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The aim of this paper was to present a user friendly numerical algorithm based on homotopy perturbation transform method for solving various linear and nonlinear convection-diffusion problems arising in physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. The homotopy perturbation transform method is a combined form of the homotopy perturbation method and Laplace transform method. The nonlinear terms can be easily obtained by the use of He's polynomials. The technique presents an accurate methodology to solve many types of partial differential equations The approximate solutions obtained by proposed scheme in a wide range of the problem's domain were compared with those results obtained from the actual solutions. The comparison shows a precise agreement between the results. ª 2015 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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This paper investigates steady two dimensional flow of an incompressible magnetohydrodynamic (MHD) boundary layer flow and heat transfer of nanofluid over an impermeable surface in presence of thermal radiation and viscous dissipation. By using similarity transformation, the arising governing equations of momentum, energy and nanoparticle concentration are transformed into coupled nonlinear ordinary differential equations, which are than solved by homotopy analysis method (HAM). The effect of different physical parameters, namely, Prandtl number Pr, Eckert number

Current article is devoted with the study of MHD 3D flow of Oldroyd B type nanofluid induced by bi-directional stretching sheet. Expertise similarity transformation is confined to reduce the governing partial differential equations into ordinary nonlinear differential equations. These dimensionless equations are then solved by the Differential Transform Method combined with the Padé approximation (DTM-Padé). Dealings of the arising physical parameters namely the Deborah numbers β1 and β2, Prandtl number Pr, Brownian motion parameter Nb and thermophoresis parameter Nt on the fluid velocity, temperature and concentration profile are depicted through graphs. Also a comparative study between DTM and numerical method are presented by graph and other semi-analytical techniques through tables. It is envisage that the velocity profile declines with rising magnetic factor, temperature profile increases with magnetic parameter, Deborah number of first kind and Brownian motion parameter while decreases with Deborah number of second kind and Prandtl number. A comparative study also visualizes comparative study in details.

The key aim of the present work is to analyze the magnetohydrodynamic 2D flow of Williamson type nanofluid. Heat and mass transfer impacts are carried out in the manifestation of nonlinear thermal radiation, Cattaneo-Christov heat and mass flux models and varying thicker surface. By applying the appropriate similarity transformations, the mathematical equations of velocity, temperature and volume fraction transform to NODEs. An analytical scheme is pragmatic to estimate the convergence solutions in terms of power series. The dimensionless velocity profile, temperature profile and nanoparticle volume fraction with the administrative physical aspects are depicted through graphs. It is evidently ostensible that the dimensionless velocity declines for the augmented index parameter and wall thickness while cumulative values of M and , the horizontal fluid velocity decreases. Temperature specie upsurges with rising of Nb, Nt, n, , R d , w and M. Consequently demotes with the higher values of Pr and De 1 . Nanoparticle volumetric specie escalates with the growing effects of Nt, while it diminishes with Nb, Sc and De 2 . Comparison is the key procedure for validation our results with the earlier literature.

In this paper, we obtain the analytical solutions of linear and non-linear space-time fractional reaction-diffusion equations on a finite domain by the application of homotopy perturbation transform method (HPTM). The HPTM is a combined form of the Laplace transform method with the homotopy perturbation method. Some examples are also given. Numerical results show that the HPTM is easy to implement and accurate when applied to linear and non-linear space-time fractional reaction-diffusion equations. IntroductionIn recent years, it has turned out that many phenomena in engineering, physics, chemistry and other sciences can be described very successfully by models using mathematical tools from fractional calculus. For example, the nonlinear oscillation of earthquake can be modeled with fractional derivatives and the fluid-dynamic traffic model with fractional derivatives can eliminate the deficiency arising from the assumption of continuum traffic flow. Fractional derivatives are also used in modeling of many chemical processes, mathematical biology and many other problems in physics and engineering. These findings invoked the growing interest of studies of the fractional calculus in various fields such as physics, chemistry and engineering. Fractional differential equations have gained importance and popularity during the past three decades or so, mainly due to exact description of nonlinear phenomena, especially in fluid mechanics, e.g. nano-hydrodynamics, where continuum assumption does not well, and fractional model can be considered to be a best candidate. Hence, great attention has been given to finding solutions of fractional differential equations. Most fractional differential equations do not have exact analytical solutions, therefore approximate and numerical techniques must be used. Variational iteration method (VIM) [16] was first proposed to solve fractional differential equations with greatest success. Many authors found VIM as an effective way to solving linear and non-linear fractional differential equations [7,33]. The VIM was also used by many authors to study the various physical problems [23,26,36] In this paper, we use the homotopy perturbation transform method (HPTM) [22] for solving linear and non-linear space-time fractional reaction-diffusion equations on a finite domain. It is worth mentioning that this method is an elegant combination of the Laplace transformation, the HPM and He's polynomials and is mainly due to Ghorbani [10,11]. The use of He's polynomials in the nonlinear term was first introduced by Ghorbani [10,11]. This algorithm provides the solution in a rapid convergent series which may lead to the solution in a closed form. The advantage of this method is its capability of combining two powerful methods for obtaining exact solutions for nonlinear equations. In recent years, fractional reaction-diffusion models are studied due to their usefulness and importance in many areas of science and engineering. The reaction-diffusion equations arise naturally as description mode...