## An implicit finite difference approximation for the solution of the diffusion equation with distributed order in time

N. J. Ford, M. L. Morgado, and M. Rebelo

### Abstract

In this paper we are concerned with the numerical solution of a diffusion equation in which the time derivative is of non-integer order, i.e., in the interval $(0,1)$. An implicit numerical method is presented and its unconditional stability and convergence are proved. Two numerical examples are provided to illustrate the obtained theoretical results.

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### Key words

Caputo derivative, fractional differential equation, subdiffusion, finite difference method, distributed order differential equation

### AMS subject classifications

35R11, 65M06, 65M12

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