Convergence Control Parameter Research Articles

The Optimal Homotopy Analysis Method for solving linear optimal control problems

In this paper, an Optimal Homotopy Analysis Method (Optimal HAM) is applied to solve the linear optimal control problems (OCPs), which have a quadratic performance index. This approach contains at most two convergence-control parameters which depend on the control system and is computationally rather efficient. A squared residual error for the system is defined, which can be used to find the unknown optimal convergence-control parameters by using Mathematica package BVPh (version 2.0). The results of comparisons among the proposed method, the homotopy perturbation method (HPM), the Adomian decomposition method (ADM), the differential transform method (DTM) and the homotopy analysis method (HAM) provide verification for the validity of the proposed approach. Moreover, numerical results are presented by several examples involving scalar and 2nd-order systems to clarify the efficiency and high accuracy of the proposed approach.

Computational analysis for velocity slip and diffusion species with carbon nanotubes

This article addresses a computational study for carbon nanotubes with diffusion species. Mathematical analysis and modeling are formulated in the presence of slip effect, Darcy porous medium and chemical species. Diffusion coefficients are accounted as equal amount. Optimal Homotopy Analysis Method (OHAM) is implemented in taking care for convergence control parameters. Residual errors and its graphs are plotted for CNTs nanofluids. The current data is compared with the previous published work. The results are found in favorable agreement. Physically interesting parameters are highlighted. The velocity is dominated for MWCNT than SWCNT. The velocity profile increases for power index when m>1 while opposite trend is observed for power index when m<1. Permeability parameter and volume fraction reduce the skin friction coefficient and reverse trend is noticed for slip parameter and wall thickness parameter. Opposite behavior is noticed for concentration profile at surface for homogenous-heterogeneous reactions parameters.

Optimal q-homotopy analysis method for time-space fractional gas dynamics equation

It is well known that the homotopy analysis method is one of the most efficient methods for obtaining analytical or approximate semi-analytical solutions of both linear and non-linear partial differential equations. A more general form of HAM is introduced in this paper, which is called Optimal q-Homotopy Analysis Method (Oq-HAM). It has better convergence properties as compared with the usual HAM, due to the presence of fraction factor associated with the solution. The convergence of q-HAM is studied in details elsewhere (M.A. El-Tawil, Int. J. Contemp. Math. Sci. 8, 481 (2013)). Oq-HAM is applied to the non-linear homogeneous and non-homogeneous time and space fractional gas dynamics equations with initial condition. An optimal convergence region is determined through the residual error. By minimizing the square residual error, the optimal convergence control parameters can be obtained. The accuracy and efficiency of the proposed method are verified by comparison with the exact solution of the fractional gas dynamics equation. Also, it is shown that the Oq-HAM for the fractional gas dynamics equation is equivalent to the exact solution. We obtain graphical representations of the solutions using MATHEMATICA.

An improved homotopy analysis method with accelerated convergence for nonlinear problems

In this paper we propose an accelerated convergence method, which is combined with the homotopy analysis method (HAM), to solve nonlinear problems. The HAM is applied to obtain approximate expressions. According to the numbers of terms in the approximations, some ratio-control parameters are introduced in the solution expressions. By solving simultaneous algebraic equations, all artificial parameters can be optimally identified, including the so-called convergence-control parameter ℏ. Twoexamples are given to illustrate the validity of the new method. Comparison with L-P perturbation method and Runge-Kutta method reveals that the improved HAM is better than the standard HAM and applies especially to the problems with complicated nonlinear terms.

Modified Homotopy Analysis Method for Nonlinear Aeroelastic Behavior of Two Degree-of-Freedom Airfoils

In this paper, the homotopy analysis method (HAM) is extended to deal with the nonlinear aeroelastic problem of a two degree-of-freedom (DOF) airfoil. To avoid determination of the parameter for the complicated high-order minimization problem, a new modified HAM is proposed based on the idea of minimizing the squared residual. Using this method, the convergence-control parameter is determined by the low order squared residual of the governing equations, and then the problem is solved in a way similar to the basic HAM. The proposed method is used to solve the nonlinear aeroelastic behavior of a supersonic airfoil, with the unsteady aerodynamic load evaluated by the piston theory. Two examples are prepared, for which the frequencies and amplitudes of the limit cycles are obtained. The approximate solutions obtained are demonstrated to agree excellently the numerical solutions, meanwhile, the convergence-control parameter can be easily determined using the present approach.

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.

Improved First-Order Approximate Models of Temperature and Thermal Stresses for Online Fatigue Monitoring

Online monitoring of temperature and thermal stresses is an important way to ensure the safety of power plants considering fatigue and creep damages. The effect of online monitoring is determined by the accuracy and calculating time of monitoring models. In this paper, the improved first-order analytical models of temperature and thermal stresses considering temperature-dependent material properties have been derived by using homotopy analysis method (HAM) and superposition principle. The optimal convergence control parameters are obtained by calculating the mean-square residual errors. The validity and accuracy of the proposed models were proved by results comparisons with finite element method (FEM) and artificial parameter method.

Solution of Higher Order Nonlinear Time-Fractional Reaction Diffusion Equation

The approximate analytical solution of fractional order, nonlinear, reaction differential equations, namely the nonlinear diffusion equations, with a given initial condition, is obtained by using the homotopy analysis method. As a demonstration of a good mathematical model, the present article gives graphical presentations of the effect of the reaction terms on the solution profile for various anomalous exponents of particular cases, to predict damping of the field variable. Numerical computations of the convergence control parameter, used to evaluate the convergence of approximate series solution through minimizing error, are also presented graphically for these cases.

3D stagnation point flow of Maxwell fluid towards an off-centered rotating disk

Purpose – The purpose of this paper is to study the three dimensional, steady and incompressible flow of non-Newtonian rate type Maxwell fluid, for stagnation point flow toward an off-centered rotating disk. Design/methodology/approach – The governing partial differential equations are transformed to a system of non-linear ordinary differential equations by conventional similarity transformations. The non-perturbation technique, homotopy analysis method (HAM) is employed for the computation of solutions. And, the solution is computed by using the well-known software Mathematica 10. Findings – The effects of rotational parameter and Deborah number on radial, azimuthal and induced velocity functions are investigated. The results are presented in graphical form. The convergence control parameter is also plotted for velocity profiles. The comparison with the previous results is also tabulated. The skin friction coefficients are also computed for different values of Deborah number. Originality/value – This paper studies the effect of rotation and Deborah number on off-centered rotating disk has been observed and presented graphically.

Effects of homogeneous and heterogeneous reactions in flow of nanofluids over a nonlinear stretching surface with variable surface thickness

This article addresses the effects of homogeneous-heterogeneous reactions in boundary layer flow of nanofluids with different base fluids. The flow is induced due to an impermeable nonlinear stretching sheet with unfixed surface shape. Investigation for CNTs (single and multi walled carbon nanotubes) is considered in water, kerosene and engine oils. Equal diffusion coefficients are taken for both reactant and auto catalyst. Boundary layer problems for velocity and concentration functions are determined. Optimal homotopy analysis method (OHAM) is used to examine the convergence control parameters for series solutions and residual errors for water, kerosene and engine oils in the presence of single wall (SWNTs) and multi wall (MWNTs) carbon nanotubes. Outcomes for the velocity and concentration distributions are calculated and analyzed. It is observed that the homogenous and heterogeneous parameters have opposite behaviors for concentration profile. Skin friction coefficient for the pertinent parameters is analyzed through graphs.

Application of HAM to seventh order KdV equations

In this paper, the homotopy analysis method (HAM) is applied to obtain the approximate solutions of KdV equations of seventh order, which are Sawada Kotera Ito equation, Lax equation, and Kaup–Kuperschmidt equation, respectively. The convergence of the homotopy analysis method is discussed with the help of auxiliary parameter h, which controls the convergence of the method and also called as the convergence control parameter. The results obtained by the HAM are compared with the exact solutions, by fixing the value of arbitrary constant. It is found that HAM is very robust and elegant method, and by choosing a suitable value of h we can get the approximate solution in very few iteration. Computations are performed with the help of symbolic computation package MATHEMATICA.

Nonlinear dynamics of MEMS/NEMS resonators: analytical solution by the homotopy analysis method

Due to various sources of nonlinearities, micro/nano-electro-mechanical-system (MEMS/NEMS) resonators present highly nonlinear behaviors including softening- or hardening-type frequency responses, bistability, chaos, etc. The general Duffing equation with quadratic and cubic nonlinearities serves as a characterizing model for a wide class of MEMS/NEMS resonators as well as lots of other engineering and physical systems. In this paper, after brief reviewing of various sources of nonlinearities in micro/nano-resonators and discussing how they contribute to the Duffing-type nonlinearities, we propose a Homotopy Analysis Method (HAM) approach for derivation of analytical solutions for the frequency response of the resonators. Toward this aim, we first apply the HAM to the proposed Duffing equation, and through this procedure, we derive the first-order and second-order HAM-based analytical solutions for the frequency response of the resonator. As the main novelty, we show that the second-order solution benefits from a tunable parameter, known as the convergence-control parameter, which is a distinguishing aspect of the HAM and plays a key role in enhancing the accuracy of the obtained analytical expressions in strongly nonlinear problems. We use the obtained analytical solutions for the study of nonlinear dynamics in two types of electrostatically actuated MEMS resonators proposing hardening, softening or mixed behaviors near their primary resonance frequency. Numerical simulations are performed to validate the analytical results.

Homotopy Analysis of the Radiation Effect on MHD Flow with Heat and Mass Transfer due to a Point Sink

An analytical solution of the magnetohydrodynamic, steady, and incompressible laminar boundary layer flow in the presence of heat and mass transfer as well as magnetic field on a cone due to a point sink by using the homotopy analysis method (HAM) has been studied under the radiative fluid properties. The HAM produces an analytical solution of the governing self-similar nonlinear two-point boundary layer equations. The effects of the suction/injection, magnetic, and radiation parameters over the obtained solution have been discussed. The effects of Prandtl number on temperature and Schmidt number on concentration profiles have also been studied. It has been observed that the temperature profiles exhibit an increasing trend with radiation in case of injection while an opposite trend is observed in case of suction. The results obtained in the present study have also been compared numerically as well as graphically with the corresponding results obtained by using other methods. An excellent agreement has been found between them. The analytical solution obtained by the HAM is very near to the exact solution for a properly selected initial guess, auxiliary, and convergence control parameters and for higher orders of deformations.

Melting heat transfer in stagnation point flow of carbon nanotubes towards variable thickness surface

This work concentrates on the mathematical modeling for stagnation point flow of nanofluids over an impermeable stretching sheet with variable thickness. Carbon nanotubes [single-wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes (MWCNTs)] as the nanoparticles are utilized. Water and kerosene oil are taken as the base fluids. Heat transfer through melting effect is discussed. Transformation procedure is adapted to obtain the non-linear ordinary differential equations from the fundamental laws of mass, linear momentum and energy. The optimal values of convergence control parameters and corresponding individual and total residual errors for SWCNTs and MWCNTs are computed by means of homotopy analysis method (HAM) based BVPh 2.0. Characteristics of different involved parameters on the velocity, temperature, skin friction coefficient and Nusselt number are discussed. Higher velocity profile is observed for wall thickness parameter in case of water carbon nanotubes when compared with the kerosene oil carbon nanotubes.

AN OPTIMAL HOMOTOPY ANALYSIS METHOD BASED ON PARTICLE SWARM OPTIMIZATION: APPLICATION TO FRACTIONAL-ORDER DIFFERENTIAL EQUATION

This paper describes a new problem-solving mentality of finding optimal parameters in optimal homotopy analysis method (optimal HAM). We use particle swarm optimization (PSO) to minimize the exact square residu- al error in optimal HAM. All optimal convergence-control parameters can be found concurrently. This method can deal with optimal HAM which has fi- nite convergence-control parameters. Two nonlinear fractional-order differen- tial equations are given to illustrate the proposed algorithm. The comparison reveals that optimal HAM combined with PSO is effective and reliable. Mean- while, we give a sufficient condition for convergence of the optimal HAM for solving fractional-order equation, and try to put forward a new calculation method for the residual error.

An effective approach for evaluation of the optimal convergence control parameter in the homotopy analysis method

A rapid and effective way of working out the optimum parameter of convergence control in the homotopy analysis method (HAM) is introduced in this paper. As compared with the already known ways of evaluating the convergence control parameter in HAM either through the classical h-level curves with h being the convergence control parameter or from the classical squared residual formula as adopted in the HAM society, an elegant way of calculating the convergence control parameter yielding the same optimum values is offered. In most cases, the new method is shown to perform quicker and better against the residual error method when integrations are much harder to evaluate or even by numerical means. Examples originating from real life applications selected from the literature demonstrate the validity and usefulness of the introduced technique.

Mixed convection flow due to a vertical plate in the presence of heat source and chemical reaction

Abstract In this paper, the hydromagnetic heat and mass transfer by mixed convection flow due to a vertical flat plate is considered for analysis. The governing equations are solved both analytically and numerically. The analytical solutions are obtained using the Homotopy Analysis Method (HAM) while the numerical solutions are computed using Keller–Box method (K–B). Convergence of the Homotopy solutions for the governing non-dimensional equations are derived. A detailed error analysis is done to compute the average squared residual errors for flow, temperature and concentration. The optimal values of the convergence control parameter are computed for velocity and temperature. This study includes the effects of various parameters such as magnetic parameter, Grashof number, chemical reaction parameter, heat source parameter and Biot number on skin friction, heat and mass transfer rates as well on velocity, temperature and concentration profiles. Comparison of the HAM and K–B methods shows a very good agreement.

Heat transfer near the stagnation point of an unsteady three-dimensional flow

In this paper, the unsteady laminar three-dimensional flow of an incompressible viscous fluid in the neighbourhood of a stagnation point is studied. The magnetic field is applied normal to the surface and the effects of viscous dissipation and Ohmic heating are taken into account. The unsteadiness in the flow is caused by the external free stream varying arbitrarily with time. The governing equations are solved both analytically and numerically. An approximate analytical solution has been obtained for flow and heat transfer in the form of series solution using Homotopy Analysis Method while the numerical solutions are computed using the Runge–Kutta–Fehlberg Method with a shooting technique. The influence of various parameters such as the viscous dissipation, unsteadiness, ratio of the velocity gradient and magnetic field effects on flow and heat transfer parameters are studied. A detailed error analysis is done to compute the averaged square residual errors for velocity and temperature. The optimal values of the convergence control parameter are computed for the flow which is used for obtaining all the other results presented.

Stagnation Point Solution Due to a Continuously Stretching Surface with Applied Magnetic Field using HAM

In the present study, we consider the steady two-dimensional stagnation point flow due to a stretching surface in an ambient fluid. The fluid is viscous, incompressible and electrically conducting near the stagnation region on a stretching surface. A uniform magnetic field of strength B is applied in the positive y.direction normal to the stretching surface. The equations governing the flow are solved using Homotopy Analysis Method (HAM). The flow variables are computed in the form of a series with its coefficients containing the parameters such as the magnetic field and the ratio of stretching velocity so that the effect of these parameters on the flow variables can be computed without much computational effort. The optimal values of the convergence control parameters and the the averaged squared residual errors are computed for the flow variables. The software Mathematica is used to perform all the semi-analytical calculations. It was observed that the effect of ratio of stretching velocity V0 on flow velocity is very significant in the sense that zero skin friction occurs at V0 = 0.5. and the trend in the variation of velocity profiles as well as skin friction coefficients are just opposite for V0 0.5.

Study of fractional order Van der Pol equation

In this article, Homotopy analysis method is successfully used to find the approximate solution of fractional order Van der Pol equation. The fractional derivative is described in the Caputo sense.The numerical computations of convergence control parameters for the acceleration of convergence of approximate series solution are obtained by the analysis of minimization of error for different particular cases and the results are depicted through graphs. The salient feature of the article is the graphical presentation of achieving limit cycles for different parameters.