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Information Teaching Model of Preschool Art Education in Colleges and Universities Based on Finite Element Higher-Order Fractional Differential Equation

Online veröffentlicht: 22 Nov 2021
Volumen & Heft: AHEAD OF PRINT
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Eingereicht: 17 Jun 2021
Akzeptiert: 24 Sep 2021
Zeitschriftendaten
License
Format
Zeitschrift
eISSN
2444-8656
Erstveröffentlichung
01 Jan 2016
Erscheinungsweise
2 Hefte pro Jahr
Sprachen
Englisch
Abstract

In order to meet the social demand for preschool education professionals, based on the finite element higher-order fractional differential equation, this paper studies the information teaching model of preschool education of fine arts in colleges and universities. Through the scientific and effective fine arts education in the preschool education major in colleges and universities, the students’ fine arts ability can be effectively improved, and the application of information means can make this goal can be better realised so that the teaching needs can be better satisfied and they promote the better development of fine arts education in the preschool education major.

MSC 2010

Introduction

The finite element method is one of the main tools for numerical solutions of differential equations. It was developed in the middle of the past century and grew up gradually. The finite element method has been widely used in the engineering field, especially in solving elliptic equations. The basic idea is to transform the differential equations into equivalent variational problems or functional extremum problems to approximate them by piecewise polynomials and to determine these equations by orthogonal projection. Its advantages are not high demand on the region, flexible mesh division and strong adaptability. Moreover, using functional analysis as a tool, the requirements for the smoothness of solutions are relaxed [1].

Finite element higher-order fractional differential equation and teaching model are the core courses of most colleges and universities for the preschool education of fine arts, and applied mathematics and information and computing science. Therefore, it is of great value and significance to integrate the teaching model into the teaching or research of the higher-order fractional differential equations of finite elements. In recent years, in the process of promoting the curriculum reform of basic education in China, countries around the world have begun to pay attention to art education and have gradually realised that the unique role and advantages of art education in the whole education are irreplaceable and have formed a general consensus that “education without art education is incomplete education”. In the process of actively promoting art education, more and more people realise that art plays an important role in the process of education and teaching [2].

Many students do not know the use of differential equations, they lack motivation and interest in learning them and, finally, they gradually think that mathematics is a very boring subject. This inspires us to combine the teaching of finite element high-order fractional differential equations with the teaching model and infiltrate the modelling idea into the teaching of finite element high-order fractional differential equations. In this way, students can understand the background, method and practical significance of the establishment of finite element high-order fractional differential equation. Moreover, it can also improve students’ ability to apply the knowledge of finite element high-order fractional differential equations and computers to practice and improve students’ ability to teach models [3].

Fractional order partial differential equation is relative to the integer order differential equation and has been widely used in many engineering fields, such as anomalous diffusion of the solute transport process, signal and image processing, the viscoelastic material and non-Newtonian fluid, gas transport process in an inhomogeneous medium, hydrology and water conservancy, bone remodelling and bone tumours simulation, fine arts preschool education in colleges and universities and other fields. In summary, the literature published by scholars mainly focuses on the following aspects.

Huo et al. by analysing the current situation of quality education of fine arts students in colleges and universities found the gap between quality education of fine arts college students and the talent-training goal of social demand. This paper discusses the scientific nature and connotation of quality-oriented education in art colleges and universities from the angle of educational mode and improves the innovative curriculum system of art teaching in art colleges. In addition, it also explores the cultivation and life of creative ability and the concept of life-long learning and the education of the university student’s world view [4].

It has played an important role in the reform and development of higher education since it was put forward. At present, as private universities grow in number and size, so does the number of faculty members. In order to improve the teaching and research level of teachers in private colleges and improve the professional progress of teachers of different subjects, these universities have established a variety of school teaching and research forms and matching systems. However, the implementation process is characterised by low participation and poor management, which seriously deviates from the original intent and affects quality. The reason lies in the lack of effective and scientific management. Therefore, this paper shows that private universities should, according to their specific conditions, improve scientific management norms and clear management objectives of training modern faculty in order to realise the original initiative and improve the quality of teaching and research in schools [5].

Research methods
Efficient solution algorithm and numerical study of the time-fractional phase-field model

Two-phase flow problems for immiscible and incompressible fluids. Let’s call it a phase function, which is a function of space and time. Let us usually take two kinds of constants, such as 1 and −1 to represent two different kinds of fluid, and the part between 1 and −1 represents the interface of two kinds of immiscible fluid. By observing the change of phase function, the phase-field model can indirectly reflect the change of the interface of two kinds of fluids so as to avoid directly tracking the location of the interface in the process of microstructure evolution [6]. Therefore, the phase-field model has a natural advantage in simulating the situation where the width of the interface is equivalent to the characteristic length of the system. Make F(Φ)=14η2(φ2φ)2F(\Phi)=\frac{1}{4 \eta^{2}}\left(\varphi^{2}-\varphi\right)^{2} to Ginzburg-Landau double potential well and define the following mixing energy: E(φ)=Ω(12|φ|2+F(Φ))dx E(\varphi)=\int_{\Omega}\left(\frac{1}{2}|\nabla \varphi|^{2}+F(\Phi)\right) d x

The potential function F (ϕ) has two minima, and they represent two pure phases. Let V represent the Hilbert space, whose inner product is defined as (·, ·)v. Then the phase function ϕ satisfies the following gradient flow: φt=γvE(φ) \varphi_{t}=\gamma \nabla v \mathrm{E}(\varphi)

where γ represents permeability parameter and ∇vE(ϕ) represents the gradient of energy functional E(ϕ). The gradient of E(ϕ) at point ϕV is: (vE(φ),ψ)v=ddsE(φ+sψ)|s=0 (\nabla v \mathrm{E}(\varphi), \psi) v=\left.\frac{d}{d s} \mathrm{E}(\varphi+s \psi)\right|_{s}=0

According to Formula (1), if V is selected as L2 (Ω), then we can get: ddsE(φ+sψ)|s=0=[(φ,ψ)L2+(f(φ),ψ)L2]=(φ+f(φ),ψ)L2 \left.\frac{d}{d s} \mathrm{E}(\varphi+s \psi)\right|_{s=0}=\left[(\nabla \varphi, \nabla \psi)_{L^{2}}+(f(\varphi), \psi)_{L^{2}}\right]=(-\nabla \varphi+f(\varphi), \psi)_{L^{2}}

where f (ϕ) = F (ϕ). Therefore, we can obtain the following classical Allen-Cahn equation of integer order (ignoring the boundary conditions): φt+γ(φ+f(φ))=0,(x,t)Ω×(0,T) \varphi_{t}+\gamma(-\nabla \varphi+f(\varphi))=0,(x, t) \in \Omega \times(0, T)

For the Allen-Cahn equation, we have: ddtΩφdx=Ωφtdx=γΩ(φ+f(φ))dx \frac{d}{d t} \int_{\Omega} \varphi d x=\int_{\Omega} \varphi_{t} d x=-\gamma \int_{\Omega}(-\nabla \varphi+f(\varphi)) d x

In general, we cannot guarantee that the right end of Equation (6) is always 0, so the Allen-Cahn equation generally does not satisfy the conservation of mass.

The Cahn-Hilliard equation was originally used to describe the time evolution of a conserved field that is a continuous, spatially sufficiently differentiable function. If we pick V as H−1, then the classical Cahn-Hilliard equation can be obtained as follows: φt=γΔ(f(φ)Δφ) \varphi_{t}=\gamma \Delta(f(\varphi)-\Delta \varphi)

In general, periodic boundary conditions or the following non-flux boundary conditions are used for closure (7): nφ=0.n(f(φ)Δφ)=0 n \cdot \nabla \varphi=0 . n \cdot \nabla(f(\varphi)-\Delta \varphi)=0

where n is the unit normal vector. For the Cahn-Hilliard equation, we have: ddtΩφdx=Ωφtdx=γΩΔ(f(φ)Δφ)dx=0 \frac{d}{d t} \int_{\Omega} \varphi d x=\int_{\Omega} \varphi_{t} d x=\gamma \int_{\Omega} \Delta(f(\varphi)-\Delta \varphi) d x=0

This means that the Cahn-Hilliard equation satisfies the conservation of mass. For both Allen-Cahn and Cahn-Hilliard equations, we have: dE(φ)dt=δE(φ)δφφt=λ|vE(φ)|20 \frac{d \mathrm{E}(\varphi)}{d t}=\frac{\delta \mathrm{E}(\varphi)}{\delta \varphi} \frac{\partial \varphi}{\partial t}=-\lambda|\nabla v \mathrm{E}(\varphi)|^{2} \leq 0

Therefore, both Allen-Cahn and Cahn-Hilliard equations satisfy the law of energy dissipation. However, if we consider the transport process with a large number of pores in the porous media, then the molecular mean free path is equivalent to the size of the characteristic pores, thus leading to a large Knudsen number. The resulting flow behaviour deviates from the traditional macroscopic flow behaviour and makes molecular diffusion play a dominant role in the transport process. The analysis of this phenomenon can be started from the assumption that Einstein and Pearson shared in the original derivation of Fick’s diffusion equation: (i) there is a mean free path and (ii) there is an average waiting time for particle motion. Based on these assumptions, it is rare for a particle to travel long distances in the same direction, and the motion of the particle can be represented by an independent jump that has no spatial correlation. In highly heterogeneous and space-limited porous media, particles may be absorbed by the porous media due to the heterogeneity of the porous media or the dominant interaction between fluid and rock, resulting in a sub-diffusion process which can be described by a fractional partial differential equation of time [7].

Finite element high-order fractional Allen-Cahn equation and its numerical study

In this section, we study, for example, the finite element high-order fractional Allen-Cahn equation: {0CDtαφ=γ(Δφf(φ)),(x,t)Ω×(0,T),φ(x,0)=φ0(x),xΩ,φ(x,t)=0,(x,t)Ω×[0,T] \left\{\begin{array}{l} { }_{0}^{C} D_{t}^{\alpha} \varphi=\gamma(\Delta \varphi-f(\varphi)),(x, t) \in \Omega \times(0, T), \\ \varphi(x, 0)=\varphi_{0}(x), x \in \Omega, \\ \varphi(x, t)=0,(x, t) \in \partial \Omega \times[0, T] \end{array}\right.

where Ω is a two-dimensional square region, ϕ0 (x) is the initial condition. 0CDtα{ }_{0}^{C} D_{t}^{\alpha} represents the Caputo fractional derivative of order 0〈α ≤ 1. The Riemann-Liouville fractional derivative and Caputo fractional derivative are related as follows: 0RDt1α0CDtαφ(x,t)=0RDt1α0It1βφt(x,t)=φt(x,t) { }_{0}^{R} D_{t}^{1-\alpha}{ }_{0}^{C} D_{t}^{\alpha} \varphi(x, t)={ }_{0}^{R} D_{t}^{1-\alpha}{ }_{0} I_{t}^{1-\beta} \varphi_{t}(x, t)=\varphi_{t}(x, t)

Applying the fractional Riemann-Liouville derivative of order 1 − α to both sides of equation (11), we can get: ddtΩφdx=dφdtdx=γ0RLDt1α(Δφf(φ))dx \frac{d}{d t} \int_{\Omega} \varphi d x=\int \frac{d \varphi}{d t} d x=\gamma_{0}^{R L} D_{t}^{1-\alpha} \int(\Delta \varphi-f(\varphi)) d x

It can be seen that, consistent with the Allen-Cahn equation of integer order, the above equation is not always 0 in general, which indicates that the time-fractional Allen-Cahn equation cannot maintain the conservation of mass [8].

Numerical experiments

Several numerical experiments are conducted to verify the accuracy of the fast finite difference (FFD) scheme for solving the time-fractional Allen-Cahn equation. Meanwhile, we will also test the energy dissipation problem of the time-fractional Allen-Cahn equation numerically.

Example 1. Let the reference solution of Equation (11) be ϕ (x,t) = t2 sin (x), other parameters are Ω = [0π], T = 1, γ = 1.0, η = 1.0. In this case, the right-hand side of the equation can be given accordingly f(x,t)=2sin(x)t2αΓ(3α)+γsin(x)t2+γη2φ(φ1)(φ12) f(x, t)=\frac{2 \sin (x)^{t^{2-\alpha}}}{\Gamma(3-\alpha)}+\gamma \sin (x) t^{2}+\frac{\gamma}{\eta^{2}} \varphi(\varphi-1)\left(\varphi-\frac{1}{2}\right)

Let’s say Δt = 2−16, h = 2−12, to test the spatial and temporal convergence order of the numerical scheme.

In Tables 1 and 2, we give the cortical error of the numerical solution and the corresponding order of convergence (CoV). It can be seen that the finite-difference scheme has the first order precision in time and the second-order precision in space. We compare the performance of the FFD scheme and the time-catch-up method in solving the equation in Table 3.3, wherein FFD we use the fast method of finding the Caputo fractional derivative in Section 3.3.2. In Figure 3.1, fix M = 2" as well. = 0.7, we show the relationship between CPU time and time step for the FFD algorithm and direct finite difference (DFD) scheme to solve the equation. We can clearly see that the FFD algorithm consumes almost linear CPU time as time steps decrease, so it is much more efficient than the DFD method [9].

L2 error and corresponding spatial convergence order of finite difference decomposition at Δt = 2−16

N α = 0.2E(ht) Cov α = 0.5E(ht) Cov α = 0.8E(ht) Cov
25 4.3385e−04 1.98 3.8225e−04 2.00 3.1800e−04 1.99
26 1.1000e−04 1.92 9.5800e−05 1.99 7.9804e−05 1.98
27 2.9043e−05 1.72 2.4189e−05 1.94 2.0254e−05 1.91
28 8.8305e−06 6.3168e−06 5.3865e−06

L2 error and corresponding time convergence order of finite difference decomposition at h = 2−12

M α = 0.2E(ht) Cov α = 0.5E(ht) Cov α = 0.8E(ht) Cov
25 3.4200e−03 0.77 2.0717e−03 1.03 7.8969e−03 1.21
26 2.0022e−03 0.90 1.0116e−03 1.06 3.4059e−03 1.22
27 1.0715e−03 0.90 4.8658e−04 1.06 1.4612e−03 1.23
28 5.5260e−04 2.3374e−04 6.2398e−04

L2 error of numerical solutions obtained by DFD and FFD when Δt = 2−16 and α = 0.7 are fixed

FFD DFD
N E(ht) Cov CPU (s) E(ht) Cov CPU (s)
25 3.4024e−04 2.00 6.69 3.4024e−04 2.00 2652.36
26 8.4964e−05 2.01 10.16 8.4964e−05 2.01 2822.55
27 2.1146e−05 2.02 36.31 2.1146e−05 2.02 3140.38
28 5.2182e−06 - 103.89 5.2182e−06 - 3651.16

FFD, fast finite difference; DFD, direct finite difference.

When M = 26, α = 0.7 and the time and space step size is b, and the initial value of h = 2−9 is: φ0(x)=12(1+tanh(0.5|x|2η)) \varphi_{0}(x)=\frac{1}{2}\left(1+\tanh \left(\frac{0.5-|x|}{\sqrt{2 \eta}}\right)\right)

We found that when the permeability parameters γ were 1.0, 1.2, 1.5 and 1.8, the greater the γ, the decay behaviour of the time-graded Allen-Calm equation appears earlier, which is consistent with the case of its integer order.

In addition, we also investigate the effects of interface parameter cutters and volume ratio of two different phases (VOF) on the attenuation behaviour of the time-fractional Allen-Calm equation. In Figure 2, fix γ = 5.0, we show how the energy changes over time for different VOFs (20%, 50% and 70%) and different knives, where. Take 0.5, 0.7, 0.9 and 1.0. In Figure 3.3(d), the Fourier spectrum method is used to verify the other three results obtained by the finite difference method. For different VOFs, it is observed that the time-fractional Allen-Calm equation still exhibits variable attenuation behaviour for the small ding, while for slightly larger η, this behaviour disappears. We have observed an interesting phenomenon that when the fractional order α becomes smaller, the energy dissipation of the time-fractional order Allen-Calm equation becomes faster. In Figure 2, for random initial values and fixed γ = 1.0, when η is 0.09, 0.08, 0.05 and 0.02, we give the time evolution process of energy and mass of one-dimensional time-fractional Allen-Cahn equation. From Figure 2, we find that the smaller the fractional order of α is the energy dissipation process corresponding to the time-fractional Allen-Cahn equation is accelerated, which seems physically unreasonable since the time-fractional derivative describes a sub-diffusion process. The specific reasons need further research [10].

Fig. 1

Relationship between CPU time consumed by DFD and FFD and time step in log-log coordinates

Fig. 2

Mass and time evolution diagram of one-dimensional time-fractional Allen-Calm equation

To verify the conclusion in Example 1, we next consider the two-dimensional time-fractional Allen-Cahn equation. The data corresponding to the results in Figure 2 are as follows: η=0.05γ=0.1Δt=1×104,Nx=Ny=256,Ω=[0,π]2 \eta=0.05 \gamma=0.1 \Delta t=1 \times 10^{-4}, N_{x}=N_{y}=256, \Omega=[0, \pi]^{2}

We take the initial value as an ellipse centred at (π2π2)\left(\frac{\pi}{2} \frac{\pi}{2}\right), and its half-length and half-short axes are α = 0.2π and b = 0.1π. At the same time, we rotate the ellipse so that its principal axis lies on the diagonal y = x. Therefore, this initial value can be expressed as: φ0(x,y)=0.9tanh(1R(x,y)2η) \varphi_{0}(x, y)=0.9 \tanh \left(\frac{1-R(x, y)}{\sqrt{2 \eta}}\right)

In which R(x,y)=((xx0)cosθ+(yy0)sinθ)2a2 R(x, y)=\frac{\left(\left(x-x_{0}\right) \cos \theta+\left(y-y_{0}\right) \sin \theta\right)^{2}}{a^{2}}

In Figure 3, with fixed γ = 0.1 and η = 0.05, we have drawn the outline of the phase function and the energy changes when α is 0.5, 0.7, 0.9 and 1.0. Meanwhile, let γ = 0.1 and η = 0.5. We can clearly see that the fractional order can affect the interface width of the equation and the speed of energy dissipation, and the smaller the order a is, the faster the energy decay is, which is consistent with the one-dimensional result.

Fig. 3

Phase function of two-dimensional time-fractional Allen-Calm equation and change of energy with time

Here, 0CDtα{ }_{0}^{C} D_{t}^{\alpha} is the fractional derivative of Caputo of order 0〈α ≤ 1 defined in the equation. Unlike the Allen-Calm equation, the Cahn-Hilliard equation contains the fourth derivative term of space, and so it is relatively tough to solve. This indicates that the time-fractional Cahn-Hilliard equation also satisfies the conservation of mass, which is consistent with the result of its integer order form.

Cranknicolson-Galerkin finite element algorithm for higher-order fractional parabolic equations

The traditional particle diffusion motion is generally simulated by integer-order Brown motion, that is, the particle diffusion flow and particle displacement satisfy the following relation: J(x,t)=αcx+bc J(x, t)=-\alpha \frac{\partial c}{\partial x}+b c

where J is the diffusion flow of the particle, c is the displacement of the particle, a is the diffusion coefficient and b is the convection coefficient. Substitute the above equation into the mass conservation relation of particles: JX=ct \frac{\partial J}{\partial X}=\frac{\partial c}{\partial t}

So we get the classical convection-diffusion equation; and from a physical point of view, this model is derived using the random walk method. Will start to time t asked when the entire section is decomposed into a series of independent random step length, the displacement of the particles (whether every step is the average square displacement or the average displacement within the whole time) can be decomposed into corresponding to its a series of step length asked displacement and, in every step and the average square displacement and the average time is limited. The mean square growth of its displacement is proportional to time (cc¯)2t\left\langle(c-\bar{c})^{2}\right\rangle-t, that is, the relationship between displacement and time satisfies Fick’s law.

However, the diffusion process in the fractal medium is related to the whole time and space, which indicates that this phenomenon is non-local activity, and this diffusion movement is called anomalous diffusion. Systems in turbulent environments, plasma environments, growing surfaces and cells also show deviations from normal Brown motion. In these phenomena, the motion of a free particle no longer satisfies Fick’s law, and its mean square displacement is proportional to the fractional power of time after a long time and is a nonlinear function of time; that is, if 0 〈α 〈1 has (cc¯)2t2α\left\langle(c-\bar{c})^{2}\right\rangle-t^{2 \alpha} but not Fick’s Law, so that is the case with α=12\alpha=\frac{1}{2}. Depending on the diffusion, abnormal diffusion is further divided into overdiffusion (cc¯)2t2α\left\langle(c-\bar{c})^{2}\right\rangle-t^{2 \alpha}, among them α12\alpha\rangle \frac{1}{2} and subdiffusion (cc¯)2t2α\left\langle(c-\bar{c})^{2}\right\rangle-t^{2 \alpha} and 0<α<120<\alpha<\frac{1}{2}. The fractional convection-diffusion equation of space in the form of the following can be obtained using Fourier transform: ut=D(a(u)cDα,xβu)+b(u)Du+(x,t,u)0β1 \frac{\partial u}{\partial t}=D\left(a(u){ }_{c} D_{\alpha, x}^{\beta} u\right)+b(u) D u+\int(x, t, u) \cdot 0\langle\beta\langle 1

Fractional anomalous diffusion has become a hot topic because it is widely used in the development of models of dynamic systems under random external forces. For example, with or without the effect of external force and external velocity field, fractional diffusion is a good tool to describe anomalous transmission behaviour. For example, surface water propagates along different amplitudes and is accompanied by diffusion-type dissipation. Fraction anomalous diffusion is an important tool to characterise hydrology. Fractional order can also be used to describe transport processes with chemical reactions and diffusion, including neural cell signalling, crowd evacuation and chemical wave propagation.

Conclusions

Preschool education is the foundation of the information construction of the preschool education of fine arts in colleges and universities. After its completion, it can provide educational institutions, educators and community residents with sharing network information services needed for various educational activities. The key to the modernisation of preschool education of fine arts lies in the improvement of the information level of preschool education of fine arts. The informationisation degree of art preschool education determines the quality of the construction of the life-long education system of learning society in China. Based on finite element method of higher-order fractional differential equation of the institutions of higher learning, the art of architecture design, preschool education informatisation will help to break through the bottleneck of fine arts preschool education informatisation development, solve the education information resources distribution, update the slow speed and low sharing degree, thus improving the overall level of fine arts preschool education informatisation of our country.

Fig. 1

Relationship between CPU time consumed by DFD and FFD and time step in log-log coordinates
Relationship between CPU time consumed by DFD and FFD and time step in log-log coordinates

Fig. 2

Mass and time evolution diagram of one-dimensional time-fractional Allen-Calm equation
Mass and time evolution diagram of one-dimensional time-fractional Allen-Calm equation

Fig. 3

Phase function of two-dimensional time-fractional Allen-Calm equation and change of energy with time
Phase function of two-dimensional time-fractional Allen-Calm equation and change of energy with time

L2 error of numerical solutions obtained by DFD and FFD when Δt = 2−16 and α = 0.7 are fixed

FFD DFD
N E(ht) Cov CPU (s) E(ht) Cov CPU (s)
25 3.4024e−04 2.00 6.69 3.4024e−04 2.00 2652.36
26 8.4964e−05 2.01 10.16 8.4964e−05 2.01 2822.55
27 2.1146e−05 2.02 36.31 2.1146e−05 2.02 3140.38
28 5.2182e−06 - 103.89 5.2182e−06 - 3651.16

L2 error and corresponding spatial convergence order of finite difference decomposition at Δt = 2−16

N α = 0.2E(ht) Cov α = 0.5E(ht) Cov α = 0.8E(ht) Cov
25 4.3385e−04 1.98 3.8225e−04 2.00 3.1800e−04 1.99
26 1.1000e−04 1.92 9.5800e−05 1.99 7.9804e−05 1.98
27 2.9043e−05 1.72 2.4189e−05 1.94 2.0254e−05 1.91
28 8.8305e−06 6.3168e−06 5.3865e−06

L2 error and corresponding time convergence order of finite difference decomposition at h = 2−12

M α = 0.2E(ht) Cov α = 0.5E(ht) Cov α = 0.8E(ht) Cov
25 3.4200e−03 0.77 2.0717e−03 1.03 7.8969e−03 1.21
26 2.0022e−03 0.90 1.0116e−03 1.06 3.4059e−03 1.22
27 1.0715e−03 0.90 4.8658e−04 1.06 1.4612e−03 1.23
28 5.5260e−04 2.3374e−04 6.2398e−04

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