Numerical modelling transient current in the time-of-flight experiment with time-fractional advection-diffusion equations

Abstract

In this work we report the development of an implicit finite difference numerical method for the one space dimension time-fractional advection-diffusion equation, on a bounded domain, to model the transient electrical current of the time of flight experiment of disordered (e.g. organic) semiconductors. Some numerical experiments and simulation of experimental data are carried out showing that the presented model describes accurately the transient electrical current.

Modelling Time-of-Flight Transient Currents with Time-Fractional Diffusion Equations

“…One possible justification for this fact is based on the assumption that the energy distribution of deep traps is in fact truncated leading to a truncated waiting time distribution ( [11]). This paper continues the investigation initiated in [4], where the following class of initialboundary value problem was considered:…”

“…For the numerical approximation of the Caputo derivative of order α on the interval [0, T ], we will use the non-uniform mesh (6), and use the following approximation for the Caputo derivative (see [4]):…”

“…We assume that g, f , φ 0 and φ L are continuous functions in their respective domains. In [4], an implicit numerical scheme on a time graded mesh was developed to approximate the solution of the initial-boundary value problem above. The reason for using a graded mesh in the time variable, is due to the fact that, as it is known ( [5]), the solution of this type of problems exhibits a sharp behavior near the origin (which is in agreement with the fact that, usually, fractional differential equations have singularities at t = 0, in the sense that the derivatives of the solution may become unbounded near that point), and therefore in order to obtain reasonable accurate results it is convenient to use small step-sizes near that point.…”

“…, n − 1), while if r > 1, the grid-points are more densely placed in the left-hand side of the interval [0, T ], as desired. In [4], the proper choice of the grading exponent was left for future work, and this is one of the aspects we want to address here, basing us on the results of [8]. Moreover, the scheme presented in [4] was first order convergent in space, while here we develop a second order convergent method with respect to the space variable.…”

Theoretical and numerical analysis of unsteady fractional viscoelastic flows in simple geometries

“…In order to realize this response, at each time step we have to take into account all the deformation history from t 0 (beginning of the experiment) to t s (actual time). The computational method becomes more expensive, and therefore, the use of a graded mesh [81,82,83,84] in time allows a huge reduction in simulation time, especially for this case where the gradients of velocity and stress drastically reduce after the jump in deformation. For more details please see Appendix B.…”