Research on tourism income index based on ordinary differential mathematical equation

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 ⊆ R^d(d = 1,2) is bounded and has a smooth boundary. The time interval J = [0,T]. Let us remember that X and Y are separable Hilbert spaces. Consider the following abstract parabolic equation: (1) {ut+Au=f(x,t),t∈J,x∈ΩBu=0,t∈J,x∈∂Ωu(0,x)=u0(x),x∈Ω \left\{\begin{array}{l} u_{t}+A u=f(x, t), t \in J, x \in \Omega \\ B u=0, t \in J, x \in \partial \Omega \\ u(0, x)=u_{0}(x), x \in \Omega \end{array}\right.

where A is a self-adjoint, positive definite and compact inverse elliptic operator with a domain of D(A) ⊂ X. B is the boundary operator. For this, we first discretise the time interval J. We set 0 = t⁰ < t¹ < ⋯ < t^N = T,I_n = (tⁿ,tⁿ⁺¹) in the article. We also set Tnn={τ}T_{n}^{n}=\{\tau\} to be a regular division of the space region Ω. Here, hτ represents the length of unit τ and −hn=max⁡τ∈Tnnhτ,h=max⁡nhn- h_n = \max _\tau \in T_n^n h_\tau ,h = \max _n h_n. The division of each space–time slice Qⁿ can be different, and TnnT_n^n and Tnn−1T_n^{n - 1} can take different nodes on the interface t = tⁿ(1 ≤ n ≤ N − 1).

We assume that ∥∥_Y and (·,·)_Y are the norm and inner product of space Y, respectively. ∥∥_X is the norm of space X. When X = H^s(Ω),Y = L²(Ω), the above norm and inner product are abbreviated as ∥∥_S, respectively [3].

We define L²(I_n;Z) as the set of all strongly measurable functions υ : I_n → Z. We make ‖υ‖Z,n2=∫In‖υ(t)‖Z2dt\left\| \upsilon \right\|_{Z,n}^2 = \int\limits_{I_n } {\left\| {\upsilon \left( t \right)} \right\|_Z^2 dt}, where Z = X (or Y), I_n ⊆ J. When Z = H^s(Ω) is abbreviated as ∥∥_S,n, we assume that C(I¯n;Z)C\left( {\overline I _n ;Z} \right) is the set of all continuous functions υ : I_n → Z. We make ‖υ‖Z,n2=max⁡tn≤t≤tn+1‖υ(t)‖Z\left\| \upsilon \right\|_{Z,n}^2 = \max _{tn} \le t \le t^{n + 1} \left\| {\upsilon \left( t \right)} \right\|_Z, where Z = X (or Y) I_n ⊆ J.

For s,m = 0,1,⋯ and υ ∈ H^m(Ω), we define the discrete norm ‖hnsυ‖m,h=(∑r∈Thnhτ2s‖υ‖m,τ2)12\left\| {h_n^s \upsilon } \right\|_{m,h} = \left( {\sum\nolimits_{r \in T_h^n } {h_\tau ^{2s} } \left\| \upsilon \right\|_{m,\tau }^2 } \right)^{{1 \over 2}}. Let u(0,x) = u₀(x) have the following equation: (2) −∫J(u,φt)Ydt+∫JA(u,φ)dt=(u,φ(0))Y+∫J(f,φ)Ydt|A(v,ω)|≤c∥v∥x∥ω∥x,∀v,ω∈XA(v,ω)≥c∥v∥X2,∀v∈X; \begin{gathered} -\int_{J}\left(u, \phi_{t}\right) Y d t+\int_{J} A(u, \phi) d t=(u, \phi(0)) Y+\int_{J}(f, \phi) Y d t \\ |A(v, \omega)| \leq c\|v\| x\|\omega\| x, \forall v, \omega \in X \\ A(v, \omega) \geq c\|v\|_{X}^{2}, \forall v \in X ; \end{gathered}

Space D(J;C^∞) := {ϕ : ϕ ∈ C¹(J;C^∞(Ω)) and ϕ(T ) = 0}.

We define the finite element space (3) Shn={ψ∈X:ψ|τ∈Pr(τ),∀r∈Thn} S_{h}^{n}=\left\{\psi \in X:\left.\psi\right|_{\tau} \in P_{r}(\tau), \forall_{r} \in T_{h}^{n}\right\} (4) Vhk={φ:φ|Qn=∑j=0q−1tjψj(x),∀ψj∈Shn} V_{h k}=\left\{\phi:\left.\phi\right|_{Q n}=\sum_{j=0}^{q-1} t^{j} \psi_{j}(x), \forall \psi_{j} \in S_{h}^{n}\right\} (5) Vhkn={φ|Qn:φ∈Vhk} V_{h k}^{n}=\left\{\left.\phi\right|_{Q n}: \phi \in V_{h k}\right\}

by replacing u in the first term at the left end of the weak form (2) with a function U ∈ V_hk. The article is divided into points on each time slice I_n to get (6) −∫J(U,φt)Ydt=−∑n=0N−1((U,φ)Y∣t+n+1−∫In(Ut,φ)Ydt)=∫J(Ut,φ)Ydt+∑n=1N−1([Un],φn)Y+(U+0,φ0)Y -\int_{J}\left(U, \phi_{t}\right)_{Y} d t=-\sum_{n=0}^{N-1}\left((U, \phi)_{Y} \mid t_{+}^{n+1}-\int_{I_{n}}\left(U_{t}, \phi\right)_{Y} d t\right)=\int_{J}\left(U_{t}, \phi\right)_{Y} d t+\sum_{n=1}^{N-1}\left(\left[U^{n}\right], \phi^{n}\right)_{Y}+\left(U_{+}^{0}, \phi^{0}\right)_{Y}

where ϕⁿ = ϕ(·,tⁿ) ∈ D(J;C^∞). U±n=lims→0±U(x,tn+s)(0≤n≤N−1)⋅[Un]=U±n−U−nU_{\pm}^{n}=\lim _{s \rightarrow 0 \pm} U\left(x, t^{n}+s\right)(0 \leq n \leq N-1) \cdot\left[U^{n}\right]=U_{\pm}^{n}-U_{-}^{n}. (7) {∫J[(Ut,φ)Y+A(U,φ)]dt+∑n=1N−1([Un],φ+n)Y+(U±n,φ+n)Y=(u0,φ+n)Y+∫J(f,φ)Ydt,∀φ∈VhknU0=u0,n=0,1,⋯,N−1 \left\{\begin{array}{l} \int_{J}\left[\left(U_{t}, \phi\right)_{Y}+A(U, \phi)\right] d t+\sum_{n=1}^{N-1}\left(\left[U^{n}\right], \phi_{+}^{n}\right)_{Y}+\left(U_{\pm}^{n}, \phi_{+}^{n}\right)_{Y}=\left(u_{0}, \phi_{+}^{n}\right)_{Y}+\int_{J}(f, \phi)_{Y} d t, \forall \phi \in V_{h k}^{n} \\ U^{0}=u_{0}, n=0,1, \cdots, N-1 \end{array}\right.

Also, note that values of φ∈Vhkn\phi \in V_{h k}^{n} are independent of each other in each subinterval I_n(0 ≤ n ≤ N − 1); so, in subinterval I_n = (tⁿ,tⁿ⁺¹), correspondingly (8) ∫In[(Ut,φ)Y+A(U,φ)]dt+([Un],φ+n)Y=∫In(f,φ)Ydt \int_{I_{n}}\left[\left(U_{t}, \phi\right)_{Y}+A(U, \phi)\right] d t+\left(\left[U^{n}\right], \phi_{+}^{n}\right)_{Y}=\int_{I_{n}}(f, \phi)_{Y} d t

Further, the first item on the left end of the equal sign in Eq. (8) can be obtained after partial integration of time t to obtain the equivalent format (II), so that U ∈ V_hk is obtained for n = 0,1,⋯ ,N − 1 (9) (Un+1,φn+1)Y+∫In[−(U,φt)Y+A(U,φ)]dt=(Un,φ+n)Y+∫In(f,φ)Ydt \left(U^{n+1}, \phi^{n+1}\right)_{Y}+\int_{I_{n}}\left[-\left(U, \phi_{t}\right)_{Y}+A(U, \phi)\right] d t=\left(U^{n}, \phi_{+}^{n}\right)_{Y}+\int_{I_{n}}(f, \phi)_{Y} d t

This article assumes that there are K and L input indicators and output indicators for M cities’ input and output efficiencies, respectively. x_mk and y_ml(x_mk,y_ml > 0) respectively represent the standardised value of input and output of the kth and lth elements of tourism in the mth city. ξ is the non-Archimedean infinitesimal quantity. λ_m(λ_m ≥ 0) is the calculation weight variable. s⁻(s⁻ ≥ 0) is the slack variable, which represents the amount of input reduced when relative efficiency is achieved. s⁺(s⁺ ≥ 0) is the residual variable, which represents the increased output when effective. Therefore, the DEA model for measuring input–output efficiency of urban tourism is represented as follows [4]: (10) {min(θ−ε(∑k=1Ks−+∑l=1Ls+))s.t. ∑m=1Mxmkλm+s−=θxkmk=1,2,⋯,K∑m=1Mymlλm−s+=ylml=1,2,⋯,Lλm≥0m=1,2,⋯,M \left\{\begin{array}{l}\min \left(\theta-\varepsilon\left(\sum_{k=1}^{K} s^{-}+\sum_{l=1}^{L} s^{+}\right)\right) \\\text {s.t. } \sum_{m=1}^{M} x_{m k} \lambda_{m}+s^{-}=\theta x_{k}^{m} k=1,2, \cdots, K \\\sum_{m=1}^{M} y_{m l} \lambda_{m}-s^{+}=y_{l}^{m} l=1,2, \cdots, L \\\lambda_{m} \geq 0 m=1,2, \cdots, M\end{array}\right.

Eq. (10) is the DEA model under constant returns to scale (CRS). Assuming θ_m = 1, the tourism industry of the mth city is at the best production frontier. The overall efficiency of the city’s tourism input and output is the best. Conversely, θ_m < 0.5 indicates that the overall efficiency is invalid [5]. If 0.5 ≤ θ_m < 0.7 and 0.7 ≤ θ_m < 1, it indicates that the overall efficiency is medium and good, respectively. This model makes θ_m = θ_PTE × θ_SE (θ_PTE and θ_SE are both > θ_m). θ_PTE and θ_SE are the pure technical efficiency indices [6]. According to the research of Ma Zhanxin et al., we set θ_PTE (θ_SE) = 1 as the most efficient; 0.8 ≤ θ_PTE (θ_SE) < 1 implies good efficiency; 0.6 ≤ θ_PTE (θ_SE) < 0.8 indicates medium efficiency and θ_PTE (θ_SE) < 0.6 is invalid for efficiency. The principle of the model is to fit the original data through sampling and obtain its process situation. (11) {θ̂kb∗,b=1,2,⋯,B} \left\{\widehat{\theta}_{k b}^{*}, b=1,2, \cdots, B\right\}

In this paper, the value of B is taken as 1,000. This involves the confidence interval range of the city tourism efficiency measurement after correction [7]. To directly compare the changes in the correction measures of urban tourism comprehensive efficiency and its decomposition efficiency in different periods, we introduce the Malmquist total factor index model. Assuming that any time t is T, the mth city uses k types of the tourism input factor xmktx_{m k}^{t}. We obtain the first tourism output factor ymkty_{m k}^{t}. This article borrows the distance function concept of the reciprocal of technical efficiency proposed by American scholar Farrell to obtain the input distance function under technology L^t(y^t|C,S): (12) Dit(ymlt,xmkt)=1/Fit(ymlt,xmkt∣C,S) D_{i}^{t}\left(y_{m l}^{t}, x_{m k}^{t}\right)=1 / F_{i}^{t}\left(y_{m l}^{t}, x_{m k}^{t} \mid C, S\right)

This function is understood as the ratio of the input-output array (ymlt,xmkt)(y_{m l}^{t}, x_{m k}^{t}) of any city’s tourism industry close to the lowest input point in the general state [8]. Changes in technical efficiency from time t to t +1 are represented as follows: (13) Mit+1=Dit+1(xmkt,ymlt)/Dit+1(xmkt+1,ymlt+1) M_{i}^{t+1}=D_{i}^{t+1}\left(x_{m k}^{t}, y_{m l}^{t}\right) / D_{i}^{t+1}\left(x_{m k}^{t+1}, y_{m l}^{t+1}\right) (14) Mit=Dit(xmkt,ymlt)/Dit(xmkt+1,ymlt+1) M_{i}^{t}=D_{i}^{t}\left(x_{m k}^{t}, y_{m l}^{t}\right) / D_{i}^{t}\left(x_{m k}^{t+1}, y_{m l}^{t+1}\right)

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: (15) θTFP′=M0(xmkt+1,ymlt+1,xmkt,ymlt) \theta_{T F P}^{\prime}=M_{0}\left(x_{m k}^{t+1}, y_{m l}^{t+1}, x_{m k}^{t}, y_{m l}^{t}\right)

It is generally believed that each efficiency index >1 indicates that each efficiency is improved. If it is ≤ 1, there is no change or decrease in efficiency.

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.