通过将超大型浮式结构物(a very large floating structure,简称VLFS)模拟为黏弹性薄板,本工作对VLFS的非线性水弹性响应进行了解析研究。运用同伦分析方法(the homotopy analysis method,简称HAM),计算出速度势和板挠度的近似迭代解,...通过将超大型浮式结构物(a very large floating structure,简称VLFS)模拟为黏弹性薄板,本工作对VLFS的非线性水弹性响应进行了解析研究。运用同伦分析方法(the homotopy analysis method,简称HAM),计算出速度势和板挠度的近似迭代解,并根据计算结果着重探究了几个重要的物理参数对黏弹性板形变的影响。结果发现:黏弹性板的挠度随着黏弹性时间、杨氏模量和板厚度增加而减少,而板挠度随着入射波波幅的增加而增加。最后,还对非线性色散关系和波幅之间的联系进行了探讨。展开更多
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 s...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.展开更多
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 t...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.展开更多
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 model...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.展开更多
Large deformation of a cantilever axially functionally graded (AFG) beam subject to a tip load is analytically studied using the homotopy analysis method (HAM). It is assumed that its Young’s modulus varies along the...Large deformation of a cantilever axially functionally graded (AFG) beam subject to a tip load is analytically studied using the homotopy analysis method (HAM). It is assumed that its Young’s modulus varies along the longitudinal direction according to a power law. Taking the solution of the corresponding homogeneous beam as the initial guess and obtaining a convergence region by adjusting an auxiliary parameter, the analytical expressions for large deformation of the AFG beam are provided. Results obtained by the HAM are compared with those obtained by the finite element method and those in the previous works to verify its validity. Good agreement is observed. A detailed parametric study is carried out. The results show that the axial material variation can greatly change the deformed configuration, which provides an approach to control and manage the deformation of beams. By tailoring the axial material distribution, a desired deformed configuration can be obtained for a specific load. The analytical solution presented herein can be a helpful tool for this procedure.展开更多
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 appro...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.展开更多
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 a...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.展开更多
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 the...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.展开更多
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 approxima...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.展开更多
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 effect...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.展开更多
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 equatio...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.展开更多
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 ...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.展开更多
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 homotop...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.展开更多
In this paper, the homotopy analysis method (HAM) has been employed to obtain the approximate analytical solution of the nonlinear Harry-Dym (HD) equation, which is one of the most important soliton equations. Utilizi...In this paper, the homotopy analysis method (HAM) has been employed to obtain the approximate analytical solution of the nonlinear Harry-Dym (HD) equation, which is one of the most important soliton equations. Utilizing the HAM, thereby employing the initial approximation, variations of the 7th-order approximation of the Harry-Dym equation is obtained. It is found that effect of the nonzero auxiliary parameter on convergence rate of the series solution is undeniable. It is also shown that, to some extent, order of the fractional derivative plays a fundamental role in the prediction of convergence. The final results reported by the HAM have been compared with the exact solution as well as those obtained through the other methods.展开更多
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, th...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.展开更多
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 m...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.展开更多
文摘通过将超大型浮式结构物(a very large floating structure,简称VLFS)模拟为黏弹性薄板,本工作对VLFS的非线性水弹性响应进行了解析研究。运用同伦分析方法(the homotopy analysis method,简称HAM),计算出速度势和板挠度的近似迭代解,并根据计算结果着重探究了几个重要的物理参数对黏弹性板形变的影响。结果发现:黏弹性板的挠度随着黏弹性时间、杨氏模量和板厚度增加而减少,而板挠度随着入射波波幅的增加而增加。最后,还对非线性色散关系和波幅之间的联系进行了探讨。
文摘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.
文摘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.
基金Project supported by the National Natural Science Foundation of China(Nos.51276014 and51476191)the Fundamental Research Funds for the Central Universities(No.FRF-BR-12-004)
文摘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.
基金Project supported by the China Postdoctoral Science Foundation(No.2018M630167)
文摘Large deformation of a cantilever axially functionally graded (AFG) beam subject to a tip load is analytically studied using the homotopy analysis method (HAM). It is assumed that its Young’s modulus varies along the longitudinal direction according to a power law. Taking the solution of the corresponding homogeneous beam as the initial guess and obtaining a convergence region by adjusting an auxiliary parameter, the analytical expressions for large deformation of the AFG beam are provided. Results obtained by the HAM are compared with those obtained by the finite element method and those in the previous works to verify its validity. Good agreement is observed. A detailed parametric study is carried out. The results show that the axial material variation can greatly change the deformed configuration, which provides an approach to control and manage the deformation of beams. By tailoring the axial material distribution, a desired deformed configuration can be obtained for a specific load. The analytical solution presented herein can be a helpful tool for this procedure.
基金Project supported by the National Natural Science Foundation of China(Nos.51476191 and51406008)
文摘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.
文摘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.
文摘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.
基金supported by Fundamental Research Funds for the Central Universities of China (Grant No. N090405009)
文摘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.
基金Project supported by the National Natural Science Foundation of China （No. 10972136） and the Doctoral Fund for New Teachers of Higher Eduation of China （No. 20090073120014）
文摘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.
文摘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.
基金Project supported by the National Natural Science Foundation of China (Nos. 51004013,50936003,51174028,and 50905013)the Research Foundation of Engineering Research Institute of University of Science and Technology Beijing (No. Yj2011-015)the Fundamental Research Funds for the Central Universities (No. T-RF-TP-12-108A)
文摘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.
基金Supported by the National Key Basic Research Development Program of China(973 Program,Grant No.2014CB046203)the National Natural Science Foundation of China(Grant No.11472166)the Natural Science Founda-tion of Shanghai(Grant No.14ZR1416200)
文摘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.
文摘In this paper, the homotopy analysis method (HAM) has been employed to obtain the approximate analytical solution of the nonlinear Harry-Dym (HD) equation, which is one of the most important soliton equations. Utilizing the HAM, thereby employing the initial approximation, variations of the 7th-order approximation of the Harry-Dym equation is obtained. It is found that effect of the nonzero auxiliary parameter on convergence rate of the series solution is undeniable. It is also shown that, to some extent, order of the fractional derivative plays a fundamental role in the prediction of convergence. The final results reported by the HAM have been compared with the exact solution as well as those obtained through the other methods.
基金supported by the Program for New Century Excellent Talents in University of China(No.NCET-12-0347)
文摘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.
基金supported by the National Research Foundation(NRF) of South Africa for research grant
文摘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.
