence of the solutions.

Duffing equations with cubic and quintic nonlinearities

In this study, an accurate analytical solution for Duffing equations with cubic and quintic nonlinearities is obtained using the Homotopy Analysis Method (HAM) and Homotopy Pade technique. Novel and accurate analytical solutions for the frequency and displacement are derived. Comparison between the obtained results and numerical solutions shows that only the first order approximation of the Homotopy Pade technique leads to accurate solution with a maximum relative error less than 0.4%.

Nonlinear free vibration of conservative oscillators with inertia and static type cubic nonlinearities using homotopy analysis method

In this study, an accurate analytical solution for the nonlinear free vibration of a conservative oscillator with inertia and static type cubic nonlinearities is derived. This solution has been obtained using homotopy analysis method (HAM). Then, homotopy Pade technique is applied to accelerate the convergence rate of the series solution. This study shows that the HAM leads to an accurate analytical solution, which is valid for a wide range of considered system parameters. Unlike the other analytical methods, HAM can control and adjust the convergence region and rate of the approximation series solution. The excellent accuracy of the current results is demonstrated by comparing with the available analytical and numerical results.