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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]

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]