In this paper, we introduce a class of time-fractional diffusion model with singular source term. The derivative employed in this model is defined in the Caputo sense to fit the conventional initial condition. With assistance of corresponding linear fractional differential equation, we verify that the solution of such model may not be globally well-defined, and the dynamics of this model depends on the order of fractional derivative and the volume of spatial domain. In simulation, a finite difference scheme is implemented and interesting numerical solutions of model are illustrated graphically. Meanwhile, the positivity, monotonicity, and stability of the proposed scheme are proved. Numerical analysis and simulation coincide the theoretical studies of this new model.

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