This article combines ordinary differential equations’ theoretical and practical characteristics to explore how to integrate mathematical modeling ideas in teaching materials and teaching methods. We apply the fractional differential equation algorithm to the mathematical modeling of Terman diffusion. At the same time, the article proposes a framework for obtaining accurate solutions of a class of nonlinear Sinai stochastic models with power-law diffusion through Mittag-Leffler function transformation. In this way, we have obtained great explicit and accurate solutions of some important Mittag-Leffler function power-law diffusion physical processes.

#### Keywords

- Fractional differential equations
- Very slow diffusion
- Mathematical modeling ideas
- Sinai stochastic model
- Fractional structural derivatives
- Teaching

#### MSC 2010

- 33E12

The diffusion process is usually described by the power-rate proportional relationship between the mean square displacement and time, and its mathematical form is as follows:

When the parameter α<1, the above formula represents the “slow diffusion” process. When the parameter α>1, the above formula represents the “fast diffusion” process. Normal diffusion is a special case when the parameter α=1.

In recent years, researchers have done a lot of work on modeling abnormal diffusion [1]. However, many ultra-slow diffusion phenomena are much slower than the above-mentioned slow diffusion process. Such as the diffusion of atoms in amorphous alloy melts, the aging of high-density colloids, and the diffusion of chemical solvents in polymerization media. Sinai first proposed the following stochastic model of logarithmic relationship very slow diffusion in 1982:

The Sinai model is a statistical model that describes the very slow diffusion of the logarithmic relationship, but it has the following two problems:

1) When t changes monotonically from 0 to 1, 〈^{2}(^{2}(

2) The scope of application is very small, and it is not ideal to fit many very slow diffusion experimental data [2]. Some scholars generalize Sinai logarithmic diffusion to obtain a general logarithmic diffusion model, as follows:

But the above two problems have not been solved yet. This paper uses the inverse function _{α}

Where 0 <

There are two commonly used definitions of fractional derivatives [3]. That is, Caputo derivative and Riemann-Liouville (derivative. Caputo derivative operator with order

The definition of the single-parameter Mittag-Leffler function (M-L function for short) is as follows:

Among them ^{z}_{α}

Assume that the real-valued smooth function _{1}(_{2}(_{1}(0) = _{2}(0). If
_{1}(_{2}(

For any _{1}(_{2}(

The condition
_{1}(_{2}(

The certificate is complete.

When _{α}^{x}

When _{α}^{0} = 1,

For any positive integer n, there is

Then _{α}^{x}_{α}^{x}

When _{α}_{α}

1) When _{1} = _{α}_{2} = ln _{α}_{1}) = ^{y}^{2}. Because of _{α}_{1}) > 1 = _{α}_{1} > 0 is obtained from the monotonicity of _{α}_{α}_{1}) > ^{y}^{1} has ^{y}^{2} = _{α}_{1}) > ^{y}^{1}. _{2} > _{1} get I from the monotonicity of the exponential function. The conclusion is proved.

2) When

The anomalous diffusion law in the proportional form of the mean square displacement and the time function can be derived from the theory of the probability density function in the continuous-time random walk (CTRWs) [5]. It can be derived from the following relationship:

Where

Where

1) The movement of each particle in the diffusion area is independent of the movement of other particles.

2) When the considered time region is long enough, the movement of the same particle in the diffusion region in different time regions is an independent process.

Einstein derived the normal diffusion equation:

Assuming that the particle is at the origin position in the unbounded region at time 0, the solution to the above diffusion equation has the following probability density function:

The mean square displacement (the second moment of the probability density function) can be obtained from the above formula:

Equation (12) is a stochastic model of normal diffusion. The diffusion equation (10) can also be derived from the continuous random time walk theory. In the random walk theory, the Einstein hypothesis is further quantified [6]. The waiting time probability distribution

The corresponding Laplace transform and Fourier transform are

Some scholars have pointed out that for any waiting time probability density

Einstein assumes that the time and space of each particle are independent [7]. We generally think that abnormal diffusion is a non-local process with memory and path dependence. Some scholars have pointed out that if the waiting time probability distribution

Then their Laplace transform and Fourier transform respectively satisfy:

From this, the corresponding anomalous diffusion equation is derived as follows:

Applying the second-moment operator
^{α}^{−1} in the Riemann-Liouville fractional derivative operator

Where _{α}

Where _{α}_{a}^{a}_{α}_{a}

Add the proportional coefficient _{α}

In the above formula, both sides of t are derived at the same time to obtain

Take _{α}_{α}_{α}

In the above formula, the integrals on both sides from 0 to t are obtained at the same time

In the above formula, perform Laplace transformation on both sides at the same time and arrange to get

In the above formula, the Laplace inverse transformation is performed on both sides at the same time to obtain the singular kernel in the form of

This section introduces the following two probability assumptions to derive the very slow diffusion law proposed in this paper (4).

Hypothesis 1: The probability distribution of the coverage length of the offset field in the diffusion area is the following power-law relationship:

Among them

Hypothesis 2: The rate of change of the offset field received by the particles in the diffusion region concerning time is controlled by the derivative of the inverse function of the desingular M-L function, namely

Among them,

If you limit ^{E}

Both sides act on the M-L operator _{a}

Where _{a}

Substituting the above formula into formula (29) and finishing to get

In

First, we observe the difference between the very slow diffusion and Sinai logarithmic diffusion proposed in this paper based on the M-L inverse function from the image [10]. Firstly, when

The ordinate R in Figure 2 is the root mean square displacement. Inverse M-L(0.5) diffusion represents the extremely slow diffusion with the M-L inverse function parameter α=0.5. Inverse M-L(0.8) diffusion means very slow diffusion when α=0.8. Modified Sinai diffusion means that when α=1, the extremely slow diffusion degenerates into Sinai logarithmic diffusion after adding the initial parameter δ=1. The extremely slow diffusion is much slower than the Sinai logarithmic diffusion to characterize the Sinai Logarithmic diffusion as slower diffusion.

We compare and analyze the long-term diffusion behavior of the extremely slow diffusion of the M-L inverse function and the long-term logarithmic diffusion of Sinai [11]. When we take α=0.9, 0.95, δ=1, and the migration field distribution factor β=2, the long-term behaviors of very slow diffusion and Sinai logarithmic diffusion based on the M-L inverse function are shown in Fig. 3.

Inverse M-L (0.9) diffusion means very slow diffusion when α=0.9. Inverse M-L (0.95) diffusion means very slow diffusion when α=0.95. Modified Sinai diffusion means that when α=1, the very slow diffusion degenerates into Sinai logarithmic diffusion with the initial parameter δ=1. Figure 3 shows that the very slow diffusion of the inverse M-L function gradually converges to the Sinai logarithmic diffusion when α is constantly close to 1.

This paper uses the inverse function of the M-L function to extend the Sinai diffusion to a more general, very slow diffusion. We introduce initial state parameters to solve the singularity problem of Sinai diffusion near the initial moment. On the other hand, our very slow diffusion model has a wider application range than Sinai diffusion. This article also introduces the concept of fractional structural derivatives. At the same time, a fractional structural derivative model of super slow diffusion is established. Then the expression of the singular nucleus of the governing equation of the super slow diffusion is derived.

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