In this paper, we mainly study the solution and properties of the multiterm time-fractional diffusion equation. First, we obtained the stochastic representation for this equation, which turns to be a subordinated process. Based on the stochastic representation, we calculated the mean square displacement (MSD) and time average mean square displacement, then proved some properties of this model, including subdiffusion, generalized Einstein relationship, and nonergodicity. Finally, a stochastic simulation algorithm was developed for the visualization of sample path of the abnormal diffusion process. The Monte Carlo method was also employed to show the behavior of the solution of this fractional equation.

Recently, the diffusion equations that generalize the usual one have received considerable attention due to the broadness of their physical applications, in particular, to the anomalous diffusion. In fact, fractional diffusion equations and the nonlinear fractional diffusion equations have been successfully applied to several physical situations such as percolation of gases through porous media [

The multiterm time-fractional advection-diffusion equations are linear integrodifferential equations. They are obtained from their corresponding classical multiterm advection-diffusion equations by replacing the first-order time derivative by fractional derivative, which reads

There are growing interests in studying these equations because of their importance in modeling many physical, biological, medical, chemical, and many other fields. For example, the over diagnostic ultrasound frequencies, acoustic absorption in biological tissue exhibits a power law with a noninteger frequency [

Many analytic and numeric methods are employed to solve this equation. Daftardar-Gejji and Bhalekar considered the multiterm time-fractional diffusion-wave equation using the method of separation of variables [

In this paper, we introduce the stochastic representation method to solve this multiterm time-fractional diffusion equation. This paper is organized as follows. In Section

In this section, we will give the stochastic representation of the multiterm time-fractional diffusion equation.

Let

The subordinated process

Following the same procedure shown in [

So, the Laplace transform of

Here, the

Thus, the following relation between

Since the process

The Laplace transform of the above equation with respect to

By changing the variable

Since the Laplace transform of the Caputo fractional derivative is given by

The superiority of the stochastic representation approach to the fractional differential equation is that it not only helps us to understand the physical process by providing a description of the dynamical system governed by the fractional differential equation but also provides a way to get the properties of the corresponding equations. For simplicity, we retain two terms for the time-fractional operator, i.e.,

The subordinated process governed by the multiterm time-fractional diffusion equation (

Taking advantage of the relation between

Here,

In order to calculate the mean square displacement, we can first compute its Laplace transform. By using the Laplace transform of

From Equation (

The generalized Einstein relation holds for the multiterm time-fractional diffusion equation (

With the help of the stochastic representation, we can calculate the first moment of

Comparing the above result with Equation (

To connect to single-particle tracking experiments, we now turn to the time-averaged MSD of the stochastic process, defined by

The time series

The subordinated process

From Equation (

According to the definition of time-averaged MSD, we calculate the

The disparity between the ensemble and the -averaged MSD exists. Even in the limit of long measurement times

The stochastic representation provides two ways to get the solution of the multiterm time-fractional advection-diffusion equation (

The algorithm of simulation of the subordinated process

This step is aimed at simulating the subordinator

Here,

This step is aimed at simulating the process

The sample paths of the stochastic process

The corresponding waiting times with

The sample paths of the stochastic process

The approximated solutions (up) of Equation (

In this paper, an advection-diffusion equation with multiterm time-fractional derivatives is employed. We obtained its stochastic representation, which is driven by the Brown motion and the inverse time of the sum of Lévy motions with different parameters. Then, the mean square displacement indicates the model is subdiffusive and the generalized Einstein relation is also retained, but weak ergodicity is breaking. At last, an algorithm is constructed to simulate the sample paths of the stochastic process. With the help of stochastic representation, the Monte Carlo method is employed to approximate the solution of the corresponding equation. We find that the solution is heavy tailed and sharp peaked, which is common in statistical physics and finance. So, we expect that the results obtained here may be useful for the discussion of the anomalous diffusion systems.

The data used to support the findings of this study are included within the article.

The authors declare that no conflicts of interest exist regarding this manuscript.

We are grateful to Professor Zhongdi Cen for useful comments and suggestions. This work is supported by the National Natural Science Foundation of China No. 11801288, the Natural Science Foundation of Ningbo No. 2017A610130, and the Natural Science Foundation of Zhejiang Province No. LY17A010020.