Application Of Homotopy Analysis Method Research Articles

Parametric resonance of multi-frequency excited MEMS based on homotopy analysis method

The nonlinear dynamics and control of micro electromechanical systems (MEMS) is an important topic at present and homotopy analysis method (HAM) is an effective semi-numerical and semi-analytical method for solving strongly nonlinear problems. In this paper, the parametric resonance of MEMS with multi-frequency excitation is studied by HAM. Firstly, the differential equation of electrostatically driven microbeam is processed by Taylor expansion, and it is transformed into the parametric motion model with multi-frequency excitation. Then, the approximate solution and amplitude–frequency response equation of the system are obtained by HAM, and compared with the numerical solution. Finally, the influence of direct current (DC) and alternating current (AC) on principal parametric resonance and superharmonic resonance is discussed. The results show that HAM is an effective method to analyze the parametric vibration of multi-frequency excitation system, and the amplitude–frequency response curve of microbeam about parametric motion depends on the time scale in high DC and high AC state. This study effectively extends the application of HAM in parametric resonance, which is of great significance to the study of nonlinear vibration of MEMS.

Analysis of solitons structure of the damped KdV equation arising in superthermal plasmas: Application of homotopy analysis method

AbstractThe aim of the proposed work is to analyze the soliton structures of dust‐ion acoustic waves obtained in the framework of the Korteg‐de Vries (KdV) equation with the presence of a damping term. The concept of electron acoustic solitary wave in an unmagnetized plasma consisting of superthermal electrons has been taken into consideration. The KdV equation with the presence of a damping term has been derived with the help of the reductive perturbation technique and solved by using the well‐known homotopy analysis method. The considered method approximates all problems in a straightforward and simplified manner. The method computes the series solution efficiently and provides a simple way to ensure its convergence. The approximate analytical solution obtained from the present analysis is compared with available results in the literature for a different choice of pertinent parameters. The upshots specified that the amplitude of solitary waves increases for increasing values of the damping parameter. This study would in a way to demonstrate the potential and effectiveness of the homotopy analysis method to evaluate the various kinds of nonlinear equations arising in the soliton theory.

Application of HAM for Nonlinear Integro-Differential Equations of Order Two

In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable; therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type; then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.

APPLICATION OF HOMOTOPY ANALYSIS METHOD (HAM) TO THE NON-LINEAR KDV EQUATIONS

In this work, approximate analytic solutions for different types of KdV equations are obtained using the homotopy analysis method (HAM). The convergence control parameter h helps us to adjust the convergence region of the approximate analytic solutions. The solutions are obtained in the form of power series. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of non-linearity. We have compared the approximate analytical results which are determined by HAM, with the exact solutions and shown graphically with their absolute errors. By choosing an appropriate value of the convergence control parameter, we can obtain the solution in few iterations. All the computations have been performed using the software package MATHEMATICA.

Homotopy analysis approach to study the dynamics of fractional deterministic Lotka-Volterra model

The population dynamics of two species that is governed by the deterministic Lotka-Volterra model concerned with the interaction of predator and prey is investigated in this article as an application of homotopy analysis method. The analytical approximate solution in the form of convergent infinite series is obtained by considering the time-fractional derivatives in the Caputo sense. The simulation of obtained results exhibit the effect of variation in fractional parameters, auxiliary parameters and auxiliary linear operators on the mass concentration of both the biological species which in turn affects the structure of the system.

Time-delay feedback control of a cantilever beam with concentrated mass based on the homotopy analysis method

In this paper, the strongly nonlinear vibration of a cantilever beam system with concentrated mass at an intermediate position controlled by displacement and velocity time delay is investigated. The nonlinear governing equation is studied using the homotopy analysis method. The effects of the time delay, displacement and velocity feedback gain coefficients, as well as frequency on the amplitude of the system were studied in detail. The results indicate that the velocity feedback control is not necessarily superior to displacement feedback control. Reasonable selection of the displacement and velocity time delay parameters can avoid the Hopf bifurcation, adjust the number, amplitude and stability of periodic solutions, and exhibit the softening behavior at low frequency and hardening spring characteristic at high frequency. The theoretical research in this paper will promote the application of homotopy analysis method in the field of time delay control, and serve as a theoretical reference for the design and optimization of cantilever beam control systems with concentrated mass.

On the Application of Homotopy Analysis Method to Fractional Differential Equations

In this paper, an attempt has been made to obtain the solution of linear and nonlinear fractional differential equations by applying an analytic technique, namely the homotopy analysis method (HAM). The fractional derivatives are described by Caputo’s sense. By this method, the solution considered as the sum of an infinite series, which converges rapidly to exact solution with the help of the nonzero convergence control parameter ℏ. Some examples are given to show the efficiently and accurate of this method. The solutions obtained by this method has been compared with exact solution. Also our graphical represented of the solutions have been given by using MATLAB software.

Application of HAM to laminar boundary layer flow over a wedge with an external magnetic field

AbstractThe incompressible laminar along with injection or suction over a wedge for a boundary layer flow is studied. Falkner–Skan transformations reduce the governing partial differential equations (PDEs) to two nonlinear coupled PDEs and are solved by the homotopy analysis method (HAM). The variation of a dimensionless temperature and velocity profiles and for , , and to the nonidentical values of the parameter for injection/suction has been shown in the graph. The results hence obtained show that the flow field is determined by the existence of the applied magnetic field. The finite difference method is applied to the reduced PDEs and the velocity and temperature profiles are compared with the HAM solutions and depicted graphically. To distinguish singularities in the graph, we have applied Pade for the HAM series solution, which is depicted in graphs for all three cases. We have also estimated the radius of convergence of HAM solutions by Domb–Sykes plot for injection, no suction, and suction respectively. The important observation made by us through HAM and numerical solution is the existence of flow separation for injection, which is not shown by previous authors.

Application of homotopy analysis method to the determination of vertical sediment concentration distribution with shear-induced diffusivity

Application of homotopy analysis method to the determination of vertical sediment concentration distribution with shear-induced diffusivity

An analytical investigation of the mixed convective Casson fluid flow past a yawed cylinder with heat transfer analysis

Abstract The hydrodynamic flow of an incompressible and isotropic Casson fluid through a yawed cylinder is investigated by employing continuity, momentum, and energy equations satisfying suitable boundary conditions. The density variation is governed by Boussinesq approximation. The model equations consisting of coupled partial differential equations (PDEs) are transformed by applying non-similar transformation relations. The set of transformed PDEs is solved using the analytical technique of homotopy analysis method (HAM). The impacts of varying yaw angle, and mixed convection and Casson parameters over fluid velocity (chordwise and spanwise components), its temperature, Nusselt number, and skin friction coefficients are investigated and explained through various graphs. It is found that the enhancing yaw angle, Casson parameter, and convection parameter augment the fluid velocity, heat transfer rate, and skin friction and reduce the fluid temperature. The agreement of present and published results justifies the application of HAM in modeling the mixed convective Casson fluid flow past a yawed cylinder.

Application of Homotopy Analysis Method for Investigating Nonlinear Oscillations

Application of Homotopy Analysis Method for Investigating Nonlinear Oscillations

The Application of Homotopy Analysis Method to Determine the Thermal Response of Convective-Radiative Porous Fins with Temperature-Dependent Properties

In this paper, the Homotopy Analysis Method (HAM) is utilized to investigate the thermal response of a circular convective-radiative porous fin having a rectangular cross-section. It is assumed that internal heat generation depends linearly on the temperature for the solid part of the porous fin, while temperature-dependent functions are used for thermal conductivity and convective terms. The thermal conductivity and convective coefficient of heat transfer are supposed to change respectively as a nonlinear hyperbolic and as a power-law function of the temperature all through the fin length. The function employed for the thermal conductivity can be applied for the thermal analysis if for example fins are made of semiconductor materials. Additionally, the general form of convective function enables us to account for the convection heat transfer in different processes. The heat transfer analysis in the porous media is conducted through passage velocity in Darcy’s model. HAM is utilized to study the effects of different pertinent parameters and nondimensional numbers on the described problem. The accuracy and convergence of HAM are validated with numerical methods (i.e., Runge–Kutta method) as well as other published works that reveal an excellent agreement.

Homotopy Analysis Method Applied SABR and XVA

The SABR models are very popular in the financial industry. Their evaluation using asymptotic formulas for implied volatilities is fast and widely used but becomes less accurate and exhibits negative probability distribution values around zero strikes for long maturities. In addition, alternative version for more accurate valuation exists but involves numerical integration. In this paper, we apply the homotopy analysis method (HAM) to derive approximated option prices under a SABR model. This scheme is simple and our numerical examples demonstrate that the derived price can be evaluated easily and gives a good approximation with a computational cost that is only several times heavier than that of the Black-Scholes model. We also discuss the application of HAM to XVA evaluation.

Application of HAM to the von Kármán Swirling Flow with Heat Transfer Over a Rough Rotating Disk

In this study, analytical solution to the classical von Karman swirling viscous flow with heat transfer is obtained. Using similarity transformations the governing partial differential equations are reduced to a set of coupled, nonlinear ordinary differential equations and the conventional no-slip boundary conditions are replaced by partial slip boundary conditions due to the roughness of the disk. An effective analytical method for fully coupled and highly nonlinear differential equations, called Homotopy Analysis Method (HAM) is adopted and the solutions are obtained in the form of an convergent Taylor series. The effect of azimuthally anisotropic roughness, radially anisotropic roughness and isotropic roughness on the velocity and temperature profiles are studied in detail. Result shows that HAM is very efficient and easy to implement.

Weakly resonant double Hopf bifurcation in coupled nonlinear systems with delayed freedback and application of homotopy analysis method

In this paper, we study the dynamical behaviors of coupled nonlinear systems with delay coupling by the multiple scales method and the homotopy analysis method. Firstly, we analyze the distribution...

Application of Homotopy Analysis Method for the Pull-In and Nonlinear Vibration Analysis of Nanobeams Using a Nonlocal Euler–Bernoulli Beam Model

Abstract In this investigation, the homotopy analysis method (HAM) is utilized for the pull-in and nonlinear vibration analysis of nanobeams based on the stress-driven model (SDM) of nonlocal elasticity theory. The physical properties of nanobeams are assumed not to vary through the thickness. The nonlinear equation of motion and the corresponding boundary condition are derived on the basis of the Euler–Bernoulli beam theory. For the solution purpose, the Galerkin method is employed for reducing the nonlinear partial differential equation to a nonlinear ordinary differential equation in the time domain, and then, the resulting equation is analytically solved using the HAM. In the results section, the influences of different parameters, including nonlocal parameter, electrostatic and intermolecular van der Waals forces and fringing field effect changes on the pull-in and nonlinear vibration response are investigated.

Comment on the paper “Application of homotopy analysis method to solve MHD Jeffery–Hamel flows in non-parallel walls, S.M. Moghimi, G. Domairry, Soheil Soleimani, E. Ghasemi, H. Bararnia, Advances in Engineering Software 42 (2011) 108–113”

The present comment concerns some doubtful results included in the above paper.

Application of homotopy analysis method to the solution of ninth order boundary value problems in AFTI-F16 fighters

This paper is devoted to the study of ninth order boundary value problems that emerge in the mathematical modeling of AFTI-F16 fighter. An augmented longitudinal and lateral dynamics of the AFTI-F16 fighter is described by ninth order differential equations, containing unknown parameters which can be determined using automated system identification algorithms. The solution of the boundary value problem is obtained in terms of a convergent series using homotopy analysis method (HAM). The method is effectively applied on numerical examples and the results are compared with those given in the literature, revealing that the presented method gives better approximations to the exact solution.

Solution of Riccati Type Nonlinear Fractional Differential Equation by Homotopy Analysis Method

The present paper deals with the application of Homotopy Analysis Method (HAM) to solve Riccati type nonlinear fractional differential equation. After the applications of various analytical methods in different forms to solve many linear and nonlinear problems (see Ref.: Liao and Shijun, Homotopy Analysis Method in Nonlinear Differential Equations. Springer-Verlag Berlin, 2012), a new method known as HAM which has a convergence control parameter  introduced in the deformation equation to reach the corresponding series solution in a much easier way. The present analysis is accompanied by numerical examples to justify its validity and efficiency. The solution obtained by this method has been compared with those obtained by Power Series Method (PSM) and Adomian Decomposition Method (ADM). The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus. The graphical representations of the solutions obtained by different methods are also presented for comparison of the solutions.

Application of Homotopy Analysis Method to Solve Fuzzy Volterra Integro-Differential Equations

Application of Homotopy Analysis Method to Solve Fuzzy Volterra Integro-Differential Equations