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Calculus Logic Function in Integrated Manufacturing Automation of Single Chip Microcomputer

Publié en ligne: 15 Jul 2022
Volume & Edition: AHEAD OF PRINT
Pages: -
Reçu: 13 Feb 2022
Accepté: 10 Apr 2022
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Magazine
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2444-8656
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01 Jan 2016
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Anglais
Introduction

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.

Steady-state power-law heat conduction implicit calculus equation model

The fractional Laplacian operator (−Δ)s/2 is a typical differential-integral operator. It can use a single parameter s to characterize the spatial non-locality of the physical, mechanical system [6]. As a general form of the classic integer-order Laplace operator, it can model physical and mechanical problems such as energy dissipation, turbulent diffusion, groundwater solute transport, the electromagnetic field in fractal space, and non-local heat conduction of sound wave propagation in soft matter. Operator (−Δ)s/2 satisfies the Fourier transform F{(Δ)s/2u()}=ksF{u()} F\left\{{{{\left({- \Delta} \right)}^{s/2}}\,u\left({} \right)} \right\} = {\left\| k \right\|^s}\,F\left\{{u\left({} \right)} \right\} k is the wave number in the frequency domain. It is very difficult to directly derive the explicit expression of the operator by using the inverse Fourier transform. At the same time, the explicit definitions of the existing two-dimensional and three-dimensional fractional Laplacian operators are not unified. We use the implicit calculus modeling method [7]. The author does not consider the specific expression form of the fractional Laplacian but directly constructs the basic solution of the fractional Laplacian from its inverse operator. The kernel function of the fractional Ries potential in three-dimensional space is defined as u*(x,ξ)=1xξ3s {u^*}\left({x,\xi} \right) = {1 \over {{{\left\| {x - \xi} \right\|}^{3 - s}}}}

xξ ‖ represents the Euclidean distance between points x and ξ. s is the order of the fractional potential. The basic solution of the classical Laplacian of integer order is a special case of the fractional order. u*(x,ξ)=1xξ {u^*}\left({x,\xi} \right) = {1 \over {\left\| {x - \xi} \right\|}}

We take formula (2) as the basic solution of the fractional Laplacian (−Δ)s/2. The fractional Laplace order s of general physics problems is a real number from 1 to 2. It can be proved that the fractional Laplacian defined in this way satisfies the definition of Fourier transform. Complex media often have discontinuities. It causes the partial derivative at the discontinuity point to lose its physical meaning [8]. The calculus equation model of the classical integer-order derivative is no longer suitable for describing the heat conduction in this kind of complex medium. The fractional Laplace equation can describe this kind of power-law (nonlocal) heat conduction behavior more accurately, and its steady-state equation is (Δ)s/2u(x)=0,s(1,2],xΩR3 - {\left({- \Delta} \right)^{s/2}}u\left(x \right) = 0,\,\,s \in \left({1,2} \right],\,x \in \Omega \subset {R^3} u is a dimensionless temperature function. s characterizes the non-locality of the material and characterizes the power law. Ω is the calculation area. As shown in Figure 1, the length of the cylinder is 6, and the radius of the bottom surface is 1. The center of the cylinder coincides with the origin of the coordinates [9]. Assume that the temperature distribution on the cylinder boundary satisfies u¯(x,y,z)=excos(22y)cos(22z)+1 \bar u\left({x,\,y,\,z} \right) = {e^x}\,\cos \left({{{\sqrt 2} \over 2}y} \right)\cos \left({{{\sqrt 2} \over 2}z} \right) + 1

Figure 1

Calculation area for steady-state non-local heat conduction example

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.

Figure 2

Numerical and exact solutions of integer order Laplace equation (s = 2)

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 {(x, y, z) | x = 0, y = 0, −3 ≤ z ≤ 3} of the cylinder with the order s of the fractional Laplacian in equation (4). The numerical results are shown in Figure 3. Under the same boundary conditions, when s tends to 2, the solution of the implicit fractional Laplace equation monotonously approaches the solution of the integer-order Laplace equation. In addition, the smaller the s, the stronger the non-locality of the material.

Figure 3

The temperature on the central axis of a cylinder varies with the fractional Laplace order

The implicit calculus equation model of the non-steady abnormal diffusion problem based on the Levi statistical distribution

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 u(x,t)t=D(Δ)s/2u(x,t),xRn {{\partial u\left({x,t} \right)} \over {\partial t}} = - D{\left({- \Delta} \right)^{s/2}}u\left({x,t} \right),\,\,x \in {R^n} u is the dimensionless concentration function of the diffuser. D is the diffusion coefficient. s is the order of the fractional derivative. (−Δ)s/2 is the fractional Laplacian. Rn stands for n dimensional space.

Case of s = 2

When s = 2, Eq. (6) degenerates to the classical integer-order Fick diffusion equation describing normal diffusion u(x,t)t=DΔu(x,t) {{\partial u\left({x,t} \right)} \over {\partial t}} = D\Delta u\left({x,t} \right)

The basic time-space solution of integer-order diffusion equation (7) is G(x,y,t)=H(t)(4πDt)n/2exy2/(4Dt) G\left({x,y,t} \right) = {{H\left(t \right)} \over {{{\left({4\pi Dt} \right)}^{n/2}}}}{e^{- {{\left\| {x - y} \right\|}^2}/\left({4\,Dt} \right)}} x is the coordinate of the source point. y is the field point coordinates. H(t) is the Heaviside function. n is the space dimension. ‖xy‖ is the Euclidean distance between x and y. From the basic solution (8), it can be seen that the basic solution of the integer-order Fick diffusion equation (7) has a corresponding relationship with the probability density function of the following n dimensional Gaussian distribution Φ(η)=1(2πσ2)n/2eημ2/(2σ2),η,μRn \Phi \left(\eta \right) = {1 \over {{{\left({2\pi {\sigma ^2}} \right)}^{n/2}}}}{e^{- {{\left\| {\eta - \mu} \right\|}^2}/\left({2{\sigma ^2}} \right)}},\eta ,\mu \in {R^n}

η is a random variable. σ is the scale parameter. μ is the positional parameter. If σ=t1/22D \sigma = {t^{1/2}}\sqrt {2D} , η=x/t \eta = x/\sqrt t , μ=y/t \mu = y/\sqrt t is G(x,y,t)=Φ(η)H(t) G\left({x,\,y,\,t} \right) = \Phi \left(\eta \right)H\left(t \right)

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.

Case of s ≠ 2

At this time, we fully consider the fractional order of spatial fractional diffusion equation (6) at s ≠ 2. In the one-dimensional case (n = 1), the following equation (6) is simplified to u(x,t)t=D(2x2)s/2u(x,t) {{\partial u\left({x,t} \right)} \over {\partial t}} = - D{\left({- {{{\partial ^2}} \over {\partial {x^2}}}} \right)^{s/2}}\,u\left({x,\,t} \right)

The basic solution of equation (11) can be expressed as G(x,y,t)=H(t)t1/sL(|xy|t1/s) G\left({x,\,y,\,t} \right) = {{H\left(t \right)} \over {{t^{1/s}}}}L\left({{{\left| {x - y} \right|} \over {{t^{1/s}}}}} \right)

Where: L is the probability density function of Levi's steady-state distribution. L(|xy|t1/s)=12π+eik|xy|t1/sφ(k)dk L\left({{{\left| {x - y} \right|} \over {{t^{1/s}}}}} \right) = {1 \over {2\pi}}\int_{- \infty}^{+ \infty} {{e^{{{ik\left| {x - y} \right|} \over {{t^{1/s}}}}}}} \varphi \left(k \right)dk s is the steady-state index of the Levi distribution. The Fourier transform of the probability density function of the one-dimensional s steady-state Levitic distribution is φ(k)=eDks \varphi \left(k \right) = {e^{- D{k^s}}} φ is the characteristic function of Levi's probability density function L(|xy| / t1/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 s− steady-state Levitic distribution. The probability density function of the one-dimensional Levitic steady-state statistical distribution is shown in Figure 4. When s = 2, the Levi distribution degenerates to a Gaussian distribution. When s ≠ 2, the Levi distribution has obvious tailing phenomenon. This is in line with the distribution law of abnormal diffusion and fast diffusion process.

Figure 4

Probability density function of one-dimensional Levitic steady-state distribution

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 n dimensional space fractional diffusion equation obtained from the n dimensional s − steady-state Levitic distribution probability density function is G(x,y,t)=H(t)tn/sL(xyt1/s) G\left({x,\,y,\,t} \right) = {{H\left(t \right)} \over {{t^{n/s}}}}L\left({{{\left\| {x - y} \right\|} \over {{t^{1/s}}}}} \right)

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.

Discussion

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.

Conclusion

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.

Figure 1

Calculation area for steady-state non-local heat conduction example
Calculation area for steady-state non-local heat conduction example

Figure 2

Numerical and exact solutions of integer order Laplace equation (s = 2)
Numerical and exact solutions of integer order Laplace equation (s = 2)

Figure 3

The temperature on the central axis of a cylinder varies with the fractional Laplace order
The temperature on the central axis of a cylinder varies with the fractional Laplace order

Figure 4

Probability density function of one-dimensional Levitic steady-state distribution
Probability density function of one-dimensional Levitic steady-state distribution

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