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In this paper, Haar wavelet operational matrix(HWOM) is proposed to solve initial-boundary value problems for a class of time-space fractional partial differential equations of Caputo sense with variable coefficients in both time and space (1) ∑ i = 1 n θ i ∂ γ i u ( x , t ) ∂ t γ i = v ( x , t ) ∂ α u ( x , t ) ∂ x α + d ( x , t ) ∂ β u ( x , t ) ∂ x β + q ( x , t ) , 0 < x < 1 , 0 < t ≤ 1 , as an extension of Rehman and Khan's (2013) work. We obtain a matrix L instead of Q α in Rehman and Khan (2013). when dealing with boundary conditions. By utilizing the operational matrix of fractional integration and Hadamard product,we made an improvement of algorithm to deal with time-space coefficients and gave the error analysis of the HWOM for space-time dimensions. Some numerical results are paralleled with exact solutions to show the efficiency and precision of the presented technique. [ABSTRACT FROM AUTHOR]

In this paper, the flow of a micropolar fluid through a porous channel with deforming walls is investigated. The Lie group method is applied to determine symmetry reductions of partial differential equations. Compared with previous work, the similar variable for the microrotation velocity also is obtained, which is an extension of the case corresponding to the Newtonian fluids (Boutros et al., 2007). This problem we study here can be considered as an extension of the model discussed in Boutros et al. (2007). The effect of the permeation Reynolds number Re , the micropolar parameter K and the expansion ratio α are studied numerically and the results are plotted. Furthermore, the multiple solutions also can be found for some values of parameters. [ABSTRACT FROM AUTHOR]

Abstract: Considering the influence of the viscous dissipation in the energy equation, we investigate the flow and heat transfer of an incompressible micropolar fluid in a channel with expanding or contracting walls. By introducing suitable similarity transformations, the nonlinear partial differential equations are reduced to a system of ordinary differential equations. Homotopy analysis method (HAM) is employed to obtain the series solutions for the velocity, microrotation and temperature distribution. In order to analyze the influence of the micropolar parameter, the Reynolds number and the deformation of the wall, the graphs for velocity components, microrotation, temperature, couple stress and shear stress in the channel are presented and discussed in detail. [Copyright &y& Elsevier]

The finite difference schemes for equations depicting the fluid flow, heat and mass transfer of viscoelastic fluid in porous media based on fractional constitutive model are analyzed, when Soret and Dufour effects are taken into account. Different from the conventional fractional partial differential equations, the governing momentum equation of this system not only contains convection terms, but also a term: D t β 0 (∂ 2 u / ∂ y 2) , which are great challenges when we do the stability analysis. In addition, the temperature equation and concentration equation are coupled with each other, which increases the difficulty of numerical analysis. By using the energy method, the stability and convergence of the finite difference schemes employed for the governing equations are obtained. The accuracy of the scheme for momentum equation is O (τ + h x 2 + h y 2) , while the one for temperature and concentration equations is O (τ 2 + h x 2 + h y). Finally, numerical examples are presented to verify methods and demonstrate the analysis effectiveness. [ABSTRACT FROM AUTHOR]