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.

#### Keywords

- Preschool education
- Finite element higher-order fractional derivative
- Informatisation
- The teaching model

#### MSC 2010

- 97B10

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].

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

The potential function

where

According to Formula (1), if V is selected as ^{2} (Ω), then we can get:

where ^{′} (

For the Allen-Cahn equation, we have:

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 ^{−1}, then the classical Cahn-Hilliard equation can be obtained as follows:

In general, periodic boundary conditions or the following non-flux boundary conditions are used for closure (7):

where

This means that the Cahn-Hilliard equation satisfies the conservation of mass. For both Allen-Cahn and Cahn-Hilliard equations, we have:

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].

In this section, we study, for example, the finite element high-order fractional Allen-Cahn equation:

where Ω is a two-dimensional square region, _{0} (

Applying the fractional Riemann-Liouville derivative of order 1 −

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].

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.

^{2} sin (

Let’s say Δ^{−16}, ^{−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].

^{2} error and corresponding spatial convergence order of finite difference decomposition at Δ^{−16}

N | |||
---|---|---|---|

2^{5} |
4.3385e−04 1.98 | 3.8225e−04 2.00 | 3.1800e−04 1.99 |

2^{6} |
1.1000e−04 1.92 | 9.5800e−05 1.99 | 7.9804e−05 1.98 |

2^{7} |
2.9043e−05 1.72 | 2.4189e−05 1.94 | 2.0254e−05 1.91 |

2^{8} |
8.8305e−06 | 6.3168e−06 | 5.3865e−06 |

^{2} error and corresponding time convergence order of finite difference decomposition at ^{−12}

M | |||
---|---|---|---|

2^{5} |
3.4200e−03 0.77 | 2.0717e−03 1.03 | 7.8969e−03 1.21 |

2^{6} |
2.0022e−03 0.90 | 1.0116e−03 1.06 | 3.4059e−03 1.22 |

2^{7} |
1.0715e−03 0.90 | 4.8658e−04 1.06 | 1.4612e−03 1.23 |

2^{8} |
5.5260e−04 | 2.3374e−04 | 6.2398e−04 |

^{2} error of numerical solutions obtained by DFD and FFD when Δ^{−16} and

N | E( |
Cov | CPU (s) | E( |
Cov | CPU (s) |
---|---|---|---|---|---|---|

2^{5} |
3.4024e−04 | 2.00 | 6.69 | 3.4024e−04 | 2.00 | 2652.36 |

2^{6} |
8.4964e−05 | 2.01 | 10.16 | 8.4964e−05 | 2.01 | 2822.55 |

2^{7} |
2.1146e−05 | 2.02 | 36.31 | 2.1146e−05 | 2.02 | 3140.38 |

2^{8} |
5.2182e−06 | - | 103.89 | 5.2182e−06 | - | 3651.16 |

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

When ^{6}, ^{−9} is:

We found that when the permeability parameters

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

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:

We take the initial value as an ellipse centred at

In which

In Figure 3, with fixed

Here,

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:

where

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

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 〈

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.

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.

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

N | E( |
Cov | CPU (s) | E( |
Cov | CPU (s) |
---|---|---|---|---|---|---|

2^{5} |
3.4024e−04 | 2.00 | 6.69 | 3.4024e−04 | 2.00 | 2652.36 |

2^{6} |
8.4964e−05 | 2.01 | 10.16 | 8.4964e−05 | 2.01 | 2822.55 |

2^{7} |
2.1146e−05 | 2.02 | 36.31 | 2.1146e−05 | 2.02 | 3140.38 |

2^{8} |
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 | |||
---|---|---|---|

2^{5} |
4.3385e−04 1.98 | 3.8225e−04 2.00 | 3.1800e−04 1.99 |

2^{6} |
1.1000e−04 1.92 | 9.5800e−05 1.99 | 7.9804e−05 1.98 |

2^{7} |
2.9043e−05 1.72 | 2.4189e−05 1.94 | 2.0254e−05 1.91 |

2^{8} |
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 | |||
---|---|---|---|

2^{5} |
3.4200e−03 0.77 | 2.0717e−03 1.03 | 7.8969e−03 1.21 |

2^{6} |
2.0022e−03 0.90 | 1.0116e−03 1.06 | 3.4059e−03 1.22 |

2^{7} |
1.0715e−03 0.90 | 4.8658e−04 1.06 | 1.4612e−03 1.23 |

2^{8} |
5.5260e−04 | 2.3374e−04 | 6.2398e−04 |

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