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.

中文翻译：

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含弱奇异性的时间分数阶抛物-椭圆问题的L1格式分析

考虑了混合抛物线-椭圆型的时间分数初始边界值问题。这种问题的领域分为两个子领域。在第一个域中考虑了反应扩散抛物线问题，在第二个域中考虑了对流扩散椭圆型问题。该问题在初始时间t  = 0处具有轻微的奇异性。引入经典的L1方案来近似时间导数，并且使用二阶标准有限差分方案来近似空间导数。用两个方向上的均匀网格离散域。结果表明，随着t  = 0，收敛阶数更高。而不是整个域上的收敛顺序。为了显示该方案的效率，提供了数值结果。