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A time fractional initial boundary value problem of mixed parabolic-elliptic type is considered. The domain of such problem is divided into two subdomains. A reaction-diffusion parabolic problem is considered on the first domain, and on the second, a convection-diffusion elliptic type problem is considered. Such problem has a mild singularity at the initial time t = 0. The classical L1 scheme is introduced to approximate the temporal derivative, and a second order standard finite difference scheme is used to approximate the spatial derivatives. The domain is discretized with uniform mesh for both directions. It is shown that the order of convergence is more higher away from t = 0 than the order of convergence on the whole domain. To show the efficiency of the scheme, numerical results are provided.

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This article deals with an efficient numerical technique to solve a class of multiterm time fractional Volterra-Fredholm partial integro-differential equations of first kind. The fractional derivatives are defined in Caputo sense. The Adomian decomposition method is used to construct the scheme. For simplicity of the analysis, the model problem is converted into a multi-term time fractional Volterra-Fredholm partial integro-differential equation of second kind. In addition, the convergence analysis and the condition for existence and uniqueness of the solution are provided. Several numerical examples are illustrated in support of the theoretical analysis.

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