Economic benefits continue to increase with the rapid development of the tourism industry. The level of tourism development is an indicator to measure the maturity of a country’s tourism industry. The article takes China’s coastal provinces as the research object and uses finite element ordinary differential mathematical equations to explore the income indicators of the tourism industry in coastal cities. In the dynamic process, the article portrays the development trend of tourism in different regions. It analyses the income levels and indicators of tourism development in coastal cities under different time and space conditions. The research results show that we can divide the time zone to count the base period value and average growth rate of domestic tourism in each province to classify the regional response type of domestic tourism development.

#### Keywords

- finite element
- ordinary differential mathematical equation
- tourism economy
- income index
- time and space difference
- coastal area

#### AMS 2010 codes

- 74S05

Tourism is an integral part of the tertiary industry and consumer economy. It plays an essential role in international trade and the national economy. Many countries or regions have adopted tourism as an effective means to reduce regional differences. The Chinese economy is highly concentrated in the coastal areas [1], which have superior locations, rich tourism resources and unique climatic conditions. The development of inbound tourism is conducive to the industrial transformation of the region. It promotes the industrial transfer of the three significant east, middle and west areas to achieve coordinated regional development [2].

The study of regional differences in inbound tourism by foreign academic circles began in the mid-1970s. Domestic scholars started the study of regional differences in inbound tourism at a later time. They mainly analysed time and space differences, regional differences in tourism, regional differences in the provincial inbound tourism economy and development models, endowments of tourism resources and spatial distribution of tourism flows. This article studies the temporal and spatial differences of inbound tourism in coastal areas.

We look forward to a valuable exploration of inbound tourism from this aspect.

We assume that the spatial region Ω ^{d}

where ^{0} ^{1} < ^{N}_{n}^{n},t^{n+1}) in the article. We also set ^{n}^{n}

We assume that ∥∥_{Y}_{Y}_{X}^{s}^{2}(Ω), the above norm and inner product are abbreviated as ∥∥_{S}

We define ^{2}(_{n}_{n} → Z_{n} ⊆ J^{s}_{S,n}_{n} → Z_{n} ⊆ J

For ^{m}_{0}(

Space ^{∞}) := {^{1}(^{∞}(Ω)) and

We define the finite element space

by replacing _{hk}_{n}

where ^{n}^{n}^{∞}).

Also, note that values of _{n}_{n}^{n},t^{n+1}), correspondingly

Further, the first item on the left end of the equal sign in Eq. (8) can be obtained after partial integration of time _{hk}

This article assumes that there are _{mk}_{ml}_{mk},y_{ml}_{m}_{m} ≥^{−}^{−} ≥^{+}(^{+}

Eq. (10) is the DEA model under constant returns to scale (CRS). Assuming _{m}_{m}_{m}_{m}_{m}_{PTE} × θ_{SE}_{PTE}_{SE}_{m}_{PTE}_{SE}_{PTE}_{SE}_{PTE}_{SE}_{PTE}_{SE}_{PTE}_{SE}

In this paper, the value of ^{t}^{t}|C,S

This function is understood as the ratio of the input-output array

According to the geometric mean of the measurement results of Eqs (13) and (14), we write the Malmquist index model in different periods as follows:

It is generally believed that each efficiency index >1 indicates that each efficiency is improved. If it is

The evolution trend of the overall difference in tourism economy in China’s coastal areas is shown in Figure 1. It can be roughly divided into two stages: The first stage is the slow expansion of the absolute difference from 2006 to 2012. The difference increased from 765.18 in 2006 to 1,452.52 in 2012, and the value of the absolute difference increased by 89.83%. The second stage is from 2013 to 2021, and the absolute difference expands at a relatively rapid rate, increasing from 1,238.43 in 2013 to 3838.61 in 2021. The increase was as high as 209.96%.

Although the overall trend is slow, it is still pronounced [9]. All coefficients of variation >1 indicate that the economic development of inbound tourism in coastal areas is still uneven. Furthermore, the differences between provinces and cities are still quite obvious, and there are still significant differences in the overall coordinated development of regions.

Through the analysis of the Herfindahl index (see Figure 1), it can be seen that the overall concentration of the inbound tourism economy of the coastal provinces gradually decreases, and the phenomenon of differentiation becomes more and more apparent. It can be known by calculating the ratio of the foreign exchange income of tourism in various provinces and cities relative to the average level in coastal areas. From a horizontal perspective, the foreign exchange income index of tourism is >1. Before 2005, it was concentrated in Jiangsu Province, Shanghai and Guangdong Province [10]. After 2005, it was concentrated in the three provinces and cities of Shanghai, Fujian and Guangdong. The remaining provinces and cities generally had index <0.5. From a vertical perspective, the tourism foreign exchange indexes of Hebei, Shanghai, Fujian, Guangdong, Guangxi and Hainan show a fluctuating and rising trend. Other places showed volatility and decline [11]. However, the rate of increase or decrease in the index of each province and city is different. This shows that the inbound tourism economy in China’s coastal areas is polarised. For example, tourism’s foreign exchange income index in Guangdong in 2021 is 4.957, while that of Hebei is only 0.139. The difference between the two is as much as 35 times. Thus, the polarisation of the inbound tourism economy is evident.

Figure 2 shows the evolution of the differences between provinces, cities and regions of the inbound tourism economy in China’s coastal areas from 2006 to 2021. The regional differences in the inbound tourism economy show a trend of rising volatility. For example, the regional difference index increased from 0.052 in 2006 to 0.063 in 2021. At the same time, there are two high periods and two low periods.

The regional differences from 2006 to 2008 showed a low-level downward trend. Regional differences mainly caused the reason why TWR was higher than TBR. It can be seen from Figure 3 that the internal differences in the coastal areas of the Yangtze River Delta are higher than those in the Bohai Rim and the Pan-Pearl River Delta. Although the difference in the coastal area of the Yangtze River Delta shows a downward trend, it is still significantly higher than the other two regions. During this period, the total number of inbound tourists in Shanghai accounted for 45.98% of the total coastal area of the Yangtze River Delta.

From 1999 to 2012, regional differences (TP) showed a high upward trend. TWR and TBR were roughly the same, and regional differences were composed of intra-regional and inter-regional differences (see Figure 2).

It can be seen from Table 1 that the contribution rate of the three significant regional and intra-regional differences to the regional differences in China’s coastal inbound tourism from 2006 to 2021 shows the following characteristics. The inter-regional difference first increased and then decreased, while the intra-regional difference first decreased and then increased. The inter-regional differences showed a fluctuating upward trend, and the intra-regional differences showed a fluctuating downward trend. The inter-regional differences and interregional differences constitute the main aspects that affect the regional differences of inbound tourism in coastal areas.

Theil Index of China’s Coastal Inbound Tourism Scale

Years | 2006 | 2015 | 2021 |
---|---|---|---|

TP | 0.052 | 0.061 | 0.063 |

TBR | 0.016 | 0.037 | 0.025 |

TWR | 0.036 | 0.024 | 0.038 |

TBR contribution rate | 30.77 | 60.66 | 39.68 |

TWR contribution rate | 69.23 | 39.34 | 60.32 |

The economic foundation is the foundation of the inbound tourism economy. The economic foundation of various provinces and cities affects and restricts the development of inbound tourism to a certain extent [12]. Guangdong Province ranks first in the country in terms of gross domestic product, and economic income from inbound tourism is always far ahead, followed by the three provinces and cities of Shanghai, Jiangsu and Zhejiang. In addition, tourism is an essential part of the tertiary industry. From the overall industrial structure, the proportion of tertiary industries in the Yangtze River Delta is as high as 45.88%, followed by the Pan-Pearl River Delta with 42.49% and, finally, the Bohai Rim with only 37.94%.

Taking 5A-level scenic spots as an example, there are 61 national 5A-level tourist attractions in coastal areas. There are 27 in the Yangtze River Delta, accounting for 44% of the total. However, from a regional perspective, high-level tourism resources are not an absolute condition that determines tourism income, and natural factors also play a huge role. Take the Jiangsu coast as an example. Although Jiangsu Province has the most significant number of 5A-level scenic spots, it lags behind Shanghai in terms of the overall inbound tourism economy [13]. Natural conditions are also one of the reasons. The coastal areas of Jiangsu are large areas of wetlands and tidal flats, and there are few beaches with better quality for recreation. This affects its inbound tourism development to a certain extent.

From the perspective of spatial interaction theory, location is the basis of regional development. The Bohai Rim is close to Japan and South Korea, and the coastal areas of the Pan-Pearl River Delta take advantage of being close to Hong Kong, Macao and Taiwan. In addition, an open policy is also a prerequisite for the development of inbound tourism. Affected by government policies, the opening hours and levels of the provinces and cities in China’s coastal areas are not consistent. Opening to the outside world affects the city’s popularity to a certain extent, and the city’s popularity harms the regional inbound tourism economy.

The absolute difference in the inbound tourism economy of the coastal provinces and cities has an expanding trend of fluctuations and increases. On the other hand, the relative difference shows a downward trend. As a result, the degree of concentration is gradually decreasing, and the phenomenon of polarisation is gradually becoming apparent. From the perspective of different stages, the differences between and within regions affected by multiple factors fluctuate. In-depth analysis of the reasons for the above differences has found that the economic foundation, industrial structure, resource endowments, location conditions, the level of opening to the outside world and essential events all have a more significant impact on regional differences.

#### Theil Index of China’s Coastal Inbound Tourism Scale

Years | 2006 | 2015 | 2021 |
---|---|---|---|

TP | 0.052 | 0.061 | 0.063 |

TBR | 0.016 | 0.037 | 0.025 |

TWR | 0.036 | 0.024 | 0.038 |

TBR contribution rate | 30.77 | 60.66 | 39.68 |

TWR contribution rate | 69.23 | 39.34 | 60.32 |

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