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The flow as well as the chemically reactive solute transfer problem in a viscous fluid over an exponentially stretching sheet are analytically investigated in this work. The nonlinear differential equations associated to the motion equations are approximately solved by means of the modified Optimal Homotopy Asymptotic Method (OHAM). The dual approximate analytical solutions are obtained. By comparison with the corresponding numerical solutions a good agreement is performed and show that this procedure is effective, accurate and easy to use in applications. [ABSTRACT FROM AUTHOR]

Enhancing the nanosized‐electrolyte's characteristics in Lithium‐driven micro‐batteries (LIMBs) is indispensable to improve the overall efficiency, security, and lifespan of these energy devices, designing nano‐sized electrolyte with a wide electrochemical stability window while keeping them compatible with electrode materials is one of the improvement goals. Battery technologies must go through this optimization process in order to be used practically. A sensing mechanism to keep an eye on the health of Li‐ion energy devices through the magnetization. Magnetic micro‐fluidic patterns that change could be a sign of battery deterioration or other problems with performance. Li‐ion battery health is one application of magnetic sensing that you can do with magnetic sensing. Battery health variations and other performance problems can be found using magnetic mass transport patterns. Present study examines the effects of magnetic field on Eyring–Powel mass transport in nano‐porous channels over a stretching sheet. The principal equations exhibiting the phenomenon are transformed into non‐linear differential equation by second‐order approximation by using a similarity transformation. Furthermore, a semi‐analytic technique named optimal homotopy asymptotic method (OHAM) is used to solve the transformed Eyring–Powell model. The numerical results demonstrated the impact of variations in velocity, skin‐friction coefficient and Sherwood number for the proposed scheme. [ABSTRACT FROM AUTHOR]

In this study, a new physical model has been created to look into the behaviour of transient incompressible unsteady flow between two infinite parallel plates exposed to high temperatures. The model takes into consideration thermal radiation flux, chemical reaction, and mass diffusion at the boundaries. To handle non-integer behaviour, the model incorporates the Caputo notion of time fractional derivative. To solve this complex physical fractional order fluid model, a novel optimal homotopy asymptotic method and semi-analytical methodology is extended and utilized successfully. This method provides a third-order highly approximate solution, offering valuable insights into the behaviour within the system. The study comprehensively examines the effects of varied flow characteristics and fractional order on the dynamics of the system. The results are visually presented through graphs, offering a clear understanding of the system's response under different conditions. The effectiveness and ease of use of the optimal homotopy asymptotic method make it a valuable tool for solving boundary value fractional order problems encountered in scientific fields. The developed physical model and its fractional extension contribute significantly to the understanding of unsteady flow phenomena with thermal and chemical effects, advancing knowledge in this area of research. [ABSTRACT FROM AUTHOR]

This work focuses on solving and analyzing two-point fuzzy boundary value problems in the form of fractional ordinary differential equations (FFOBVPs) using a new version of the approximation analytical approach. FFOBVPs are useful in describing complex scientific phenomena that include heritable characteristics and uncertainty, and obtaining exact or close analytical solutions for these equations can be challenging, especially in the case of nonlinear problems. To address these difficulties, the optimal homotopy asymptotic method (OHAM) was studied and extended in a new form to solve FFOBVPs. The OHAM is known for its ability to solve both linear and nonlinear fractional models and provides a straightforward methodology that uses multiple convergence control parameters to optimally manage the convergence of approximate series solutions. The new form of the OHAM presented in this work incorporates the concepts of fuzzy sets theory and some fractional calculus principles to include fuzzy analysis in the method. The steps of fuzzification and defuzzification are used to transform the fuzzy problem into a crisp problem that can be solved using the OHAM. The method is demonstrated by solving and analyzing linear and nonlinear FFOBVPs at different values of fractional derivatives. The results obtained using the new form of the fuzzy OHAM are analyzed and compared to those found in the literature to demonstrate the method's efficiency and high accuracy in the fuzzy domain. Overall, this work presents a feasible and efficient approach for solving FFOBVPs using a new form of the OHAM with fuzzy analysis. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate effective and accurate dual analytic approximate solutions, while taking into account thermal effects. The heat and mass transfer problem in a viscous fluid flow are analytically explored by using the modified Optimal Homotopy Asymptotic Method (OHAM). By using similarity transformations, the motion equations are reduced to a set of nonlinear ordinary differential equations. Based on the numerical results, it was revealed that there are dual analytic approximate solutions within the mass transfer problem. The variation of the physical parameters (the Prandtl number and the temperature distribution parameter) over the temperature profile is analytically explored and graphically depicted for the first approximate and the corresponding dual solution, respectively. The advantage of the proposed method arises from using only one iteration for obtaining the dual analytical solutions. The presented results are effective, accurate and in good agreement with the corresponding numerical results with relevance for further engineering applications of heat and mass transfer problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new method to create a series solution to the time-fractional Navier–Stokes equations using a combination of the Laplace transform with the optimal homotopy asymptotic method. Caputo's derivative was applied in the present technique. Three test problems are carried out to validate and demonstrate the method's efficacy. The system has been found to be a very reliable, effective and powerful technique for solving the proposed model as well as a variety of engineering and science problems. [ABSTRACT FROM AUTHOR]

This paper emphasizes some geometrical properties of the Maxwell–Bloch equations. Based on these properties, the closed-form solutions of their equations are established. Thus, the Maxwell–Bloch equations are reduced to a nonlinear differential equation depending on an auxiliary unknown function. The approximate analytical solutions were built using the optimal homotopy asymptotic method (OHAM). These represent the ε -approximate OHAM solutions. A good agreement between the analytical and corresponding numerical results was found. The accuracy of the obtained results is validated through the representative figures. This procedure is suitable to be applied for dynamical systems with certain geometrical properties. [ABSTRACT FROM AUTHOR]

In this paper, a powerful semianalytical method known as optimal homotopy asymptotic method (OHAM) has been formulated for the solution of system of Volterra integro‐differential equations. The effectiveness and performance of the proposed technique are verified by different numerical problems in the literature, and the obtained results are compared with Sinc‐collocation method. These results show the reliability and effectiveness of the proposed method. The proposed method does not require discretization like other numerical methods. Moreover, the convergence region can easily be controlled. The use of OHAM is simple and straightforward. [ABSTRACT FROM AUTHOR]

A comparative study between optimal homotopy asymptotic method (OHAM) and some other well-known methods such as the Homotopy perturbation, Adomian decomposition and Runge-Kutta fehlberg fourth-fifth order methods has been presented in the current paper. These methods have been applied in solving the system of the second order decomposition of ozone in aqueous solutions. The obtained results indicate these methods are appropriate to solve the equations whereas applying OHAM is more simpler with the same accuracy in comparison to other methods. [ABSTRACT FROM AUTHOR]

The effects of nanoparticle diameter on the magnetohydrodynamic flow and heat transfer of TiO2-H2O nanofluid over an exponentially stretching sheet are studied in this paper. It is assumed that the effective viscosity of TiO2-H2O nanofluid is a function of nanofluid concentration and nanoparticle diameter by use of new accurate correlation. Therefore, the flow and heat characteristics of TiO2-H2O nanofluid near a surface can be found under the effects of nanoparticle diameter for the first time. For this purpose, the governing partial differential equations are transformed to the nonlinear ordinary differential equations using similarity transformations. The resulting equations are solved analytically using Optimal Homotopy Asymptotic Method (OHAM) which is most applicable in the analysis of nonlinear problems. The effects of nanoparticle diameter, nanofluid concentration, and magnetic field on the flow and heat transfer of nanofluid are investigated in detail. The results show that the reduced skin friction is a descending function of nanoparticle diameter. The reduced Nusselt number is a complicated function of magnetic field, nanofluid concentration, and nanoparticle diameter. It can be found that for the specific values of parameters, the curves of reduced Nusselt number with respect to the parameters such as magnetic field and nanofluid concentration have peaks where the maximum of heat transfer occurs. [ABSTRACT FROM AUTHOR]

In this paper, we implement the optimal homotopy asymptotic method to find the approximate solutions of the Poisson–Boltzmann equation. We also use the results of the conjugate gradient method for comparison with those of the optimal homotopy asymptotic method. Our study reveals that the optimal homotopy asymptotic method gives more effective results than conjugate gradient algorithms for the considered problems. [ABSTRACT FROM AUTHOR]

This work is devoted to the study of non-Newtonian fluid of second grade. This model assumes that the flow and heat transfer in MHD viscoelastic fluid is subject to convective boundary conditions in the presence of electromagnetic field and nonuniform source/sink. The flow is considered over a porous sheet. The governing partial differential equations are converted as ordinary differential equations using suitable similarity transformations. The resulting coupled of nonlinear differential equations are successfully solved by means of Optimal Homotopy Asymptotic Method (OHAM). The effects of different physical parameters such as: the second-grade viscoelastic parameter, thermal radiation parameter or Eckert number are analyzed. Several closed form analytical approximate solutions for dimensionless stream function and temperature are presented for some particular cases. Our procedure is based upon an original construction of the solutions using a moderate number of the convergence-control parameters. These parameters lead to very high precision, comparing our approximate solutions with numerical solutions. Let us note that every nonlinear differential equation is reduced to two linear differential equations. The values of the convergence-control parameters are determined using rigorous mathematical procedures. OHAM is an iterative approach which rapidly converges after only one approximation, is effective, explicit, simple and easy to apply. These remarks are in fact the true power of our procedure to solving nonlinear problems without the presence of small or large parameters in the governing equations or boundary/initial conditions. [ABSTRACT FROM AUTHOR]

In this article, generalised Kudryashov technique has been implemented to construct new soliton solutions of modified Benjamin–Bona–Mahony (mBBM) equation. Also, the optimal homotopy asymptotic method (OHAM) has been employed to estimate the numerical solution. The solutions thus acquired by the above methods are illustrated graphically. The techniques considered here are efficacious, plausible and can be employed in mathematical physics to compute new exact and numerical solutions of NPDEs. [ABSTRACT FROM AUTHOR]

In this work, a comparative study of seven well-known mathematical techniques for the coupled Burgers' equations is reported. The techniques involve in this comparison are as follows: Laplace transform Adomian decomposition method, Laplace transform homotopy perturbation method, Variational iteration method, Variational iteration decomposition method, Variational iteration homotopy perturbation method, the optimal homotopy asymptotic method, and OHAM with Daftardar-Jafari polynomial. Here we considered a practical example which consists of coupled Burgers' equations with the kinematic viscosity ε = 1. Convergence and stability analysis is a major part of this analysis. After a careful observation, it is found that the variational iteration method has faster convergence than all the remaining methods. Adomian decomposition method and Homotopy perturbation method show weaker stability in comparison with other involved techniques. [ABSTRACT FROM AUTHOR]

Polymer flow during wire coating in a pressure type coating die has been simulated under nonisothermal conditions. The flow is dependent on the wire velocity, geometry of the die, and the viscosity of the polymer. The constitutive equation of an Oldroyd 8-constant fluid is used to characterize the rheology of the polymer melt. The equation describing the flow of polymer melt inside the die is solved (i) analytically by applying optimal homotopy asymptotic method and (ii) numerically by shooting method with Runge–Kutta–Fehlberg algorithm. The convergence of the series solution is established. The effect of physical characteristics of the problem has been discussed in detail through graphs by assigning numerical values for several parameters of interest. At the end, this study is also compared with the published work as a particular case and good agreement is found. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the implementation of optimal homotopy asymptotic method to handle the approximate analytical solutions of fractional order partial differential equations. Fractional differential equations have great importance regarding distinct fields of science and engineering. Approximate solutions of space fractional order diffusion model and its various special cases are handled using the innovative proposed method. The space fractional derivatives are described in the Caputo sense. The results obtained by the proposed method are compared with various methods. The proposed method demonstrates excellent accuracy and strength over various methods. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the implementation of optimal homotopy asymptotic method to handle the approximate analytical solutions of fractional partial differential equations. Approximate solutions of fractional models in both 1-D and 2-D cases are handled using the innovative proposed method. The consequences show excellent accuracy and strength of the planned method. Using this method, one can easily handle the convergence of approximation series solution for the fractional partial differential equations and can adjust the convergence region when required. The method is effective and explicit. Moreover, this method is flexible with respect to geometry and ease of implementation for fractional order models of physical and biological problems. [ABSTRACT FROM AUTHOR]

Purpose The purpose of this paper is the comparative analysis of Haar Wavelet Method and Optimal Homotopy Asymptotic Method for fractional Fisher type equation. In this paper, two reliable techniques, Haar wavelet method and optimal homotopy asymptotic method (OHAM), have been presented. The Haar wavelet method is an efficient numerical method for the numerical solution of fractional order partial differential equation like the Fisher type. The approximate solutions of the fractional Fisher-type equation are compared with those of OHAM and with the exact solutions. Comparisons between the obtained solutions with the exact solutions exhibit that both the featured methods are effective and efficient in solving nonlinear problems. However, the results indicate that OHAM provides more accurate value than the Haar wavelet method.Design/methodology/approach Comparisons between the solutions obtained by the Haar wavelet method and OHAM with the exact solutions exhibit that both featured methods are effective and efficient in solving nonlinear problems.Findings The comparative results indicate that OHAM provides a more accurate value than the Haar wavelet method.Originality/value In this paper, two reliable techniques, the Haar wavelet method and OHAM, have been proposed for solving nonlinear fractional partial differential equation, i.e. fractional Fisher-type equation. The proposed novel methods are well suited for only nonlinear fractional partial differential equations. It also exhibits that the proposed method is a very efficient and powerful technique in finding the solutions for the nonlinear time fractional differential equations. The main significance of the proposed method is that it requires less amount of computational overhead in comparison to other numerical and analytical approximate methods. The application of the proposed methods for the solutions of time fractional Fisher-type equations satisfactorily justifies its simplicity and efficiency. [ABSTRACT FROM AUTHOR]

In the present work we examine the motion of an incompressible unidirectional magnetohydrodynamics thin film flow of a third grade fluid over an oscillating inclined belt embedded in a porous medium. Moreover, heat transfer analysis has been also discussed in the present work. This physical problem is modeled in terms of non-linear partial differential equations. These equations together with physical boundary conditions are solved using two analytical techniques namely optimal homotopy asymptotic method and homotopy perturbation method. The comparisons of these two methods for different time level are analyzed numerically and graphically. The results exposed that both methods are in closed agreement and they have identical solutions. The effects of various non-dimensional parameters have also been studied graphically. [ABSTRACT FROM AUTHOR]

In this paper, we study the fractional-order biological population models (FBPMs) with Malthusian, Verhulst, and porous media laws. The fractional derivative is defined in Caputo sense. The optimal homotopy asymptotic method (OHAM) for partial differential equations (PDEs) is extended and successfully implemented to solve FBPMs. Third-order approximate solutions are obtained and compared with the exact solutions. The numerical results unveil that the proposed extension in the OHAM for fractional-order differential problems is very effective and simple in computation. The results reveal the effectiveness with high accuracy and extremely efficient to handle most complicated biological population models. [ABSTRACT FROM AUTHOR]

Here, we investigate the effectiveness of the optimal homotopy asymptotic method (OHAM) with a symbolic computational method for constructing the approximate solution for quantum Zakharov-Kuznetsov equation that is derived to describe the in magnetized plasma in ion acoustic waves. The results reveal that the method is explicit, effective and easy to use. The proposed method is a strong and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The results obtained here reveal that the proposed method is very effective and simple for solving nonlinear problems. The basic ideas of this approach can be widely employed to solve other strongly nonlinear evaluation form equations arising in physics. [ABSTRACT FROM AUTHOR]

In this paper, we compared two different methods, one numerical technique, viz Legendre multiwavelet method, and the other analytical technique, viz optimal homotopy asymptotic method (OHAM), for solving fractional-order Kaup-Kupershmidt (KK) equation. Two-dimensional Legendre multiwavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is used to compute the numerical solution of nonlinear time-fractional KK equation. The approximate solutions of time fractional Kaup-Kupershmidt equation thus obtained by Legendre multiwavelet method are compared with the exact solutions as well as with OHAM. The present numerical scheme is quite simple, effective, and expedient for obtaining numerical solution of fractional KK equation in comparison to analytical approach of OHAM. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In the present paper, a wavelet method based on the Hermite wavelet expansion along with operational matrices of fractional derivative and integration is proposed for finding the numerical solution to a coupled system of nonlinear time-fractional Jaulent–Miodek (JM) equations. Consequently, the approximate solutions of fractional Jaulent–Miodek equations acquired by using Hermite wavelet technique were compared with those derived by using optimal homotopy asymptotic method (OHAM) and exact solutions. The present proposed numerical technique is easy, expedient and powerful in computing the numerical solution of coupled system of nonlinear fractional differential equations like Jaulent–Miodek equations. [ABSTRACT FROM AUTHOR]

In this article, we present the optimal homotopy asymptotic method (OHAM) and homotopy perturbation method (HPM) for solving 2-dimensional nonlinear Fredholm integral equations of the second kind. A comparison is made between these methods to solve 2-dimensional nonlinear Fredholm integral equations of the second kind. The results show that the presented methods are very powerful and simple techniques in solving 2-dimensional nonlinear Fredholm integral equations of the second kind. [ABSTRACT FROM AUTHOR]

In the present paper, the approximate solutions for steady boundary layer of the MHD viscous flow and radiative heat transfer over an exponentially porous stretching sheet are given. The nonlinear partial differential equations are reduced to an ordinary differential equations by the similarity transformations, taking into account velocity slip, thermal slip and the boundary conditions. These equations are solved approximately by means of the Optimal Homotopy Asymptotic Method (OHAM). This approach is highly efficient and it controls the convergence of the approximate solutions. OHAM is very efficient in practice, ensuring a very rapid convergence of the solutions after only one iteration. It does not need small or large parameters in the governing equations. Approximate solutions obtained through OHAM are compared with the results obtained by shooting method. It is found a very good agreement between these solutions. [ABSTRACT FROM AUTHOR]

The Painlev'e equations are second order ordinary differential equations which can be grouped into six families, namely Painlev'e equation I, II,?, VI. This paper presents the series solution of second Painlevé equation via optimal homotopy asymptotic method (OHAM). This approach is highly efficient and it controls the convergence of the approximate solution. Comparison of the obtained solution via OHAM is provided with those obtained by Homotopy Perturbation Method (HPM), Adomian Decomposition Method (ADM), Sinc-collocation and Runge-Kutta 4 methods. It is revealed that there is an excellent agreement between OHAM and other published data which confirm the effectiveness of the OHAM. [ABSTRACT FROM AUTHOR]

In this paper, a new wavelet method based on the Hermite wavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is proposed to solve time-fractional modified Fornberg–Whitham (mFW) equation. The approximate solutions of time fractional modified Fornberg–Whitham equation which are obtained by Hermite wavelet method are compared with the exact solutions as well as the solutions obtained by optimal homotopy asymptotic method (OHAM). The present numerical scheme is quite simple, effective and expedient for obtaining numerical solution of fractional modified Fornberg–Whitham equation. [ABSTRACT FROM AUTHOR]

Abstract: In this work, an approximate method called Optimal Homotopy Asymptotic Method (OHAM) is proposed and implemented for the numerical solution of Whitham–Broer–Kaup (WBK) equations with blow-up and periodic solutions. Results obtained through OHAM are compared with the exact as well as with the results available in the literature. It was revealed that only second-order OHAM solutions are sufficient to achieve the desired accuracy in comparison to the higher order solutions of the methods we have made comparison with. [Copyright &y& Elsevier]

In this article, we use the optimal homotopy asymptotic method ( OHAM) to compute the solution of two-dimensional incompressible laminar boundary layer flow over a flat plate (Blasius problem). The obtained results for the stream function and velocity profile were comparable in terms of accuracy with that obtained by Esmaeilpour and Ganji (2007) who studied the same problem using the homotopy perturbation method and results obtained by using a numerical method ( RK4). The good agreement obtained shows the effectiveness of OHAM. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(3): 197-203, 2014; Published online 19 June 2013 in Wiley Online Library (). DOI 10.1002/htj.21070 [ABSTRACT FROM AUTHOR]

Abstract: Starting from the reality that many known methods fail in the attempt to obtain analytic solutions of Blasius-type equations, in this work, a new procedure namely Optimal Homotopy Asymptotic Method (OHAM) is proposed to obtain an explicit analytical solution of the Blasius problem. Comparison with Howarth’s numerical solution, as well as the obtained residual, reveals that the proposed method is highly accurate. This proves the validity and great potential of the proposed procedure (OHAM) as a new kind of powerful analytical tool for nonlinear problems. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

In this paper, the optimal homotopy asymptotic method is applied on the Cauchy reaction-diffusion problems to check the effectiveness and performance of the method. The obtained solutions show that the OHAM is more effective, simpler and easier than other methods. Moreover, this technique does not require any discretization or linearization and therefore it reduces significantly the numerical computations. The results reveal that the method is explicit. [ABSTRACT FROM AUTHOR]

In this study, the pseudo-plastic model is used to obtain a solution for steady thin film flow on the outer surface of a long vertical cylinder for lifting and drainage problems. The non-linear governing equations subject to appropriate boundary conditions are solved analytically for velocity profiles by a modified homotopy perturbation method called the optimal homotopy asymptotic method. Expressions for the velocity profile, volume flux, average velocity, shear stress on the cylinder, normal stress differences, and force to hold the vertical cylindrical surface in position have been derived for both the problems. For the non-Newtonian parameter β = 0, we retrieve Newtonian cases for both the problems. We also plot and discuss the effects of the Stokes number, St, the non-Newtonian parameter, β, and the thickness, δ, of the the fluid film on the fluid velocities. [ABSTRACT FROM AUTHOR]

Integro-differential equations arise in modeling various physical and engineering problems. Several numerical and analytical methods have been developed to solving such equations. We introduce the OHAM (Optimal Homotopy Asymptotic Method) for solving nonlinear integro-differential equations. Several examples for solving integro-differential equations are presented to illustrate the reliability and efficiency of the proposed OHAM. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to apply a numerical technique namely the optimal homotopy asymptotic method (OHAM) for finding the approximate solutions of a class of Volterra integral equations with weakly singular kernels. This method uses simple computations with quite acceptable approximate solutions, which has close agreement with exact solutions. Illustrative examples are included to demonstrate the validity and applicability of the present method and a comparison has been made with existing results. [Copyright &y& Elsevier]

Abstract: The aim of this communication is to present the effectiveness of optimal homotopy asymptotic method (OHAM) for solving nonlinear Fredholm integral equations of second kind. Here, we consider the first, second and third order Fredholm integral equations of second kind. Comparisons are made with Adomian Decomposition Method, Discrete Adomian Decomposition Method, Homotopy Perturbation Method, Quasi Interpolation Method, Triangular Functions Method, Trapezoidal rule and Simpson rule. The results reveal that the OHAM is very effective, simple and has close agreement with exact solutions. [Copyright &y& Elsevier]

Abstract: In this study the wire coating in a pressure type die with the bath of Oldroyd 8-constant fluid with pressure gradient is investigated. The non-linear ordinary differential equation in dimensionless form is obtained, which is solved for the velocity profile using the Optimal Homotopy Asymptotic Method (OHAM). The effect of Dilatant constant , the Psendoplastic constant , and the pressure gradient on velocity distribution and shear stress is studied. Shear stress is examined under the effect of the viscosity parameter . Moreover, the volume flow rate and average velocity is carefully studied with changing the domain (thickness) of the polymer and varying the parameter and the pressure gradient. [Copyright &y& Elsevier]

Abstract: In this study, optimal homotopy asymptotic method is applied on singular initial value Lane–Emden type problems to check the effectiveness and performance of the method. It is observed that the method is easy to implement, quite valuable to handle singular phenomena and yield excellent results at minimum computational cost. Computational results of some of the test problems are presented to demonstrate the viability and practical usefulness of the method. [Copyright &y& Elsevier]

Abstract: In this paper a new approach combining the features of the homotopy concept with an efficient computational algorithm which provides a simple and rigorous procedure to control the convergence of the solution is proposed to find accurate analytical explicit solutions for some oscillators with discontinuities and a fractional power restoring force which is proportional to . A very fast convergence to the exact solution was proved, since the second-order approximation lead to very accurate results. Comparisons with numerical results are presented to show the effectiveness of this method. Four numerical applications prove the accuracy of the method, which works very well for the whole range of initial amplitudes. The obtained results prove the validity and efficiency of the method, which can be easily extended to other strongly nonlinear problems. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the effectiveness of the Optimal Homotopy Asymptotic Method (OHAM) in solving time dependent partial differential equations. To this effect we consider the homogeneous, non-homogeneous, linear and nonlinear Klein–Gordon equations with boundary conditions. The results reveal that the method is explicit, effective, and easy to use. [Copyright &y& Elsevier]

Abstract: In this paper, axisymmetric flow of two-dimensional incompressible fluids is studied. The Optimal Homotopy Asymptotic Method (OHAM) is applied to derive a solution of the reduced fourth-order nonlinear boundary value problem. For comparison, the same problem is also solved by the Perturbation Method (PM), the Homotopy Perturbation Method (HPM) and the Homotopy Analysis Method (HAM) . OHAM is parameter free and provides better accuracy at lower order of approximation. Moreover we can easily adjust and control the convergence region. As a result it is concluded that the new technique, OHAM, shows fast convergence, simplicity of application and efficiency. [Copyright &y& Elsevier]

In this paper, analytical analysis of the steady flow of an incompressible third grade fluid between two parallel plates is carried out, and the effect of heat transfer is taken into account. Three different types of flows namely, Couette flow, Poiseuille flow and generalized Couette flow, all depending upon the relative motion of the plates, are studied. The solutions are obtained for the developed non-linear equations using Optimal Homotopy Asymptotic Method (OHAM). This method provides a simple strategy to control the convergence region of the approximating solution series where ever necessary. [ABSTRACT FROM AUTHOR]

Abstract: In this work, we solve multipoint boundary value problems using the Optimal Homotopy Asymptotic Method (OHAM).The proposed method is tested upon several problems from the literature and the results are compared with the available exact solution. This method provides easy tools to control the convergence region of approximating solution series where ever necessary. [Copyright &y& Elsevier]

This paper proceeds from the perspective that most strongly nonlinear oscillators of fractional-order do not enjoy exact analytical solutions. Undoubtedly, this is a good enough reason to employ one of the major recent approximate methods, namely an Optimal Homotopy Asymptotic Method (OHAM), to offer approximate analytic solutions for two strongly fractional-order nonlinear benchmark oscillatory problems, namely: the fractional-order Duffing-relativistic oscillator and the fractional-order stretched elastic wire oscillator (with a mass attached to its midpoint). In this work, a further modification has been proposed for such method and then carried out through establishing an optimal auxiliary linear operator, an auxiliary function, and an auxiliary control parameter. In view of the two aforesaid applications, it has been demonstrated that the OHAM is a reliable approach for controlling the convergence of approximate solutions and, hence, it is an effective tool for dealing with such problems. This assertion is completely confirmed by performing several graphical comparisons between the OHAM and the Homotopy Analysis Method (HAM). [ABSTRACT FROM AUTHOR]