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# Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching

###### Aceptado: 18 Mar 2022
Detalles de la revista Formato
Revista
eISSN
2444-8656
Primera edición
01 Jan 2016
Calendario de la edición
2 veces al año
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Inglés

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.

#### MSC 2010

Introduction

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: $〈x2(t)〉∞ta$ \left\langle {{x^2}\left(t \right)} \right\rangle \infty {t^a}

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 . 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: $〈x2(t)〉∞(lnt)4$ \left\langle {{x^2}\left(t \right)} \right\rangle \infty {\left({\ln t} \right)^4}

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, 〈x2(t)〉 changes from infinity to 0 accordingly. In fact, 〈x2(t)〉 corresponding to t=0 should also be 0. Therefore, the Sinai model is only suitable for describing the long-term slow diffusion and cannot be used to describe the short-term diffusion near the initial moment;

2) The scope of application is very small, and it is not ideal to fit many very slow diffusion experimental data . Some scholars generalize Sinai logarithmic diffusion to obtain a general logarithmic diffusion model, as follows: $〈x2(t)〉~(lnt)2y$ \left\langle {{x^2}\left(t \right)} \right\rangle \sim {\left({\ln t} \right)^{2y}}

But the above two problems have not been solved yet. This paper uses the inverse function Mα of the Mittag-Leffler function to generalize the diffusion of the Sinai logarithmic relationship. This gives the following very slow diffusion random model: $〈x2(t)〉~[Mα(δ+t)]2y$ \left\langle {{x^2}\left(t \right)} \right\rangle \sim {\left[{{M_\alpha}\left({\delta + t} \right)} \right]^{2y}}

Where 0 < α < 1, δ > 0, 0 < y ≤ 2. And from this, a fractional structural derivative model of very slow diffusion is established. Note that equation (4) degenerates to Sinai model (2) when α = 1, δ = 0, y = 2 is.

Preliminary knowledge
Fractional Calculus

There are two commonly used definitions of fractional derivatives . That is, Caputo derivative and Riemann-Liouville (derivative. Caputo derivative operator with order q ∈ (0,1) and lower limit 0 is defined as $CD0,tq f(t):=dqdtqf(t)=J01−qf′(t)$ {}^CD_{0,t}^q\,f\left(t \right): = {{{d^q}} \over {d{t^q}}}f\left(t \right) = J_0^{1 - q}f{'}\left(t \right) . $J01−q$ J_0^{1 - q} . is the 1 − q order Riemann-Liouville integral operator, and its definition is as follows $J0β f(t)=1Γ(β)∫0t(t−s)β−1 f(s)ds$ {}J_0^\beta \,f\left(t \right) = {1 \over {\Gamma \left(\beta \right)}}\int_0^t {{{\left({t - s} \right)}^{\beta - 1}}\,f\left(s \right)ds} , β > 0. The order is The definition formula of the Riemann-Liouville derivative operator with q ∈ (0,1) and the lower limit is 0 is $LD0,tq f(t):Dtqf(t)=ddtJ01−qf(t)$ ^LD_{0,t}^q\,f\left(t \right):D_t^qf\left(t \right) = {d \over {dt}}J_0^{1 - q}f\left(t \right) .

Mittag-Leffler function and its inverse function

The definition of the single-parameter Mittag-Leffler function (M-L function for short) is as follows: $E(α, 1, z):=Ea(z)=∑k=0∞zkΓ(αk+1)$ E\left({\alpha,\,1,\,z} \right): = {E_a}\left(z \right) = \sum\limits_{k = 0}^\infty {{{{z^k}} \over {\Gamma \left({\alpha k + 1} \right)}}}

Among them α > 0, zC. When α=1, the M-L function degenerates into an exponential function ez. When α=1, the inverse function Mα(x) of the M-L function degenerates to a logarithmic function. Figure 1 shows the single-parameter M-L function and its inverse function . When x exceeds a value near 1, the inverse function of the M-L function monotonically increases with a time slower than the logarithmic function. When α ∈ (0,1), t ∈ [1, + ∞), the inverse function of the M-L function monotonously increases with time slower than the logarithmic function. First, we need to use the following lemma. Comparison of M-L function and its inverse function with exponential (log) function
Lemma 1

Assume that the real-valued smooth function f1(x), f2(x) of the domain on [0, + ∞) satisfies f1(0) = f2(0). If $f1(n)(0)>f2(n)(0)$ f_1^{\left(n \right)}\left(0 \right) > f_2^{\left(n \right)}\left(0 \right) holds for any positive integer n, then f1(x) > f2(x), $f1′(x)>f2′(x)$ f_1^{'}\left(x \right) > f_2^{'}\left(x \right) holds for any x ∈ (0, +∞).

Proof

For any x ∈ (0, +∞), we carry out Taylor expansion of f1(x) − f2(x) at 0 to get $f1(x)−f2(x)=f1(0)−f2(0)+[f1′(0)−f2′(0)]x+f1"(0)−f2"(0)2x2+…+f1(n)(0)−f2(n)(0)n!xn+…=[f1′(0)−f2′(0)]x+f1"(0)−f2"(0)2x2+…+f1(n)(0)−f2(n)(0)n!xn+….$ \matrix{{{f_1}\left(x \right) - {f_2}\left(x \right) = {f_1}\left(0 \right) - {f_2}\left(0 \right) + \left[{f_1^{'}\left(0 \right) - f_2^{'}\left(0 \right)} \right]x +} \hfill {{{f_1^{''}\left(0 \right) - f_2^{''}\left(0 \right)} \over 2}{x^2} + \ldots + {{f_1^{\left(n \right)}\left(0 \right) - f_2^{\left(n \right)}\left(0 \right)} \over {n!}}{x^n} + \ldots =} \hfill {\left[{f_1^{'}\left(0 \right) - f_2^{'}\left(0 \right)} \right]x + {{f_1^{''}\left(0 \right) - f_2^{''}\left(0 \right)} \over 2}{x^2} + \ldots + {{f_1^{\left(n \right)}\left(0 \right) - f_2^{\left(n \right)}\left(0 \right)} \over {n!}}{x^n} + \ldots.} \hfill } .

The condition $f1(n)(0)>f2(n)(0)$ f_1^{\left(n \right)}\left(0 \right) > f_2^{\left(n \right)}\left(0 \right) holds for any positive integer n, so we can get f1(x) − f2(x) > 0, ∀x ∈ (0, +∞). For any x ∈ (0, ∞), we perform Taylor expansion of $f1′(x)>f2′(x)$ f_1^{'}\left(x \right) > f_2^{'}\left(x \right) at 0 to get $f1'(x)−f2'(x)=[f1'(0)−f2'(0)]+[f1"(0)−f2"(0)]x+f1(3)(0)−f2(3)(0)2x2+…+f1(n+1)(0)−f2(n+1)(0)n!xn+…$ \matrix{{f_1^{'}\left(x \right) - f_2^{'}\left(x \right) = \left[{f_1^{'}\left(0 \right) - f_2^{'}\left(0 \right)} \right] + \left[{f_1^{''}\left(0 \right) - f_2^{''}\left(0 \right)} \right]x +} \hfill {{{f_1^{\left(3 \right)}\left(0 \right) - f_2^{\left(3 \right)}\left(0 \right)} \over 2}{x^2} + \ldots + {{f_1^{\left({n + 1} \right)}\left(0 \right) - f_2^{\left({n + 1} \right)}\left(0 \right)} \over {n!}}{x^n} + \ldots} \hfill } . The condition $f1(n)(0)>f2(n)(0)$ f_1^{\left(n \right)}\left(0 \right) > f_2^{\left(n \right)}\left(0 \right) holds for any positive integer n, so we get $f1′(x)>f2′(x), ∀x∈(0,+∞)$ f_1^{'}\left(x \right) > f_2^{'}\left(x \right),\,\forall x \in \left({0, + \infty} \right) .

The certificate is complete.

Corollary 1

When α ∈ (0,1), x∈ (0, +∞), is Ėα (x) > ex and.

Proof

When α ∈ (0,1), x∈ (0, +∞), it is easy to get Eα (0) = e0 = 1, $Eα(1)(0)=1Γ(α+1)>1=e0$ E_\alpha ^{\left(1 \right)}\left(0 \right) = {1 \over {\Gamma \left({\alpha + 1} \right)}} > 1 = {e^0} from the expression (5) of the M-L function.

For any positive integer n, there is $Eα(n)(0)=n!Γ(nα+1)>1=e0$ E_\alpha ^{\left(n \right)}\left(0 \right) = {{n!} \over {\Gamma \left({n\alpha + 1} \right)}} > 1 = {e^0}

Then Ėα (x) > ex. Ėα (x) > ex is obtained from Lemma 1. Holds for any x ∈ (0, +∞). The certificate is complete.

Theorem 1

When α ∈ (0,1), x∈ (1, +∞), the inverse function Mα(x) of the M-L function satisfies 1) Mα(x) < ln x; 2).

Prove

1) When α ∈ (0,1), x∈ (1, +∞), we assume y1 = Mα (x), y2 = ln x, then x = Eα (y1) = ey2. Because of Eα (y1) > 1 = Eα (0), y1 > 0 is obtained from the monotonicity of Eα (x). By inference, 1, Eα (y1) > ey1 has ey2 = Eα (y1) > ey1. y2 > y1 get I from the monotonicity of the exponential function. The conclusion is proved.

2) When α ∈ (0,1), x∈ (1, +∞), we assume y = ln x, then $Mα(x)<1Lα<1ey=1x=(lnx)$ {M_\alpha}\left(x \right) < {1 \over {{L_\alpha}}} < {1 \over {{e^y}}} = {1 \over x} = \left({\ln x} \right) is obtained from the property of the inverse function derivative of the function and Corollary 1. The conclusion is proved.

Continuous-time random walk

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) . It can be derived from the following relationship: $〈x2〉=∫−∞+∞x2P(x,t)dx$ \left\langle {{x^2}} \right\rangle = \int_{- \infty}^{+ \infty} {{x^2}P\left({x,t} \right)dx}

Where P(x, t) is the time-space probability density function of CTRWs. The probability density function of a continuous-time random walk with uncoupled time-space probability density P(x, t) = p(x)ψ(t) in Fourier-Laplace space is in the form $W(k,u)=1−φ(s)s11−λ(k)φ(s)$ W\left({k,u} \right) = {{1 - \varphi \left(s \right)} \over s}{1 \over {1 - \lambda \left(k \right)\varphi \left(s \right)}}

Where λ(k) is the Fourier transform of p(x). $λ(k)=∫−∞∞eikxP(x)dx$ \lambda \left(k \right) = \int_{- \infty}^\infty {{e^{ikx}}P\left(x \right)dx}

φ(s) is the Laplace transform of ψ(t). $φ(s)=∫0∞e−stψ(t)dt$ \varphi \left(s \right) = \int_0^\infty {{e^{- st}}\psi \left(t \right)dt}

Fractional structural derivative model of very slow diffusion

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: $∂∂tP(x,t)=K1∂2∂x2P(x,t)$ {\partial \over {\partial t}}P\left({x,t} \right) = {K_1}{{{\partial ^2}} \over {\partial {x^2}}}P\left({x,t} \right)

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: $P(x,t)=14πK1texp(−x24K1t)$ P\left({x,t} \right) = {1 \over {\sqrt {4\pi {K_1}t}}}\exp \left({- {{{x^2}} \over {4{K_1}t}}} \right)

The mean square displacement (the second moment of the probability density function) can be obtained from the above formula: $〈x2(t)〉=∫−∞+∞x2P(x,t)dx=2K1t$ \left\langle {{x^2}\left(t \right)} \right\rangle = \int_{- \infty}^{+ \infty} {{x^2}P\left({x,t} \right)dx = 2{K_1}t}

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 . The waiting time probability distribution ω(t) and the jumping step probability distribution function λ(x) conform to the Poisson (Poisson) distribution and the normal distribution, respectively: ${ω(t)=τ−1exp(−t/τ)λ(x)=(4πσ2)−1/2exp(−x2/(4σ2))$ \left\{{\matrix{{\omega \left(t \right) = {\tau ^{- 1}}\exp \left({- t/\tau} \right)} \hfill {\lambda \left(x \right) = {{\left({4\pi {\sigma ^2}} \right)}^{- 1/2}}\exp \left({- {x^2}/\left({4{\sigma ^2}} \right)} \right)} \hfill }} \right.

The corresponding Laplace transform and Fourier transform are ${ω(u)~1−uτ+O(τ2)λ(k)~1−σ2k2+O(k4)$ \left\{{\matrix{{\omega \left(u \right)\sim 1 - u\tau + O\left({{\tau ^2}} \right)} \hfill {\lambda \left(k \right)\sim 1 - {\sigma ^2}{k^2} + O\left({{k^4}} \right)} \hfill }} \right.

Some scholars have pointed out that for any waiting time probability density ω(t) and jumping step probability density function λ(x), as long as the Laplace transform. Fourier transform satisfies equation (14), the diffusion equation corresponding to ω(t) and λ(x) can be derived as equation (10).

Einstein assumes that the time and space of each particle are independent . 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 ω(t) and the jumping step probability distribution function λ(x) conform to the power-law distribution and the normal distribution, respectively as follows: ${ω(t)~Aα(τ/t)1+αλ(x)=(4πσ2)−1/2exp(−x2/(4σ2))$ \left\{{\matrix{{\omega \left(t \right)\sim {A_\alpha}{{\left({\tau /t} \right)}^{1 + \alpha}}} \hfill {\lambda \left(x \right) = {{\left({4\pi {\sigma ^2}} \right)}^{- 1/2}}\exp \left({- {x^2}/\left({4{\sigma ^2}} \right)} \right)} \hfill }} \right.

Then their Laplace transform and Fourier transform respectively satisfy: ${ω(u)~1−(uτ)α+O(τ2)λ(k)~1−σ2k2+O(k4)$ \left\{{\matrix{{\omega \left(u \right)\sim 1 - {{\left({u\tau} \right)}^\alpha} + O\left({{\tau ^2}} \right)} \hfill {\lambda \left(k \right)\sim 1 - {\sigma ^2}{k^2} + O\left({{k^4}} \right)} \hfill }} \right.

From this, the corresponding anomalous diffusion equation is derived as follows: $∂∂tP(x,t)=KαDt1−α∂2∂x2P(x,t)$ {\partial \over {\partial t}}P\left({x,t} \right) = {K_\alpha}D_t^{1 - \alpha}{{{\partial ^2}} \over {\partial {x^2}}}P\left({x,t} \right)

Applying the second-moment operator $∫−∞∞dxx2$ \int_{- \infty}^\infty {dx{x^2}} on both sides of equation (17) simultaneously, we can get the strange diffusion law of form (1). The power-rate integral kernel M(t)∞tα−1 in the Riemann-Liouville fractional derivative operator $Dt1−α(⋅)$ D_t^{1 - \alpha}\left(\cdot \right) in equation (17) shows that what is defined by equation (17) is non-Markov (Markov) anomalous diffusion behavior with memory . We speculate that the diffusion equation corresponding to the very slow diffusion law in the form of the M-L inverse function is as follows: $∂∂tP(x,t)=Hα 0Kt1−α∂2∂x2P(x,t)$ {\partial \over {\partial t}}P\left({x,t} \right) = {H_\alpha}{\,_0}K_t^{1 - \alpha}{{{\partial ^2}} \over {\partial {x^2}}}P\left({x,t} \right)

Where Hα is the undetermined coefficient and $0Kt1−α(⋅)$ _0K_t^{1 - \alpha}\left(\cdot \right) is the fractional structural derivative operator, which is defined as $0Kt1−αf(t)=Qα∂∂t∫0tka−1(δ+t−τ)u(τ)dτ$ _0K_t^{1 - \alpha}f\left(t \right) = {Q_\alpha}{\partial \over {\partial t}}\int_0^t {{k_{a - 1}}} \left({\delta + t - \tau} \right)u\left(\tau \right)d\tau

Where Qα is the undetermined coefficient. When the singular kernel is in the powder form ka(t) = ta and δ = 0, Qα = 1 / Γ(α), the above equation degenerates to the definition of the traditional Riemann-Liouville fractional derivative operator . The following is to derive the specific form of the strange nucleus ka in the $0Kt1−α$ _0K_t^{1 - \alpha} nuclear derivative in the diffusion equation corresponding to the very slow diffusion. Apply the second-order moment operator $∫−∞∞dxx2$ \int_{- \infty}^\infty {dx{x^2}} on both sides of equation (18) at the same time to obtain $ddt〈x2(t)〉=0Kt1−α2Hα$ {d \over {dt}}\left\langle {{x^2}\left(t \right)} \right\rangle {= _0}K_t^{1 - \alpha}2{H_\alpha}

Add the proportional coefficient Tα to equation (4) to convert it into an equation and take y = 2 to get $〈x2(t)〉= Tα[Mα(δ+t)]4$ \left\langle {{x^2}\left(t \right)} \right\rangle = \,{T_\alpha}{\left[{{M_\alpha}\left({\delta + t} \right)} \right]^4}

In the above formula, both sides of t are derived at the same time to obtain $ddt〈x2(t)〉=4Tα[Mα(δ+t)]3M˙α(δ+t)$ {d \over {dt}}\left\langle {{x^2}\left(t \right)} \right\rangle = 4{T_\alpha}{\left[{{M_\alpha}\left({\delta + t} \right)} \right]^3}{\dot M_\alpha}\left({\delta + t} \right)

Take Hα = (1/2)(Tα / Qα) and combine equations (20) and (22) to get $1Qα0Kt1−α(1)=4(Mα(δ+t)]3M˙α(δ+t)$ {\left. {{1 \over {{Q_{\alpha 0}}}}K_t^{1 - \alpha}\left(1 \right) = 4\left({{M_\alpha}\left({\delta + t} \right)} \right.} \right]^3}{\dot M_\alpha}\left({\delta + t} \right)

In the above formula, the integrals on both sides from 0 to t are obtained at the same time $∫0tkα−1(δ+t−τ)dτ=[Mα(δ+t)]4−[Mα(δ)]4$ \int_0^t {{k_{\alpha - 1}}\left({\delta + t - \tau} \right)d\tau = {{\left[{{M_\alpha}\left({\delta + t} \right)} \right]}^4} - {{\left[{{M_\alpha}\left(\delta \right)} \right]}^4}}

In the above formula, perform Laplace transformation on both sides at the same time and arrange to get $L[kα−1(δ+t)]=sL{[Mα(δ+t)]4}−[Mα(δ)]4$ L\left[{{k_{\alpha - 1}}\left({\delta + t} \right)} \right] = sL\left\{{{{\left[{{M_\alpha}\left({\delta + t} \right)} \right]}^4}} \right\} - {\left[{{M_\alpha}\left(\delta \right)} \right]^4}

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 $kα−1(δ+t)=4[Mα(δ+t)]3M˙α(δ+t)$ {k_{\alpha - 1}}\left({\delta + t} \right) = 4{\left[{{M_\alpha}\left({\delta + t} \right)} \right]^3}{\dot M_\alpha}\left({\delta + t} \right)

Probability of Very Slow Diffusion Law

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: $P(k)=k−β$ P\left(k \right) = {k^{- \beta}}

Among them β > 1.

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 $E˙=M˙α(δ+t)$ \dot E = {\dot M_\alpha}\left({\delta + t} \right)

Among them, α is a parameter that characterizes the activity of diffused particles (to be fitted). δ is the parameter that controls the rate of change of the offset field at the initial moment.

If you limit β > 2, α = 1, δ = 0, then Ė =1 / t. Thus, t ~ eE is obtained. Sinai's law of diffusion can be derived. Integrate both sides of equation (27) at the same time to get $E~Mα(δ+t)$ E\sim {M_\alpha}\left({\delta + t} \right)

Both sides act on the M-L operator Ea at the same time, so we generalize the relationship between the escape time and the regional inward offset field into the following modified M-L functional relationship: $texit~Ea[E]−δ$ {t_{exit}}\sim {E_a}\left[E \right] - \delta

Where Ea is the M-L function. E is the total offset field in the area. The area is composed of S string displacement points. The offset field in each series of displacement points is the same. Using the properties of Lévyflight, we can get $E~{l(β−1)/β,1<β<2l1/2,β≥2$ E\sim \left\{{\matrix{{{l^{\left({\beta - 1} \right)/\beta}},} & {1 < \beta < 2} {{l^{1/2}},} & {\beta \ge 2} }} \right.

Substituting the above formula into formula (29) and finishing to get $〈x2(t)〉~[Mα(δ+t)]2y$ \left\langle {{x^2}\left(t \right)} \right\rangle \sim {\left[{{M_\alpha}\left({\delta + t} \right)} \right]^{2y}}

In $y={β/(β−1),1<β<22,β≥2$ y = \left\{{\matrix{{\beta /\left({\beta - 1} \right),} & {1 < \beta < 2} {2,} & {\beta \ge 2} }} \right.

Very slow diffusion behavior

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 . Firstly, when α = 0.5, 0.8, δ = 1 is taken in Eq. (31) and the offset field distribution factor is β = 2, the comparison of the diffusion process of ultra-slow diffusion and Sinai logarithmic diffusion in a short time is shown in Figure 2. The behavior of very slow diffusion and Sinai logarithmic diffusion in a short period

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 . 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.

Conclusion

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. Comparison of M-L function and its inverse function with exponential (log) function The behavior of very slow diffusion and Sinai logarithmic diffusion in a short period Long-term behavior of very slow diffusion and Sinai logarithmic diffusion

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