Calculus is the watershed between modern mathematics and classical mathematics. The development and application of mathematics have undergone fundamental changes since then. Classical calculus equation modeling methods have succeeded in mechanics, acoustics, electromagnetics, heat transfer and diffusion theories, and even in modern quantum mechanics and relativity [1]. However, sociologists, economists, physicists, and mechanics have also discovered that more and more so-called “abnormal” phenomena are difficult to model with classical calculus equations.

The nonlinear differential equation model is a common method to describe complex physical processes. The basic idea is to assume that the coefficients in the constitutive relations of linear mechanics or physical laws are dependent on strain variables [2]. At present, the nonlinear models of complex problems are becoming more and more complicated. For example, the thermo-electric-chemical-mechanical coupling model in rock and soil mechanics requires more than forty parameters. The physical meaning and determination of these parameters is a big problem in itself.

The fractional calculus method that has attracted widespread attention in recent years is another powerful mathematical tool for complex modeling phenomena, and it has achieved remarkable success in some fields. But this method also has its limitations. First, the definition of the very important spatial fractional Laplacian is not uniform, and the numerical calculations are also difficult. Secondly, the physical interpretation of the fractional derivative order is still immature. Most fractional derivative models are empirical models or phenomenological models.

This paper proposes an implicit calculus equation modeling method that aims to solve the problem of simulating these complex problems. In the numerical simulation, only the calculus control equations’ basic solutions and boundary conditions can be used for numerical simulation calculations [3]. In this way, the numerical solution of the model can be obtained, and there is no need to derive the governing equation from the basic solution. Here “implicit” means that the explicit expression of the governing equation may be unnecessary or difficult to derive. In specific implementations, generalized basic solutions or statistical distribution density functions that describe a class of physical problems can be used.

This paper's main numerical technique for solving implicit calculus equation models is the collocation method based on radial basis functions [4]. This type of method takes distance as the basic variable and does not depend on the problem's dimensionality.

This article examines two types of application examples. First, consider the power-law behavior of multiphase soft matter heat conduction. Many studies have shown that the fractional Laplacian equation can effectively describe the physical and mechanical problems of this type of power-law behavior. Still, the mathematical definition of the fractional Laplacian is not uniform [5]. The existing expressions are complex, and difficult to perform numerical calculations. This paper uses the kernel function of the fractional Rees potential as the basic solution to construct its steady-state problem's implicit calculus equation model. We use the singular boundary method based on the basic solution of the radial basis function for numerical verification. The second example is to use the known statistical density function to construct the basic solution of the implicit calculus equation. The Gaussian distribution is just a special case of Levi's steady-state distribution. Recent studies have found that the steady-state statistical distribution of Levi has a much wider application range than the Gaussian distribution. It has been successfully applied to many engineering problems, especially the statistical modeling of the fast diffusion process in abnormal diffusion behavior. This paper uses Levi's density function to construct the time-space implicit calculus equation model of anomalous diffusion. The model in this paper is simpler than existing models and has clear physical and statistical concepts.

The fractional Laplacian operator (−Δ)^{s/2} is a typical differential-integral operator. It can use a single parameter ^{s/2} satisfies the Fourier transform

‖

We take formula (2) as the basic solution of the fractional Laplacian (−Δ)^{s/2}. The fractional Laplace order

Let us first examine the accuracy of the numerical solution of the integer-order Laplace equation. Figure 2 shows the values of the exact solution and the numerical solution on the central axis of the cylinder. As the number of discrete points on the boundary increases, the numerical solution gradually approaches the exact solution [10]. It can be seen that the particular boundary method has good convergence.

Under normal circumstances, we do not know the exact solution of the fractional Laplace equation (4). Still, we can investigate whether the numerical solution of the fractional equation is close to the exact solution of the integer-order equation by specifying the same boundary conditions as the integer-order equation [11]. Let us first examine the variation of the temperature on the axis {(

The phenomenon of diffusion exists widely in nature and industry. It is an extremely important physical and mechanical process of material migration and transportation. More and more studies have found that the classic diffusion equation cannot describe turbulence well, such as plasma diffusion under high temperature and high pressure, financial market changes, polymer dynamics [12]. The so-called abnormal diffusion refers to the diffusion behavior that does not comply with Fick's law of diffusion. It includes two forms of slow diffusion and fast diffusion. Usually, show a long-range time-space correlation. Recent studies have found that the spatial fractional diffusion equation can better describe the fast diffusion phenomenon in anomalous diffusion. However, the explicit expressions of the time-space unstable fractional equations are difficult to obtain or inaccurate, and difficult to calculate numerically.

This section considers using the density function of the Levi statistical distribution to construct the basic solution of the fractional anomalous diffusion equation in the unsteady space. At the same time, we carry out implicit calculus equation modeling [13]. This is different from the steady-state problem covered in Section 2. We establish the following anomalous diffusion equation of fractional Laplace operator in multi-dimensional space
^{s/2} is the fractional Laplacian. ^{n}

When

The basic time-space solution of integer-order diffusion equation (7) is

From equation (10), it can be seen that the basic solution of integer-order diffusion equation (7), equation (8), describes the Gaussian distribution characteristics of particle motion in the classical Fick diffusion (normal diffusion) process. The probability density function of Gaussian distribution is the kernel function of the basic solution of normal diffusion.

At this time, we fully consider the fractional order of spatial fractional diffusion equation (6) at

The basic solution of equation (11) can be expressed as

Where: ^{1/s}) in the Fourier transform domain. Therefore, the statistical mechanical explanation of the fractional diffusion equation (11) in one-dimensional space is the fast diffusion motion of particles under the one-dimensional

The Gaussian distribution is the basic solution kernel function of the integer-order Fick diffusion model. The one-dimensional Levitic distribution is the kernel function of the basic solution of the fractional fast diffusion model of the one-dimensional problem. The steady-state statistical distribution of Levi is two special cases of the basic solution kernel function of the classical diffusion equation and the spatial fractional diffusion equation [14]. Therefore, the basic solution of the multi-dimensional fractional time-space diffusion equation can be constructed using the probability density function of Levi's steady-state statistical distribution. We use it to build the implicit calculus modeling of the fast diffusion process. The basic solution of the

Here, the Levi distribution is the kernel function of the basic solution of the spatial fractional diffusion equation. It profoundly reveals the statistical nature and spatial correlation of the multi-dimensional fast diffusion process. We use the basic solution (15) of the implicit calculus equation model for statistical analysis. Then the numerical simulation calculation is carried out according to the boundary condition values obtained on the measurable boundary. This avoids many difficulties in explicitly expressing the calculus equation model.

The basic solution or statistical distribution of implicit calculus modeling can be quite extensive. It can greatly promote the scope of application of calculus modeling. For example, this method is different from the traditional boundary element method in which the differential equation model is first used to find the basic solution [15]. It can directly construct the general solution of the inhomogeneous medium according to the physical characteristics of the problem. It can even directly construct the basic solution of nonlinear problems without considering the expression form of the calculus equation. We can combine mathematical mechanics modeling and numerical modeling more closely.

In addition, the implicit calculus modeling method also deeply and closely integrates calculus modeling with the statistical model. It can construct the basic solution of the deterministic differential equation model from the statistical distribution of the complex problem. And establish a bridge between the deterministic model and the stochastic model. The basic solution can be understood as the influence function or potential function in the physical field.

To construct basic solutions or general solutions and other influence functions according to the physical properties or statistical distributions of complex problems is still a subject to be studied in depth.

Traditional mathematical and physical equations and numerical calculation schemes generally use mathematical calculus methods to establish control equations and boundary conditions based on the physical characteristics and theory of the problem. They then use numerical methods to solve these partial differentials or differential-integral equation problems. The model in this paper is different from the standard theoretical modeling and numerical simulation schemes. The implicit calculus modeling idea proposed in this paper is first to have the basic solution of the problem and then solve the problem directly. The expression of the differential control equation itself is no longer a vital link and object.

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