As the geological exploration data is relatively sparse, unevenly distributed, and contains many geological faults, simple geological surface reconstruction has certain limitations. Based on the fractional differential equations, the paper establishes a subsidence prediction model in exploring geological and mineral resources. The dynamic system described by the reaction-diffusion equation can be mapped to a nonlinear cellular network through space and time discretization. At the same time, the original partial differential equations can be transformed into ordinary differential equations. Furthermore, we can use the difference method to simulate its evolutionary behavior quantitatively. The research results show that the error accuracy between the prediction results of the fractional gray theory established in this paper and the actual engineering results is higher.

#### Keywords

- Fractional differential equation
- geological exploration
- settlement prediction
- mineral construction

#### MSC 2010

- 34A08

Subgrade settlement prediction method based on measured data is a hot research topic in traffic and civil engineering. It is also a major problem in theoretical research on settlement control calculation methods for soft foundations. This is because it is difficult to fit the complex form of the actual consolidation law of weak foundations with general mathematical curves [1]. The engineering community has conducted extensive research on settlement prediction methods based on measured data. The representative ones are a hyperbolic method, three-point method, exponential function method, Asaoka method, grey model prediction method, etc. The purpose is to better predict the development law of roadbed settlement through actual measured data. By comparing the prediction results of the above method with the actual settlement data, it is found that the calculation results of the prediction method still have a large error with the actual value. Therefore, the traditional forecasting model and calculation method need to be further improved.

Some scholars have improved the traditional hyperbolic prediction model and proposed a modified hyperbolic model with higher accuracy. Some scholars have studied the time interval value in the Asaoka method based on the illustrative interpolation method and the linear least square method [2]. This method has been successfully applied to the settlement prediction of multi-level load embankment engineering. Some scholars apply the adaptive neuro-fuzzy inference system (ANFIS) to the settlement prediction of weak foundations and propose the ANFIS weak foundation settlement prediction model. This model overcomes the shortcomings of local minima, and its prediction accuracy is higher than that of the traditional growth curve prediction model. Some scholars pointed out that the morbidity of the least-squares method will lead to failure in solving the settlement prediction model parameters. We can improve the prediction accuracy by using regularized unbiased estimation of the processing parameters. Some scholars established the Logistic model of the whole soft foundation settlement prediction process based on the growth curve and determined the model parameters and their influence on the foundation settlement in different stages [3]. Some scholars have improved the calculation of the single prediction model settlement and the error description method of the measured value and fitted a new target error function. In this way, a simple and practical method for predicting settlement after construction is obtained. The settlement law under complicated working conditions is not obvious, so to predict settlement more accurately, many scholars use gray theory to predict foundation settlement. Some scholars have extended the grey forecasting model based on univariate to the grey forecasting model of multivariate. Some scholars used the parameter accumulation method instead of the least square method to modify the traditional

This paper takes the gray prediction model as the research object, replaces the first-order differential with the fractional differential, and proposes a gray prediction model based on the fractional order. Compared with the traditional grey prediction model, we increase the identification of fractional-order to increase the difficulty of parameter identification. For this reason, the author established a gray model with fractional order based on the fractional-order system model [7]. Engineering examples verify the rationality of the model, and the error of the fractional gray prediction model is calculated.

There are currently three definitions of fractional order. This article uses the Caputo calculus definition. The

In the formula:

The Laplace transform can be expressed as:

The zero initial condition considered in this paper, the Laplace transform is:

The calculation of ^{a}^{a}_{b}_{h}

In the formula: 2

In the formula _{I}

Equation (6) can be converted to:

Where:
_{I}

Suppose the original time series is: ^{(0)} = [^{(0)} (1) ^{(0)} (2) … ^{(0)} (

The undetermined constant _{1}, _{2} in equation (8) is determined by the following equation:

In the formula:

The input term

We perform Laplace transformation on both sides of equation (10) to obtain the transfer function of the system as:

If the frequency domain of

Then the output variable

The above formula can be equivalently written as:

Where

One is that the model recognition with fractional order requires input from the system. Enter according to formula (11) to get the correct result. Second, compared with the conventional gray model identification method, the improved gray model only adds a parameter

Let's take the K1049+678 section of a certain expressway as an example. This section of the foundation is rich in soft soil and has high compressibility. The water in the soil is drained by driving a plastic drainage board on the foundation, and three points A, B, C on the section are selected for settlement monitoring (Figure 1). The project started in March 2019, and the frequency of settlement monitoring is 30d/time [8]. The project ends on June 22, 2020. We use the traditional hyperbolic settlement prediction model and the fractional gray theory prediction model proposed in this paper to calculate the settlement data of this section. See Table 1 for the original data of settlement observation points.

Raw data of settlement observation.

Monitoring time | Cumulative settlement at point A/mm | Cumulative settlement at point B/mm | Cumulative settlement at point C/mm | |
---|---|---|---|---|

2019 | 3 | 131 | 113 | 138 |

4 | 251 | 218 | 251 | |

5 | 347 | 311 | 358 | |

6 | 450 | 400 | 464 | |

7 | 530 | 484 | 564 | |

8 | 606 | 563 | 655 | |

9 | 699 | 629 | 752 | |

10 | 784 | 698 | 840 | |

11 | 849 | 759 | 935 | |

12 | 909 | 822 | 1016 | |

2020 | 1 | 969 | 876 | 1091 |

2 | 1024 | 924 | 1157 | |

3 | 1077 | 965 | 1214 | |

4 | 1122 | 1007 | 1267 | |

5 | 1162 | 1044 | 1316 | |

6 | 1202 | 1079 | 1357 |

We use the data from March to October 2019 as the original data for modeling and forecasting. The article predicts the settlement from November 2019 to June 2020 and compares it with the measured value [9]. We use the model proposed in this paper to select the fitting frequency range when using the fractional differential filter approximation is (0.0011000) rad/s, N=4. That is, the order of the filter is 9. The parameter estimates are _{1} = 0.135; _{2} = 1.200. ^{1.14} respectively. The expression of is

The definition formula for error analysis through the second-order norm is as follows:

In the formula: _{0} represents the cumulative settlement at time _{0}. _{0} represents time zero. _{t}

Observation point

Observation point

Observation point

Substituting,

Observation point _{1} = 0.0729; _{2} = 127.0539

Observation point _{1} = 0.0759; _{2} = 116.4983

Observation point _{1} = 0.0591; _{2} = 131.0533

Substituting the calculated parameters into equation (8) can calculate the integer-order prediction model.

The A, B, C errors calculated by the hyperbolic prediction model according to formula (17) are 282, 139, and 225, respectively [10]. The error of the integer-order model is 186, 202, 93. In this paper, the errors of the method with the fractional gray model are 116, 63, 53, respectively. It can be seen that the calculation results of the fractional-order grey prediction model proposed in this paper are better than the traditional hyperbolic and integer-order prediction models.

We summarize the actual measurement results and the calculation results of the fractional gray theory prediction model included in this paper, the traditional hyperbolic prediction model, and the integer-order in Figure 2. By comparison, it can be seen that the settlement calculation method with the fractional gray theory proposed in this paper can get better prediction results.

(1) The traditional gray model is an ordinary differential equation without input. This article makes two improvements to it. One is to increase the input item u. The second is that the ordinary differential equation is a fractional differential equation. (2) Calculate the errors of the fractional gray model, integer gray model, and hyperbolic model based on engineering examples. The error between the prediction results of the fractional-order gray theory and the actual engineering results established in this paper is higher than that of the integer-order gray theory settlement prediction model and hyperbolic prediction model.

#### Raw data of settlement observation.

Monitoring time | Cumulative settlement at point A/mm | Cumulative settlement at point B/mm | Cumulative settlement at point C/mm | |
---|---|---|---|---|

2019 | 3 | 131 | 113 | 138 |

4 | 251 | 218 | 251 | |

5 | 347 | 311 | 358 | |

6 | 450 | 400 | 464 | |

7 | 530 | 484 | 564 | |

8 | 606 | 563 | 655 | |

9 | 699 | 629 | 752 | |

10 | 784 | 698 | 840 | |

11 | 849 | 759 | 935 | |

12 | 909 | 822 | 1016 | |

2020 | 1 | 969 | 876 | 1091 |

2 | 1024 | 924 | 1157 | |

3 | 1077 | 965 | 1214 | |

4 | 1122 | 1007 | 1267 | |

5 | 1162 | 1044 | 1316 | |

6 | 1202 | 1079 | 1357 |

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