Approximate Solutions of Primary Resonance for Forced Duffing Equation by Means of the Homotopy Analysis Method

Nonlinear dynamic equation is a common engineering model.There is not precise analytical solution for most of nonlinear differential equations.These nonlinear differential equations should be solved by using approximate methods.Classical perturbation methods such as LP method,KBM method,multi-scale method and the averaging method on weakly nonlinear vibration system is effective,while the strongly nonlinear system is difficult to apply.Approximate solutions of primary resonance for forced Duffing equation is investigated by means of homotopy analysis method （HAM）.Different from other approximate computational method,the HAM is totally independent of small physical parameters,and thus is suitable for most nonlinear problems.The HAM provides a great freedom to choose base functions of solution series,so that a nonlinear problem may be approximated more effectively.The HAM provides us a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter and the auxiliary function.Therefore,HAM not only may solve the weakly non-linear problems but also may be suitable for the strong non-linear problem.Through the approximate solution of forced Duffing equation with cubic non-linearity,the HAM and fourth order Runge-Kutta method of numerical solution were compared,the results show that the HAM not only can solve the steady state solution,but also can calculate the unsteady state solution,and has the good computational accuracy.

Application of the homotopy analysis method to nonlinear characteristics of a piezoelectric semiconductor fiber

Based on the nonlinear constitutive equation,a piezoelectric semiconductor(PSC)fiber under axial loads and Ohmic contact boundary conditions is investigated.The analytical solutions of electromechanical fields are derived by the homotopy analysis method(HAM),indicating that the HAM is efficient for the nonlinear analysis of PSC fibers,along with a rapid rate of convergence.Furthermore,the nonlinear characteristics of electromechanical fields are discussed through numerical results.It is shown that the asymmetrical distribution of electromechanical fields is obvious under a symmetrical load,and the piezoelectric effect is weakened by an applied electric field.With the increase in the initial carrier concentration,the electric potential decreases,and owing to the screen-ing effect of electrons,the distribution of electromechanical fields tends to be symmetrical.

TiO_2-water nanofluid in a porous channel under the effects of an inclined magnetic field and variable thermal conductivity

The TiO2-water based nanofluid flow in a channel bounded by two porous plates under an oblique magnetic field and variable thermal conductivity is formulated as a boundary-value problem(BVP). The BVP is analytically solved with the homotopy analysis method(HAM). The result shows that the concentration of the nanoparticles is independent of the volume fraction of TiO2nanoparticles, the magnetic field intensity, and the angle. It is inversely proportional to the mass diffusivity. The fluid speed decreases whereas the temperature increases when the volume fraction of the TiO2nanoparticles increases. This confirms the fact that the occurrence of the TiO2nanoparticles results in the increase in the thermal transfer rate. The fluid speed decreases and the temperature increases for both the pure water and the nanofluid when the magnetic field intensity and angle increase. The maximum velocity does not exist at the middle of the symmetric channel, which is in contrast to the plane-Poiseuille flow, but it deviates a little bit towards the lower plate, which absorbs the fluid with a very low suction velocity. If this suction velocity is increased, the temperature in the vicinity of the lower plate will be increased.An explicit expression for the friction factor-Reynolds number is then developed. It is shown that the Hartmann number of the nanofluid is smaller than that of pure water,while the Nusselt number of the nanofluid is larger than that of pure water. However,both the parameters increase if the magnetic field intensity increases.

Heat transfer of nanofluids considering nanoparticle migration and second-order slip velocity

The heat transfer of a magnetohydrodynamics nanofluid inside an annulus considering the second-order slip condition and nanoparticle migration is theoret-ically investigated. A second-order slip condition, which appropriately represents the non-equilibrium region near the interface, is prescribed rather than the no-slip condition and the linear Navier slip condition. To impose different temperature gradients, the outer wall is subjected to q2, the inner wall is subjected to q1, and q1 〉 q2. A modified two-component four-equation non-homogeneous equilibrium model is employed for the nanofiuid, which have been reduced to two-point ordinary boundary value differential equations in the consideration of the thermally and hydrodynamically fully developed flow. The homotopy analysis method （HAM） is employed to solve the equations, and the h-curves are plotted to verify the accuracy and efficiency of the solutions. Moreover, the effects of the physical factors on the flow and heat transfer are discussed in detail, and the semi-analytical relation between NUB and NBT is obtained.

Accurate solutions for viscoelastic boundary layer flow and heat transfer over stretching sheet

In this article, we present accurate analytical solutions for boundary layer flow and heat transfer of an incompressible and electrically conducting viscoelastic fluid over a linearly stretching surface subject to a transverse uniform magnetic field using the homotopy analysis method （HAM） for two general types of non-isothermal boundary conditions. In addition, we demonstrate that the previously reported analytical solutions for the temperature field given in terms of Kummer＇s function do not converge at the boundary. We provide a graphical and numerical demonstration of the convergence of the HAM solutions and tabulate the effects of various parameters on the skin friction coefficient and wall heat transfer.

Second-order slip MHD flow and heat transfer of nanofluids with thermal radiation and chemical reaction

The effects of the second-order velocity slip and temperature jump boundary conditions on the magnetohydrodynamic （MHD） flow and heat transfer in the presence of nanoparticle fractions are investigated. In the modeling of the water-based nanofluids containing Cu and A1203, the effects of the Brownian motion, thermophoresis, and thermal radiation are considered. The governing boundary layer equations are transformed into a system of nonlinear differential equations, and the analytical approximations of the solutions axe derived by the homotopy analysis method （HAM）. The reliability and efficiency of the HAM solutions are verified by the residual errors and the numerical results in the literature. Moreover, the effects of the physical factors on the flow and heat transfer are discussed graphically.

Von Kármán rotating flow of Maxwell nanofluids featuring the Cattaneo-Christov theory with a Buongiorno model

This research paper analyzes the transport of thermal and solutal energy in the Maxwell nanofluid flow induced above the disk which is rotating with a constant angular velocity.The significant features of thermal and solutal relaxation times of fluids are studied with a Cattaneo-Christov double diffusion theory rather than the classical Fourier’s and Fick’s laws.A novel idea of a Buongiorno nanofluid model together with the Cattaneo-Christov theory is introduced for the first time for the Maxwell fluid flow over a rotating disk.Additionally,the thermal and solutal distributions are controlled with the impacts of heat source and chemical reaction.The classical von Karman similarities are used to acquire the non-linear system of ordinary differential equations(ODEs).The analytical series solution to the governing ODEs is obtained with the well-known homotopy analysis method(HAM).The validation of results is provided with the published results by the comparison tables.The graphically presented outcomes for the physical problem reveal that the higher values of the stretching strength parameter enhance the radial velocity and decline the circumferential velocity.The increasing trend is noted for the axial velocity profile in the downward direction with the higher values of the stretching strength parameter.The higher values of the relaxation time parameters in the Cattaneo-Christov theory decrease the thermal and solutal energy transport in the flow of Maxwell nanoliquids.The higher rate of the heat transport is observed in the case of a larger thermophoretic force.

Series solutions of annular axisymmetric stagnation flow and heat transfer on moving cylinder

The steady, laminar, incompressible flow and heat transfer of a viscous fluid between two circular cylinders for two different types of thermal boundary conditions are investigated. The governing Navier-Stokes and thermal equations of the flow are reduced to a nonlinear system of ordinary differential equations. The equations are solved analyt- ically using the homotopy analysis method （HAM）. Convergence of the HAM solutions is discussed in detail. These solutions are then compared with recently obtained numericM and perturbative solutions. Plots of the velocity and temperature profiles are provided for various values of the relevant parameters.

Boundary layer flow of Oldroyd-B fluid by exponentially stretching sheet

The present paper investigates the steady flow of an Oldroyd-B fluid. The fluid flow is induced by an exponentially stretched surface. Suitable transformations reduce a system of nonlinear partial differential equations to a system of ordinary dif- ferential equations. Convergence of series solution is discussed explicitly by a homotopy analysis method （HAM）. Velocity, temperature and heat transfer rates are examined for different involved parameters through graphs. It is revealed that for a larger retardation time constant, the velocity is enhanced and the temperature is lowered. It is noted that relaxation time constant and the Prandtl number enhance the heat transfer rate.

Flow of micropolar fluid between two orthogonally moving porous disks

The unsteady, laminar, incompressible, and two-dimensional flow of a mi- cropolar fluid between two orthogonally moving porous coaxial disks is considered. The extension of von Karman＇s similarity transformations is used to reduce the governing partial differential equations （PDEs） to a set of non-linear coupled ordinary differential equations （ODEs） in the dimensionless form. The analytical solutions are obtained by employing the homotopy analysis method （HAM）. The effects of various physical param- eters such as the expansion ratio and the permeability Reynolds number on the velocity fields are discussed in detail.

Mixed convection heat transfer in horizontal channel filled with nanofluids

The laminar fully developed nanofluid flow and heat transfer in a horizonal channel are investigated. Highly accurate solutions for the temperature and nanopavticle concentration distributions are obtained. The effects of the Brownian motion parameter Nb, the thermophoresis parameter Nt, and the Lewis number Le on the temperature and nanoparticle concentration distributions are discussed. The current analysis shows that the nanoparticles can improve the heat transfer characteristics significantly for this flow problem.

Three-dimensional free bio-convection of nanofluid near stagnation point on general curved isothermal surface

In this paper, the three-dimensional nanofluid bio-convection near a stagnation attachment is studied. With a set of similarity variables, the governing equations embodying the conservation of total mass, momentum, thermal energy, nanoparticles and microorganisms are reduced to a set of fully coupled nonlinear differential equations. The homotopy analysis method （HAM）-finite difference method （FDM） technique is used to obtain exact solutions. The effect of various physical parameters on distribution of the motile microorganisms and the important physical quantities of practical interests are presented and discussed.

Avoiding Small Denominator Problems by Means of the Homotopy Analysis Method

The so-called“small denominator problem”was a fundamental problem of dynamics,as pointed out by Poincar´e.Small denominators appear most commonly in perturbative theory.The Duffing equation is the simplest example of a non-integrable system exhibiting all problems due to small denominators.In this paper,using the forced Duffing equation as an example,we illustrate that the famous“small denominator problems”never appear if a non-perturbative approach based on the homotopy analysis method(HAM),namely“the method of directly defining inverse mapping”(MDDiM),is used.The HAM-based MDDiM provides us great freedom to directly define the inverse operator of an undetermined linear operator so that all small denominators can be completely avoided and besides the convergent series of multiple limit-cycles of the forced Duffing equation with high nonlinearity are successfully obtained.So,from the viewpoint of the HAM,the famous“small denominator problems”are only artifacts of perturbation methods.Therefore,completely abandoning perturbation methods but using the HAM-based MDDiM,one would be never troubled by“small denominators”.The HAM-based MDDiM has general meanings in mathematics and thus can be used to attack many open problems related to the so-called“small denominators”.

Unsteady time-dependent incompressible Newtonian fluid flow between two parallel plates by homotopy analysis method(HAM),homotopy perturbation method(HPM)and collocation method(CM)

Analytical and numerical analyses have performed to study the problem of the flow of incompressible Newtonian fluid between two parallel plates approaching or receding from each other symmetrically.The Navier–Stokes equations have been transformed into an ordinary differential equation using a similarity transformation.The powerful analytical methods called collocation method(CM),the homotopy perturbation method(HPM),and the homotopy analysis method(HAM)have been used to solve nonlinear differential equations.It has been attempted to show the capabilities and wide-range applications of the proposed methods in comparison with a type of numerical analysis as fourth-order Runge–Kutta numerical method in solving this problem.Also,velocity fields have been computed and shown graphically for various values of physical parameters.The objective of the present work is to investigate the effect of Reynolds number and suction or injection characteristic parameter on the velocity field.

A study on nonlinear steady-state waves at resonance in water of finite depth by the amplitude-based homotopy analysis method

Nonlinear steady-state waves are obtained by the amplitude-based homotopy analysis method(AHAM)when resonances among surface gravity waves are considered in water of finite depth.AHAM,newly proposed in this paper within the context of homotopy analysis method(HAM)and well validated in various ways,is able to deal with nonlinear wave interactions.In waves with small propagation angles,it is confirmed that more components share the wave energy if the wave field has a greater steepness.However,in waves with larger propagation angles,it is newly found that wave energy may also concentrate in some specific components.In such wave fields,off-resonance detuning is also considered.Bifurcation and symmetrical properties are discovered in some wave fields.Our results may provide a deeper understanding on nonlinear wave interactions at resonance in water of finite depth.

Series solution of a natural convection flow for a Carreau fluid in a vertical channel with peristalsis

An analysis has been achieved to study the natural convection of a non-Newtonian fluid （namely a Carreau fluid） in a vertical channel with rhythmically contracting walls. The Navier-Stokes and the energy equations are reduced to a system of non- linear PDE by using the long wavelength approximation. The optimal homotopy analysis method （OHAM） is introduced to obtain the exact solutions for velocity and temperature fields. The convergence of the obtained OHAM solution is discussed explicitly. Numerical calculations are carried out for the pressure rise and the features of the flow and temperature characteristics are analyzed by plotting graphs and discussed in detail.

Homotopy-based analytical approximation to nonlinear short-crested waves in a fluid of finite depth

A nonlinear short-crested wave system, consisting of two progressive waves propagating at an oblique angle to each other in a fluid of finite depth, is investigated by means of an analytical approach named the homotopy analysis method （HAM）. Highly convergent series solutions are explicitly derived for the velocity potential and the surface wave elevation. We find that, at every value of water depth, there is little difference between the kinetic energy and the potential energy for nonlinear waves. The nonlinear short-crested waves with a larger angle of incidence always contain the more potential wave energy. With the aid of the HAM, we obtain the dispersion relation for nonlinear short-crested waves. Furthermore, it is shown that the wave elevation tends to be smoothened at the crest and be sharpened at the trough as the water depth increases, and the wave pressure crests and troughs become steeper with increasing incident wave steepness.

Homotopy study of magnetohydrodynamic mixed convection nanofluid multiple slip flow and heat transfer from a vertical cylinder with entropy generation

Stimulated by thermal optimization in magnetic materials process engineering,the present investigation investigates theoretically the entropy generation in mixed convection magnetohydrodynamic(MHD)flow of an electrically-conducting nanofluid from a vertical cylinder.The mathematical model includes the effects of viscous dissipation,second order velocity slip and thermal slip,has been considered.The cylindrical partial differential form of the two-component non-homogenous nanofluid model has been transformed into a system of coupled ordinary differential equations by applying similarity transformations.The effects of governing parameters with no-flux nanoparticle concentration have been examined on important quantities of interest.Furthermore,the dimensionless form of the entropy generation number has also been evaluated using homotopy analysis method(HAM).The present analytical results achieve good correlation with numerical results(shooting method).Entropy is found to be an increasing function of second order velocity slip,magnetic field and curvature parameter.Temperature is elevated with increasing curvature parameter and magnetic parameter whereas it is reduced with mixed convection parameter.The flow is accelerated with curvature parameter but decelerated with magnetic parameter.Heat transfer rate(Nusselt number)is enhanced with greater mixed convection parameter,curvature parameter and first order velocity slip parameter but reduced with increasing second order velocity slip parameter.Entropy generation is also increased with magnetic parameter,second order slip velocity parameter,curvature parameter,thermophoresis parameter,buoyancy parameter and Reynolds number whereas it is suppressed with first order velocity slip parameter,Brownian motion parameter and thermal slip parameter.

Analytical Solution for Time-dependent Flow of a Third Grade Fluid Induced Due to Impulsive Motion of a Flat Porous Plate

The aim of the present communication is to discuss the analytical solution for the unsteady flow of a third grade fluid which occupies the space y 〉 0 over an infinite porous plate. The flow is generated due to the motion of the plate in its own plane with an impulsive velocity V（t）. Translational symmetries in variables t and y are utilized to reduce the governing non-linear partial differential equation into an ordinary differential equation. The reduced problem is then solved using homotopy analysis method（HAM）. Graphs representing the solution are plotted and discussed and proper conclusions are drawn.