One of the focuses of current social research is wastewater treatment, which is currently a hotspot for scientific research [1]. The polyvinyl alcohol (PVA) nanofiber membrane was prepared by using electrospinning technology, the adsorption of Pb^{2+} and Cu^{2+} in wastewater was studied and the effects of different preparation processes on the diameter, uniformity and fibre morphology of PVA nanofibers were studied [2]. At the same time, the adsorption performance, model and adsorption kinetics of Pb^{2+} nanofiber membrane on Pb^{2+} and Cu^{2+}, the regulation and characterisation of PVA nanofibers in water morphology and the characterisation of PVA-g-POSS nanofibers and the adsorption properties of Pb^{2+} and Cu^{2+} were also studied [3]. The research aims to address the shortcomings of traditional wastewater treatment technologies and further enhance the ability to treat wastewater.

The effects of the PVA nanofiber preparation process on fibre diameter, uniformity and fibre morphology were studied by an orthogonal experiment, and the effects of spinning voltage, spinning distance and PVA spinning solution concentration on the preparation process were analysed. The Pb^{2+} and Cu^{2+} ions in PVA nanofiber adsorption solution were studied, and the adsorption model was also proved to be a multi-layer adsorption process of the PVA nanofiber membrane.

Heavy metal ions in wastewater are highly toxic, are not easily degradable and exist in water for a long time, endangering human life and health [4]. Nowadays, the problem of water shortage in China is becoming more and more serious, and the cost of sewage treatment is relatively large. Therefore, water resources must be developed benign. In addition to water conservation, the recycling of wastewater must be paid attention to. The nanofiber membrane prepared by the adsorption of heavy metal ions in the wastewater is superior to the traditional water treatment method, achieving a high adsorption level with certain innovation [5].

The research is divided into three parts: the first part is literature review, the second part is the waste mud solidification treatment method of PVA nanofiber, including the electrospinning process of PVA nanofiber and the introduction of the adsorption performance method, and the third part is the analysis of experimental results.

Water is the source of life and the natural resource on which human beings depend. Elnemr et al. [6] believed that water pollution has become a global issue in recent years and has reached the point where it needs to be managed immediately. Wan et al. [7] proposed that the ever-changing social development has greatly improved the industrial level in human society, but this development has brought crisis to our living environment, and natural water resources are the biggest victims to this development. Mozaffari et al. [8] believed that large amounts of natural water are discharged into industrial wastewater, causing rapid accumulation of heavy metal ions such as lead, copper and chromium in natural waters. Wei and Yang et al. [9] proposed that heavy metal ions in wastewater are highly toxic, difficult to degrade and exist in water for a long time and finally enriched throughout the food chain, endangering human life and health. Therefore, water resources must be developed benign, and in addition to water conservation, the recycling of wastewater must be paid attention too. Lee et al. [10] believes that there are many methods for the treatment of heavy metal wastewater in China. The traditional methods include the precipitation method, filtration method, ion exchange method and membrane separation method. Membrane treatment is more commonly used, and the most used ones are reverse osmosis treatment and membrane treatment, which use special membranes to allow certain particles to pass smoothly and prevent some particles from flowing out. Sun et al. [11] proposed that the chemical treatment method is to effectively extend the life of the equipment and save water by effectively inhibiting scale corrosion. Piana et al. [12] proposed that the most common material used for water treatment is activated carbon, which is well known for its excellent adsorption properties. There are many adsorption methods for activating carbon, but the adsorption principle is consistent. The huge surface area of activated carbon and unique pore structure make it have strong adsorption performance. Wang et al. [13] used activated carbon fibre to remove Cr^{6+} from wastewater. The study shows that activated carbon fibres can adsorb Cr^{6+} quickly and efficiently, which is better than common activated carbon; so, it has practical application value. Wang et al. [14] proposed that, in recent years, various novel water treatment methods have emerged, such as aerogels for water treatment, which have been extensively studied. Aerogels can be used to remove toxic metals and organics such as Hg^{2+}, Pb^{2+} and nitrophenol from water. Motavalizadehkakhky et al. [15] prepared some composite aerogels for water treatment. In 2013, they prepared metal composite magnetic carbon aerogels as adsorbent materials for adsorbing heavy metal ions in wastewater; solid-liquid separation can be achieved by magnetic separation after adsorption, which is an easy-to-implement method and environmentally friendly.

Wastewater treatment is the focus of today's research, and traditional water treatment technologies have various drawbacks, such as high cost, cumbersome treatment process, inconspicuous treatment of low-concentration heavy metal ions and secondary pollution during treatment. The PVA nanofiber membrane was prepared by electrospinning technology, and the adsorption of Pb^{2+} and Cu^{2+} in wastewater was studied to further solve the shortcomings of traditional water treatment technology and enhance the ability to treat sewage.

The solution of Cu^{2+} and Pb^{2+} with 50 mL mass concentration of 100 mg/L was added to the conical bottle, and the pH was adjusted to 7.0 by NaOH or HCl. Then, 50 mg cell membrane or Cell-g-SAA membrane was added to the above solution, and the supernatant was filtered after shaking at a constant temperature in a water bath at 25°C for 10, 30, 60, 120, 240, 360, 480, 720, 960, 1200 and 1400 mins. The mass concentrations of Cu^{2+} and Pb^{2+} in the solution were measured by an atomic absorption phototypesetter, and the mass concentrations of Cu^{2+} and Pb^{2+} in the solution after adsorption were calculated. The adsorption capacity qe (mg/g) of Cu^{2+} and Pb^{2+} on the fibre membrane was calculated by Eq. (1).

In the formula, _{0} and _{e}

In order to further study the adsorption kinetics of Cu^{2+} and Pb^{2+} on nanofiber membranes, quasi-first-order and quasi-second-order models need to be used to fit the experimental data. The quasi-first-order Eq. (2) and quasi-second-order Eq. (3) kinetic equations are as follows:

The Corey diffusion equation in the solution has the following form:

The modified pseudo-first-order kinetic equation is as follows:
^{−1}·min^{−0.5}. Second-order reaction model,

The adsorption equilibrium of _{0} and _{2} is a quasi-first-order adsorption rate constant.

Initially, 50 mg fibre membrane samples were added to the solution of Cu^{2+} and Pb^{2+} ions with 50 mL concentrations of 40, 60, 80, 100, 120, 160 and 200 mg, respectively, and 1200 min was oscillated at a constant temperature in a water bath at 25°C. After the reaction, the supernatant was filtered, and the mass concentrations of Cu^{2+} and Pb^{2+} in the solution were measured by an atomic absorption phototypesetter, and the mass concentrations of Cu^{2+} and Pb^{2+} in the solution after adsorption were calculated. The parameters of the experimental data were estimated by Langmuir and Freundlich isothermal adsorption models. The equation expressions of the two models are as follows:
_{max} is the saturated adsorption capacity of the fibre membrane, _{e}_{a}_{F}

Adsorption thermodynamics mainly studies the change of energy in the adsorption process and the effect of temperature on adsorption. The following parameters can be obtained by isothermal adsorption at different temperatures: Gibbs free energy (Δ^{0}), enthalpy change (Δ^{0}) and entropy change (Δ^{0}). The relationship between them is shown in the Eqs (9)–(11):
_{D}_{e}_{e}

Initially, 50 mg Cell-g-SAA membrane samples were added to the solution of 50 mL Cu^{2+} and Pb^{2+} ions at 10°, 20°, 30°, 40° and 50°, and are left in a shaker for 1200 min. After the reaction, the supernatant was filtered, and the mass concentrations of Cu^{2+} and Pb^{2+} in the solution were measured by an atomic absorption spectrophotometer, and the mass concentrations of Cu^{2+} and Pb^{2+} in the solution after adsorption were calculated.

The PVA nanofiber membrane has a large specific surface area and has a good effect on the adsorption of Pb^{2+} and Cu^{2+}. The finer the diameter of the fibre, the larger the specific surface area and the better the theoretical adsorption efficiency, in order to obtain the PVA nanofiber with the best adsorption effect, three-factor and three-level orthogonal tests are designed based on the spinning voltage, spinning distance and PVA spinning solution concentration during spinning, and the effects of the process on the average diameter, diameter distribution and fibre morphology of PVA nanofiber membranes are investigated. The materials used in the experiment are shown in Table 1.

Name, specifications and models of experimental raw materials and manufacturers

Polyvinyl alcohol | 117 | Colerie Trading (Shanghai) Co., Ltd. |

Deionised water | - | Laboratory self-control |

The equipment and instruments used in the experiment are shown in Table 2.

Experimental equipment and instruments

Scanning electron microscope | QuantaFEG250 | American FEI Company |

Electrospinning machine | ESM-C01 | Shanghai Yuyue Nanotechnology Co., Ltd. |

The PVA powder was weighed and dissolved in deionised water and heated and stirred in a constant temperature water bath at 90°C for 2 h to obtain a uniform spinning solution of different mass concentrations for use. The spinning solution was poured into a syringe with a volume of 10 ml and a needle diameter of 0.8 mm, and air bubbles were extruded. The spinning voltage ranged from 10 kV to 20 kV, and the aluminium foil was fixed on a roller at a distance from 10 cm to 14 cm from the needle to collect the nanofibers. The nanofiber membrane was separated from the surface of the aluminium foil for further testing. A schematic diagram of electrospinning is shown in Figure 1.

The fibre structure of the PVA nanofiber membrane is observed by using a scanning electron microscope (SEM), including the average fibre diameter, diameter distribution and presence or absence of adhesion. The sample is adhered to the sample stage, and the test is carried out under high vacuum after the gold treatment, and the test condition is 10 kV. According to the SEM photograph of the PVA nanofiber membrane, the fibre diameter is measured using Image ProPlus software, and the fibre diameter measurement of not less than 100 is taken as the standard, the average diameter of the nanofibers is calculated according to Eq. (1):
_{i}

The raw materials used in the experiment are shown in Table 3.

Names, specifications and models of raw materials and manufacturers

Polyvinyl alcohol | One hundred and seventeen | Colerie Trading (Shanghai) Co., Ltd. |

Deionised water | - | Laboratory self-control |

Copper sulphate pentahydrate | Analytical purity | China Pharmaceutical Group Chemical Reagents Co., Ltd. |

Lead chloride | Analytical purity | China Pharmaceutical Group Chemical Reagents Co., Ltd. |

The equipment and instruments used in the experiment are shown in Table 4.

Experimental equipment and instruments

Electrospinning machine | ESM-C01 | Shanghai Yuyue Nanotechnology Co., Ltd. |

ICP emission spectrometer | PS3500DDII | Hitachi Manufacturing Institute |

A batch test is performed, and 50 mg of the PVA nanofiber membrane is placed in a vacuum oven at 60°C for drying for 1 h. Then, it is immersed in 100 mL of 25–100 mg/L CuSO_{4}.5H_{2}O and PbCl_{2} solution, and the adsorption at different contact times is carried out at a constant temperature of 20°C, and the supernatant is taken for dilution and acidification, and then, the amount of adsorption is tested. The supernatant is diluted and acidified, and the concentration of heavy metal ions in the solution is measured on an ICP emission spectrometer (inductively coupled plasma spectrometer), and the adsorption capacity of the nanofiber membrane at different initial ion concentrations of the adsorbed solution and at different contact times is calculated:
_{t}_{0} (mg/L) is the initial ion concentration, _{t}

In the preparation of PVA nanofibers, the concentration of PVA spinning solution plays an important role in the average diameter, diameter distribution and fibre morphology of nanofibers. When the concentration of the PVA spinning solution is less than 6%, the viscosity of the solution is extremely low, and the molecular chains are not uniformly entangled, resulting in uneven distribution of charges, and the jet becomes unstable and cannot maintain continuity, and it is easy to obtain bead-shaped nanofibers, and the distribution of the diameter is very uneven. When the concentration of the PVA spinning solution exceeds 10%, the viscosity of the solution is large, and the spinning solution is easily condensed into a block at the nozzle due to the small amount of solvent and volatilisation, which causes the nozzle to be clogged and cannot be continuously spun. As the PVA concentration increased from 6% to 10%, the average fibre diameter increased from 0.24 μm to 0.44 μm. The average diameter of PVA nanofibers increases with the concentration of the spinning solution because as the concentration increases, the viscosity of the solution increases, and the PVA content in the fluid increases, and the ability of the fluid to differentiate in the electrostatic field is greatly diminished, and the average fibre diameter becomes larger. The spinning voltage directly affects the strength of the electrostatic field, the diameter of the fibre, the distribution of the diameter and the morphology of the fibre, and when the spinning voltage of the PVA nanofibers is lower than 10 kV, the PVA solution has less charge when forming a jet from the nozzle due to insufficient electric field strength and cannot form a Taylor cone to form a filament; when the spinning voltage is higher than 20 kV, the fluid carries too much charge, the repulsive force is too large after differentiation, the liquid disperses and splashes and the filament cannot be formed. The spinning voltage is increased from 10kV to 20kV, and the average diameter of PVA nanofibers is reduced from 0.48μm to 0.19μm. When the voltage is increased and the spinning distance is constant, the electric field strength is increased, the charge of the PVA solution jet increases, the charge repulsive force increases, the differentiation ability of the PVA spinning solution after being ejected from the nozzle is enhanced and the number of fibres is increased; so, the diameter of the PVA nanofiber is reduced.

When the spinning distance ranged from 10 cm to 12 cm, the average diameter of the PVA nanofibers was reduced from 0.21 μm to 0.19 μm. Because as the spinning distance increases, the flight time of the PVA fluid in the electrostatic field increases, the solvent evaporation time becomes longer and the volatilisation is more sufficient, which optimises the differentiation environment of PVA nanofibers, and the differentiation time increases; so, the average fibre diameter decreases. However, as the spinning distance increased from 12 cm to 14 cm, the average diameter of the PVA nanofibers increased to 0.24 μm. This is because the continuous increase of the distance makes the differentiation environment of the PVA fluid superior, and the differentiation time increases.

Through orthogonal experimental analysis, the factors affecting the average fibre diameter, diameter distribution and fibre morphology were ranked as follows: PVA concentration > spinning distance > spinning voltage. The optimum level combination is determined to have a spinning voltage of 20 kV, a PVA concentration of 6% and a spinning distance of 12 cm. As the concentration of PVA spinning solution increases, the viscosity increases, and the greater the PVA content per unit volume, the weaker the ability of the fluid to differentiate in the electrostatic field, resulting in a larger average fibre diameter. With the increase of the spinning voltage, the electrostatic field strength increases when the distance is constant, the repulsive force of the static electricity in the PVA fluid increases, the tensile force of the fibre increases and the differentiation ability increases, which reduces the average diameter of the PVA nanofibers and improves the uniformity of the diameter distribution. Therefore, after the spinning solution concentration and the spinning voltage are determined, the spinning distance is not linear with the average diameter of the nanofibers and has an extreme value, which is a combination of two factors. In order to accurately measure the adsorption amount, a standard curve between the metal ion concentration and the absorption intensity of the inductively coupled plasma spectrometer (ICP emission spectrometer) is drawn; first, the standard Pb^{2+} solutions of 5.5 mg/L, 1 mg/L, 2 mg/L, 3 mg/L, 4 mg/L and 5 mg/L are prepared and acidified. Figure 2 shows the test to achieve the Pb^{2+} standard solution curve because the ICP emission spectrometer can simultaneously measure a variety of metal ions; there are different ion conversion procedures inside, and the Cu^{2+} standard curve can be converted by Pb^{2+}; so, there is no need to draw a Cu^{2+} standard solution curve.

The Pb^{2+} standard solution curve is shown in Figure 2; the supernatant after adsorption is diluted, and after acidification, the super atomising device changes into an aerosol through the conduit into the plasma flame, and most of the state immediately becomes an excited state of atoms and ions. When an excited atom or ion returns to a stable ground state, a fixed energy is released and appears as a spectrum of a certain wavelength, the emission spectrometer obtains a standard curve by measuring the specific line and intensity of each element and the larger the ion concentration, the greater the absorption intensity. The correlation coefficient R2 of the Pb^{2+} standard curve is 0.99974, which conforms to the standard and guarantees the credibility of the subsequent actual test results. Figure 3 is a graph showing the adsorption amount of Pb^{2+} on the PVA nanofiber membrane, as shown in Figure 3. With the increase of contact time, the adsorption amount of Pb^{2+} by PVA nanofiber membrane gradually increases, and it becomes stable after reaching equilibrium, and the equilibrium time is 30h, the maximum adsorption amount of 100 mg/L at the initial concentration of the adsorbed solution reaches 168.3 mg/g. Pb^{2+} is free to move in the solution. When Pb^{2+} is immobilised on the PVA nanofiber membrane, the remaining Pb^{2+} takes time to continue to approach and be adsorbed by the PVA nanofiber membrane by the driving force of the concentration difference. At the same time, a small amount of adsorbed Pb ions is desorbed and re-entered into the solution, and at the beginning of the adsorption, the adsorption rate caused by the concentration difference is much larger than the desorption rate. Therefore, as the contact time increases, a large amount of Pb^{2+} is adsorbed and fixed on the PVA nanofiber membrane, and the adsorption amount gradually increases with time, and when the adsorption rate is equal to the desorption rate, the adsorption reaches equilibrium. It can also be observed from Figure 3 that the PVA nanofiber membrane adsorbs the adsorbed liquid at different initial concentrations, and the equilibrium adsorption amount is proportional to the initial Pb^{2−} concentration of the adsorbed liquid. For the PVA nanofiber membrane, the number of adsorption active sites is sufficient, but it takes a certain time for the Pb^{2+} to be adsorbed from the adsorbed liquid gradually close to the surface of the PVA nanofiber membrane by the concentration difference driving. When the adsorption rate is equal to the desorption rate, the adsorption equilibrium is reached. It is not because the PVA nanofiber membrane adsorbs insufficient active sites but because it is difficult to drive the remaining Pb^{2+} near the surface of the PVA nanofiber membrane for the concentration difference is insufficient. Therefore, for different initial concentrations of Pb^{2+} adsorbed liquid, the adsorption amount reaching the adsorption equilibrium is basically proportional to the initial concentration of Pb^{2+}.

Figure 4 is a graph showing the equilibrium adsorption capacity of the PVA nanofiber membrane to Cu^{2+}. As shown in Figure 4, as the contact time increases, the adsorption amount of Cu^{2+} by the PVA nanofiber membrane continues to increase, and the equilibrium time is 15 h; the equilibrium adsorption amount of the initial concentration of Cu^{2+} at 100 mg/L reaches 62.3 mg/g. Similar to Pb^{2+}, Cu^{2+} moves freely in the adsorbed solution. When most of the Cu^{2+} is immobilised on the PVA nanofiber membrane, the remaining Cu^{2+} requires a longer contact time to drive closer to and be adsorbed by the PVA nanofiber membrane by the difference in concentration until the adsorption equilibrium is reached. It can also be seen from Figure 4 that the value of the equilibrium adsorption amount of the PVA nanofiber membrane to the adsorbed liquid at different initial concentrations is substantially proportional to the initial Cu^{2+} concentration of the adsorbed liquid, which is still caused by the concentration difference drive. However, compared with Pb^{2+}, the time for Cu^{2+} to reach equilibrium adsorption is only half of that of Pb^{2+}, and the adsorption amount is also far from Pb^{2+}. In order to study the differences, the adsorption mechanism and adsorption kinetics are studied.

In order to further study the adsorption mechanism of PVA nanofibers, the adsorption isotherms of Pb^{2+} and Cu^{2+} were fitted by Langmuir and Freundlich theoretical models, respectively. The Langmuir theoretical model equation is one of the commonly used adsorption isotherm equations and is widely used in the field of adsorption. Langmuir theoretical model is written as follows:
_{e}_{e}_{m}_{L}_{L}_{L}

The Freundlich theoretical model has no practical physical meaning but is applied to the adsorption theory as an empirical formula, and it satisfies the Freundlich theoretical model, which represents the PVA nanofiber membrane adsorption model as a multilayer adsorption model. Freundlich theoretical model is written as follows:
_{e}_{e}_{F}_{F}

In this study, the two adsorption models are fitted according to the equilibrium adsorption amount, and the fitting data are analysed. For the Langmuir model, the Ke values of Pb^{2+} and Cu^{2+} are 0.0129 and 0.0023, respectively, indicating that the PVA nanofiber membrane has a higher adsorption capacity for Pb^{2+} than Cu^{2+}. Both of them have R2 of 0.98 or more, indicating that they conform to the single-layer adsorption model. For the Freundlich model, Pb^{2+} and 0: the & value of 112+ is 14.88 and 1.399 respectively, indicating that the adsorption capacity of PVA nanofibers to Pb^{2+} is greater than that of Cu^{2+}, and the n values of both are about 1.1, indicating that the adsorption modes are preferential adsorption, that is, adsorption is easier to carry out. Since the R2 of the Freundlich model is larger than the R2 of the Langmuir model, the nanofiber membrane belongs to multilayer adsorption. In order to study the difference of the equilibrium adsorption amount of Pb^{2+} and Cu^{2+} at the same initial concentration, the adsorption kinetics of Pb nanofiber membranes adsorbing Pb^{2+} and Cu^{2+} are studied by using quasi-first-order and quasi-second-order kinetic equations. In the kinetic study of chemical reactions, scientists derived the chemical reaction kinetics equation based on the relationship between the reaction rate of the substance and the concentration of the participating substances. The number of chemical reactions is the sum of the individual reactant concentration indices of the kinetic equation. The quasi-first-order dynamic equation is a new model obtained by the modified method, which is equivalent to the dynamic model of the first-order reaction. According to the quasi-first-order dynamic equation, the system is mainly based on physical adsorption, that is, adsorption plays a major role in electrostatic and van der Waals forces. For Pb^{2+}, both the quasi-first-order kinetic equation and the quasi-second-order kinetic equation have R2 exceeding 0.98, indicating that both physical adsorption and chemisorption act simultaneously. For Cu^{2+}, the R2 of the quasi-second-order kinetic equation is much larger than the quasi-first order; so, chemisorption plays a major role. This is because Cu^{2+} can form a complex structure with the hydroxyl groups on the PVA molecular chain; so, Cu^{2+} is fixed on the fibre membrane mainly through complexation, and the adsorption equilibrium time is short. However, because 4 mol –OHs are required to reach the complex equilibrium with each mole of Cu^{2+} ions, some of the hydroxyl groups on the surface of the PVA nanofiber membrane can no longer be complexed with Cu^{2+}. The increase of the surface degree of the PVA nanofiber membrane weakens the driving force, and the adsorption amount is reduced. For Pb^{2+}, although it has no complexation with hydroxyl groups, it is more easily adsorbed by the PVA nanofiber membrane through the action of static electricity and Howard force, and the concentration driving force is greater than Cu^{2+} ion, but the adsorption equilibrium time is relatively long, and the adsorption amount is larger than Cu^{2+} ions.

With the increase of contact time, the adsorption amount of Pb^{2+} and Cu^{2+} showed an increasing trend, and finally, it became equilibrium. The time for Pb^{2+} to reach equilibrium was 30 h, and the time for Cu^{2+} to reach equilibrium was 15h. As the initial concentration of the adsorbed liquid increases, the amount of adsorption also increases, and the final amount of adsorption is proportional to the treated concentration. By fitting the Langmuir model, the _{L}^{2+} and Cu^{2+} for PVA nanofiber membranes were 0.0129 and 0.0023, respectively, indicating that the adsorption capacity of the PVA nanofiber membrane for Pb^{2+} is much larger than that of Cu^{2+}, and both of them have R2 of 0.98 or more, indicating that they conform to the single-layer adsorption model. For the Freundlich model, the _{F}^{2+} and Cu^{2+} are 14.88 and 1.399, respectively, indicating that the adsorption capacity of PVA nanofibers to Pb^{2+} is much larger than that of Cu^{2+}, and the n values of both are greater than 1; so, it is a multi-layer adsorption model that is easy to adsorb. However, because the R2 of Freundlich is larger than the R2 of the Langmuir model, the nanofiber membrane belongs to multilayer adsorption. For Pb^{2+}, the R2 of the quasi-first-order and quasi-second-order kinetic equations is close, and the chemisorption and physical adsorption are performed simultaneously; for Cu^{2+}, the R2 of the quasi-second-order kinetic equation is greater than that of the quasi-first order, indicating that chemisorption plays a major role.

The Pb^{2+} and Cu^{2+} in the PVA nanofiber adsorption solution were studied, and the results show that when the initial concentration of the adsorbed liquid increases, the adsorption amount also increases. The study of the adsorption model proves that the PVA nanofiber membrane belongs to multilayer adsorption, and the theoretical model is supported by the Freundlich theorem. By fitting the Langmuir model, the _{L}^{2+} and Cu^{2+} for PVA nanofiber membranes were 0.0129 and 0.0023, respectively, indicating that the adsorption capacity of the PVA nanofiber membrane for Pb^{2+} is much larger than that of Cu^{2+}. Both of them have R2 of 0.98 or more, indicating that they conform to the single-layer adsorption model. For the Freundlich model, the _{F}^{2+} and Cu^{2+} are 14.88 and 1.399, respectively, indicating that the adsorption capacity of PVA nanofibers to Pb^{2+} is much larger than that of Cu^{2+}, and the n values of both are greater than 1; so, it is a multi-layer adsorption model that is easy to adsorb. However, because the R2 of Freundlich is larger than the R2 of the Langmuir model; the nanofiber membrane belongs to multilayer adsorption. For Pb^{2+}, the R2 of the quasi-first-order and quasi-second-order kinetic equations is close, and the chemisorption and physical adsorption are performed simultaneously; for Cu^{2+}, the R2 of the quasi-second-order kinetic equation is greater than that of the quasi-first order, indicating that chemisorption plays a major role.

#### Names, specifications and models of raw materials and manufacturers

Polyvinyl alcohol | One hundred and seventeen | Colerie Trading (Shanghai) Co., Ltd. |

Deionised water | - | Laboratory self-control |

Copper sulphate pentahydrate | Analytical purity | China Pharmaceutical Group Chemical Reagents Co., Ltd. |

Lead chloride | Analytical purity | China Pharmaceutical Group Chemical Reagents Co., Ltd. |

#### Experimental equipment and instruments

Electrospinning machine | ESM-C01 | Shanghai Yuyue Nanotechnology Co., Ltd. |

ICP emission spectrometer | PS3500DDII | Hitachi Manufacturing Institute |

#### Name, specifications and models of experimental raw materials and manufacturers

Polyvinyl alcohol | 117 | Colerie Trading (Shanghai) Co., Ltd. |

Deionised water | - | Laboratory self-control |

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optimisation model based on IoT intelligent computing Conventional Algorithms in Sports Training Based on Fractional Differential Equations Dynamic Nonlinear System Based on Complex System Theory in the Development of Vocational Education LTE wireless network coverage optimisation based on corrected propagation model The Algorithm Accuracy of Mathematical Model to Improve the Transmission Speed of E-commerce Platform Study on tourism development income index calculation of finite element ordinary differential mathematical equation The Security of Database Network Model Based on Fractional Differential Equations Electric Vehicle Mechanical Transmission System Based on Fractional Differential Equations The Practice System of Physics and Electronics Courses in Higher Vocational Colleges Based on Fractional Differential Equations The Economic Model of Rural Supply and Demand Under the Data Analysis Function Based on Ordered Probit 3D Modeling System of Indoor Environment Art Landscape Design under Statistical Equation Electronic Information Security Model of Nonlinear Differential Equations The Optimization Model of College Students' Physical Exercise Motivation and Self-control Ability Based on the Mathematical Model of Probability Theory Impact of ASEAN-China free trade area on fishery value chain based on difference-in-difference method Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Application and risk assessment of the energy performance contracting model in energy conservation of public buildings Sensitivity analysis of design parameters of envelope enclosure performance in the dry-hot and dry-cold areas Criminal law imputation path for biometric information Research on composite dynamic disaster prevention and control system of mine earthquake and shock in thick and hard rock mines Research on innovative strategies of college students’ English teaching under the background of artificial intelligence Electromagnetic interference prediction technology of new energy motor drive system Research on the application of PLC technology in electrical automation engineering Research on indoor environment design of comprehensive commercial shopping center based on numerical simulation Application of matrix multiplication in signal sensor image perception Empirical analysis of the economic absolute income hypothesis based on mathematical statistics Analysing the variation of metadiscourse verb patterns in English academic papers from within and between disciplines Impact of COVID-19 policy on stock prices of listed property companies Realization of Book Collaborative Filtering Personalized Recommendation System Based on Linear Regression Equation Research on the experimental principle of deep integration of LETS software and criminal procedure under the background of artificial intelligence Study on Interactive Relations between Enterprise Social Media and Decision Style Based on a vector Autoregressive Model Research on Dynamics of Flexible Multibody System with Deployable Antenna Based on Static Lagrangian Function The Optimization of Mathematics Teaching Models in Colleges and Universities Based on Nonlinear Differential Equations Study on spatial planning and design of learning commons in university libraries based on fuzzy matrix model The Stability Model of Piano Tone Tuning Based on Ordinary Differential Equations Construction and application of automobile user portrait based on k-mean clustering model The Homework Model of Screening English Teaching Courses Based on Fractional Differential Equations VR-based computer maintenance practical training platform development design and application research Research on innovative human capital for China’s economic development based on STI model In-depth analysis of the artistic expression of paper-cut elements in the design of boat space Knowledge graph construction and Internet of Things optimisation for power grid data knowledge extraction Modeling the pathway of breast cancer in the Middle East Construction and intelligent analysis of power grid physical data knowledge graph based on Internet of Things for power system Research on industrial Internet of Things and power grid technology application based on knowledge graph and data asset relationship model Research on the effects of financial literacy on rural household credit constraint Calculus Logic Function in Integrated Manufacturing Automation of Single Chip Microcomputer Football Offense Training Strategy Based on Fractional Differential Mathematical Modeling Research on educational resource recommendation system based on MRLG Rec The Mathematical Analysis Model of Educational System in Music Courses in Colleges and Universities Continuing Education Network Data Center Model Based on Fractional Differential Mathematical Equations A study on the phenomenon of anaphoric correction in college students’ English conversation Computer Art Design Model Based on Nonlinear Fractional Differential Equations The Optimization Model of Public Space Design Teaching Reform Based on Fractional Differential Equations The Approximate Solution of Nonlinear Vibration of Tennis Based on Nonlinear Vibration Differential Equation Graphical Modular Power Technology of Distribution Network Based on Machine Learning Statistical Mathematical Equation Employment and Professional Education Training System of College Graduates Based on the Law of Large Numbers Economic Research on Multiple Linear Regression in Fruit Market inspection and Management Nonlinear Differential Equations in Preventing Financial Risks Lagrange’s Mathematical Equations in the Sports Training of College Students Simulation Research of Electrostatic Precipitator Power Supply Voltage Control System Based on Finite Element Differential Equation Research on the effect of generative adversarial network based on wavelet transform hidden Markov model on face creation and classification Research on Lightweight Injection Molding (CAE) and Numerical Simulation Calculate of New Energy Vehicle Power Flow Calculation in Smart Distribution Network Based on Power Machine Learning Based on Fractional Differential Equations Demonstration of application program of logistics public information management platform based on fuzzy constrained programming mathematical model Basketball Shooting Rate Based on Multiple Regression Logical-Mathematical Algorithm The Optimal Application of Lagrangian Mathematical Equations in Computer Data Analysis Similarity Solutions of the Surface Waves Equation in (2+1) Dimensions and Bifurcation Optimal decisions and channel coordination of a green supply chain with marketing effort and fairness concerns Game theoretic model for low carbon supply chain under carbon emissions reduction sensitive random demand Limit cycles of a generalised Mathieu differential system Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Application of data mining in basketball statistics The nonlinear effects of ageing on national savings rate – An Empirical Study based on threshold model Design of fitness walker for the elderly based on ergonomic SAPAD model AtanK-A New SVM Kernel for Classification Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Study of a linear-physical-programming-based approach for web service selection under uncertain service quality The Relationship Between College Students’ Taekwondo Courses and College Health Based on Mathematical Statistics Equations Analysis and countermeasures of cultivating independent learning ability in colleges teaching English based on OBE theory A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Fractional Differential Equations in the Exploration of Geological and Mineral Construction AdaBoost Algorithm in Trustworthy Network for Anomaly Intrusion Detection Projection of Early Warning Identification of Hazardous Sources of Gas Explosion Accidents in Coal Mines Based on NTM Deep Learning Network Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China Enterprise Financial Risk Early Warning System Based on Structural Equation Model A Study on the Application of Quantile Regression Equation in Forecasting Financial Value at Risk in Financial Markets Fractional Differential Equations in the Model of Vocational Education and Teaching Practice Environment Information transmission simulation of Internet of things communication nodes under collision free probability equation Image denoising model based on improved fractional calculus mathematical equation Random Fourier Approximation of the Kernel Function in Programmable Networks The Complexity of Virtual Reality Technology in the Simulation and Modeling of Civil Mathematical Models University Library Lending System Model Based on Fractional Differential Equations Calculation and Performance Evaluation of Text Similarity Based on Strong Classification Features Intelligent Matching System of Clauses in International Investment Arbitration Cases Based on Big Data Statistical Model Evaluation and Verification of Patent Value Based on Combination Forecasting Model Financial Institution Prevention Financial Risk Monitoring System Under the Fusion of Partial Differential Equations Prediction and Analysis of ChiNext Stock Price Based on Linear and Non-linear Composite Model Calculus Logic Function in Tax Risk Avoidance in Different Stages of Enterprises The Psychological Memory Forgetting Model Based on the Analysis of Linear Differential Equations Optimization Simulation System of University Science Education Based on Finite Differential Equations The Law of Large Numbers in Children's Education Optimization System of Strength and Flexibility Training in Aerobics Course Based on Lagrangian Mathematical Equation Data structure simulation for the reform of the teaching process of university computer courses RETRACTION NOTE Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Research on non-linear visual matching model under inherent constraints of images Good congruences on weakly U-abundant semigroups Can policy coordination facilitate unimpeded trade? An empirical study on factors influencing smooth trade along the Belt and Road Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Research on design of customer portrait system for E-commerce Research on rule extraction method based on concept lattice of intuitionistic fuzzy language Fed-UserPro: A user profile construction method based on federated learning A multi-factor Regression Equation-based Test of Fitness Maximal Aerobic Capacity in Athletes Design and evaluation of intelligent teaching system on basic movements in PE Garment Image Retrieval based on Grab Cut Auto Segmentation and Dominate Color Method Financial Risk Prediction and Analysis Based on Nonlinear Differential Equations Constructivist Learning Method of Ordinary Differential Equations in College Mathematics Teaching Multiple Effects Analysis of Hangzhou Issuing Digital Consumer Coupons Based on Simultaneous Equations of CDM Model Response Model of Teachers’ Psychological Education in Colleges and Universities Based on Nonlinear Finite Element Equations A Hybrid Computational Intelligence Method of Newton's Method and Genetic Algorithm for Solving Compatible Nonlinear Equations Pressure Image Recognition of Lying Positions Based on Multi-feature value Regularized Extreme Learning Algorithm English Intelligent Question Answering System Based on elliptic fitting equation Precision Machining Technology of Jewelry on CNC Machine Tool Based on Mathematical Modeling Application Research of Mathematica Software in Calculus Teaching Computer Vision Communication Technology in Mathematical Modeling Skills of Music Creation Based on Homogeneous First-Order Linear Partial Differential Equations Mathematical Statistics Technology in the Educational Grading System of Preschool Students Music Recommendation Index Evaluation Based on Logistic Distribution Fitting Transition Probability Function Children's Educational Curriculum Evaluation Management System in Mathematical Equation Model Query Translation Optimization and Mathematical Modeling for English-Chinese Cross-Language Information Retrieval The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization research on prefabricated concrete frame buildings based on the dynamic equation of eccentric structure and horizontal-torsional coupling Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Green building considering image processing technology combined with CFD numerical simulation Research on identifying psychological health problems of college students by logistic regression model based on data mining Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching Research on evaluation system of cross-border E-commerce platform based on the combined model A mathematical model of PCNN for image fusion with non-sampled contourlet transform Nonlinear Differential Equations in Computer-Aided Modeling of Big Data Technology The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Financial customer classification by combined model Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Design of Tennis Mobile Teaching Assistant System Based on Ordinary Differential Equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis High Simulation Reconstruction of Crowd Animation Based on Optical Flow Constraint Equation Construction of Intelligent Search Engine for Big Data Multimedia Resource Subjects Based on Partial Least Squares Structural Equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm Analysis of the Teaching Quality of Physical Education Class by Using the Method of Gradient Difference The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Animation VR scene mosaic modeling based on generalized Laplacian equation Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based on Fractional Differential Equations The influence of accounting computer information processing technology on enterprise internal control under panel data simultaneous equation Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Construction of comprehensive evaluation index system of water-saving irrigation project integrating penman Montei the quation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression Analysis of technical statistical indexes of college tennis players under the win-lose regression function equation Automatic extraction and discrimination of vocal main melody based on quadratic wave equation Analysis of wireless English multimedia communication based on spatial state model equation Optimization of Linear Algebra Core Function Framework on Multicore Processors Application of hybrid kernel function in economic benefit analysis and evaluation of enterprises Research on classification of e-commerce customers based on BP neural network The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Fractional Differential Equations in Electronic Information Models BIM Engineering Management Oriented to Curve Equation Model Leakage control of urban water supply network and mathematical analysis and location of leakage points based on machine learning Analysis of higher education management strategy based on entropy and dissipative structure theory Prediction of corporate financial distress based on digital signal processing and multiple regression analysis Mathematical Method to Construct the Linear Programming of Football Training Multimedia sensor image detection based on constrained underdetermined equation The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Application of Numerical Computation of Partial Differential Equations in Interactive Design of Virtual Reality Media Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Knowledge Analysis of Charged Particle Motion in Uniform Electromagnetic Field Based on Maxwell Equation Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Term structure of economic management rate based on parameter analysis of estimation model of ordinary differential equation Influence analysis of piano music immersion virtual reality cooperation based on mapping equation Chinese painting and calligraphy image recognition technology based on pseudo linear directional diffusion equation Label big data compression in Internet of things based on piecewise linear regression Animation character recognition and character intelligence analysis based on semantic ontology and Poisson equation Design of language assisted learning model and online learning system under the background of artificial intelligence Study on the influence of adolescent smoking on physical training vital capacity in eastern coastal areas Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering Action of Aut( G ) on the set of maximal subgroups ofp -groupsResearch on loyalty prediction of e-commerce customer based on data mining Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on e-commerce platform Evaluation and prediction of regional human capital based on optimised BP neural network Study on inefficient land use determination method for cities and towns from a city examination perspective A study of local smoothness-informed convolutional neural network models for image inpainting Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training Nonlinear Channel Estimation for Internet of Vehicles Some Necessary Conditions for Feedback Functions of de Bruijn Sequences The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model An empirical research on economic growth from industrial structure optimisation in the Three Gorges Reservoir area The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Study of agricultural finance policy information extraction based on ELECTRA-BiLSTM-CRF Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Innovative research of vertical video creation under the background of mobile communication Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model Genetic algorithm-based congestion control optimisation for mobile data network