Calculating university education model based on finite element fractional differential equations and macro-control analysis

Cost refers to the sum of all actual resources consumed to engage in an investment plan. Under the conditions of a market economy [6], the cost is a part of the value of commodities, materialised labour and living labour consumed in the production of commodities. With the transformation of the social-economic system and the transformation of people's educational concepts, the cost of education began to be involved when studying the economic benefits of educational investment, so we introduced the cost of economics into the field of education to use the concept of education cost [7]. Education cost refers to all the expenses paid by each student, including the direct and indirect expenses borne by the government, society, schools and educated individuals. The cost of higher education refers to the total cost of training each college student, that is, the sum of direct and indirect living labour and materialised labour consumed by students of various types of colleges and universities during their studies. Specifically, the cost of higher education includes the direct and indirect expenses paid by the government, society, schools and educated individuals to train students [8]. Therefore, the cost of higher education is the subordinate concept of the cost of higher education, that is, the total value of materialized labour and living labour consumed by each student in the stage of receiving higher education [9], and the value consumed to maintain personal life needs. The cost of higher education is the direct social cost of higher education, and it is the actual cost.

To conduct an in-depth analysis of the cost of a college education, many scholars have classified the cost of a college education. Since it is a classification, there must be a certain classification standard [10]. For example, according to the burden of the education cost or the source of education funding (social, personal), the nature or function of the education resource (recurring, capital), the use of the education resource, the material), the technical characteristics of the education resource (fixed, variable) And other standards for classification. Concerning the merits of various education cost classification methods, this research proposes the classification of college education costs. This classification conforms to the essential connotation of education cost and reflects the composition and its function. It should be noted that the opportunity cost here includes not only the income loss of capital assets but also the income loss of non-capital public funds and personal funds. In most of the literature, the educational opportunity cost does not include the above-mentioned non-capital public funds and personal capital loss. This research adopts the ideas of Han Zongli and Gao Jianmin and incorporates them into opportunity cost. It is because they are of the same nature as the loss of capital assets. Since the income loss of capital assets is included in the opportunity cost, they should also be included in the opportunity cost. You cannot simply add up school costs and personal costs to get the total financial cost, because there are also several duplication problems. They are mainly the duplication of tuition and miscellaneous fees and school costs in personal costs. The actual total financial cost is the addition of tuition and miscellaneous fees and the two [11].

How to reduce the waste of resources and improve the efficiency of resource utilisation based on controlling the cost in the process of running a university and ensuring the quality of talent training are the major problems facing the development of universities. However, the many problems in cost control in the process of running colleges and universities now highlight the necessity of scientific cost control.

For the sake of simplicity, under the condition that the per capita education cost shared by individuals increases with the increase in enrolment, we set γ(N) = a + bN and g(γ) = g₀γ as linear functions in model (2). So, we have γ (N) = b(−n + N), where b=γMM−n>0 b = {{{\gamma _M}} \over {M - n}} > 0 . After substituting (2), We get the following education fee model (7) R′(N)=b(δ−σg0)(−n+N) {R^\prime}\left( N \right) = b\left( {\delta - \sigma {g_0}} \right)\left( { - n + N} \right) It can be seen that when N > n, the equilibrium condition to keep the university's education fees unchanged is δ = σg₀; again, 0 < γ (N) ≤ γ_M knows, n < N ≤ M, so the size of the enrolment of colleges and universities can be expanded within the range specified by the government. Under unbalanced conditions, when g0<δσ,N>n {g_0} < {\delta \over \sigma },N > n , There is R′ > 0. This shows that when colleges and universities expand the scale of enrolment, only when the growth rate of unpaid fees per poor student is small (or large), can we consider raising (or lowering) education fees. Points (7), we the education charge of some colleges and universities is a function of the number of enrolments (8) R(N)=12b(δ−σg0)(−n+N)2 R\left( N \right) = {1 \over 2}b\left( {\delta - \sigma {g_0}} \right){\left( { - n + N} \right)^2} It can be seen that the adjustment of education fees according to (8) will be too high. If g(γ) = g₀γ² is changed to a quadratic function in model (2), then it is easy to know that the scale of enrolment of colleges and universities whose education fees are positive is (9) n<N≤n+3δ(M−n)2g0γM n < N \le n + {{3\delta \left( {M - n} \right)} \over {2{g_0}{\gamma _M}}} It can be seen that when the government stipulates that the per capita education cost is lower (γ_M → 0), it will help universities to expand the scale of autonomous enrolment. Below, we will conduct a qualitative analysis from model (4). Assume that λ = f (R) = c + dR is a linear function and g(γ) = g₀γ^1+T is a power function, where T > 0 Because in free education, there is R = 0,γ = 0; and in charging education, if the government-specified education fee is R₀, the per capita education cost shared by individuals is γ₀, then we have γ = dR, g(γ) = g₀d^1+T R^1+T, where d=γ0R0>0 d = {{{\gamma _0}} \over {{R_0}}} > 0 . After substituting (4), we Get the following education fee model (10) R′=δdR−σg0d1+TR1+T {R^\prime} = \delta dR - \sigma {g_0}{d^{1 + T}}{R^{1 + T}} Model (10) has an unstable equilibrium point R₁ = 0 (which corresponds to free) and a stable equilibrium point R2=1dδσg0T {R_2} = {1 \over d}\root T \of {{\delta \over {\sigma {g_0}}}} (which corresponds to stable charges). In particular, when T = 1 is taken, (10) is a Logistic model, and there is R2R0=δσg0γ0 {{{R_2}} \over {{R_0}}} = {\delta \over {\sigma {g_0}{\gamma _0}}} Therefore, when 0<γ0<δσg0 0 < {\gamma _0} < {\delta \over {\sigma {g_0}}} , there is R₂ > R₀, which shows that when the per capita education cost shared by individuals is within a certain limit, the stable education fees of colleges and universities will increase; when γ0>δσg0 {\gamma _0} > {\delta \over {\sigma {g_0}}} , there is R₂ < R₀, which shows that when the per capita share of the individual is When the cost of education exceeds one degree, the stable education fees of universities will be reduced due to the phenomenon of some students in arrears.

Under the condition that the per capita education cost shared by individuals decreases with the increase in the number of enrolled students, we assume γ(N)=bN \gamma \left( N \right) = {b \over N} and g(γ) = g₀γ in model (2). From the previous assumption, we have γ(N)=γMMn \gamma \left( N \right) = {\gamma _M}{M \over n} . After substituting (2), the following education charge model is obtained (11) R′(N)=(δ−σg0)MγMN {R^\prime}\left( N \right) = \left( {\delta - \sigma {g_0}} \right){{M{\gamma _M}} \over N} Points (11) have a logarithmic relationship between education fees and enrolment (12) R(N)=(δ−σg0)MγMlnNn R\left( N \right) = \left( {\delta - \sigma {g_0}} \right)M{\gamma _M}\ln {N \over n} Because of lnNn=o(N)=o(N2)(N→∞) \ln {N \over n} = o\left( N \right) = o\left( {{N^2}} \right)\left( {N \to \infty } \right) , the fee determined according to (12) is lower than that of (8), and the education fees of colleges and universities can increase slowly with the expansion of enrolment scale, and then from 0 < γ (N) ≤ γ_M, N ≥ M. It can be seen that the enrolment scale of colleges and universities can exceed the government's Regulations, and fees increase slowly, this is an education fee and expansion model worth promoting.

If g(γ) = g₀γ² is changed in model (2), then the number of enrolled students in universities that maintains the increase in education fees is (13) N≥σg0MγMδ N \ge {{\sigma {g_0}M{\gamma _M}} \over \delta } And the education fees of universities are (14) R(N)=δMγMlnNn−σg0M2γ2M(1n−1N) R\left( N \right) = \delta M{\gamma _M}\ln {N \over n} - \sigma {g_0}{M^2}{\gamma ^2}_M\left( {{1 \over n} - {1 \over N}} \right) When N → ∞, the approximate enrolment scale for which the college's education fees are positive is (15) N≥nexp[σg0δnMγM] N \ge n\exp \left[ {{{\sigma {g_0}} \over {\delta n}}M{\gamma _M}} \right] It is shown that the enrolment of colleges and universities must reach a certain scale to avoid the result of school losses.

Under the condition that the per capita education cost shared is a bounded function of the number of enrolment, considering the previous two situations, we assume γ(N)=bN+c1+N \gamma \left( N \right) = {{bN + c} \over {1 + N}} and g(γ) = g₀γ in model (2). Similarly, from the previous assumption, we have a monotonic bounded function γ(N)=b(−n+N)1+N \gamma \left( N \right) = {{b\left( { - n + N} \right)} \over {1 + N}} , where b=γM(1+M)M−n>0 b = {{{\gamma _M}\left( {1 + M} \right)} \over {M - n}} > 0 . After substituting (2), the following educational charging model is obtained (16) R′(N)=δb(−n+N)1+N−σg0b(−n+N)1+N {R^\prime}\left( N \right) = {{\delta b\left( { - n + N} \right)} \over {1 + N}} - {{\sigma {g_0}b\left( { - n + N} \right)} \over {1 + N}} Points (16) have a logarithmic relationship between education fees and enrolment (17) R(N)=(δ−σg0)b[N−n−(n+1)lnN+1n+1] R\left( N \right) = \left( {\delta - \sigma {g_0}} \right)b\left[ {N - n - \left( {n + 1} \right)\ln {{N + 1} \over {n + 1}}} \right] If g(γ) = g₀γ² is changed in model (2), then it is easy to know that when δ < σg₀b, the enrolment scale for universities to maintain their educational fee increase (18) n<N≤σg0b+δσg0b−δ n < N \le {{\sigma {g_0}b + \delta } \over {\sigma {g_0}b - \delta }} Therefore, under condition δ < σg₀b, colleges and universities should limit the scale of enrolment. Finally, we assume γ=f(R)=dR1+R,g(γ)=g0γ2 \gamma = f\left( R \right) = {{dR} \over {1 + R}},g\left( \gamma \right) = {g_0}{\gamma ^2} in model (4). If the government-specified education fee is R₀, the per capita education cost shared by individuals is γ₀, then we have γ = d₁R, where d1=1+R0R0γ0 {d_1} = {{1 + {R_0}} \over {{R_0}}}{\gamma _0} . Substituting (4), we get the following Logistic model of education charges (19) R′=δd1R−σg0d12R2 {R^\prime} = \delta {d_1}R - \sigma {g_0}d_1^2{R^2} It is easy to know that when δ < δg₀d₁, Eq. (19) has unstable education charges (20) R=δR0δR0(1+R0)−δR0 R = {{\delta {R_0}} \over {\delta {R_0}\left( {1 + {R_0}} \right) - \delta {R_0}}}

Imagine that colleges and universities consider adopting corresponding education charging models according to different enrolment numbers. Then, we can mix and connect two or more models to adjust the college education charging methods according to the number of people, such as (21) {R1′(N)=b(δ−σg0)(N−n)R2′(N)=(δ−σg0)MγMNR3′(N)=0,R2(N1)=R1(N1) \left\{ {\matrix{ {R_1^\prime\left( N \right) = b\left( {\delta - \sigma {g_0}} \right)\left( {N - n} \right)} \hfill \cr {R_2^\prime\left( N \right) = \left( {\delta - \sigma {g_0}} \right){{M{\gamma _M}} \over N}} \hfill \cr {R_3^\prime\left( N \right) = 0,{R_2}\left( {{N_1}} \right) = {R_1}\left( {{N_1}} \right)} \hfill \cr } } \right.

Imagine that colleges and universities in different (regions) all adopt the same charging model, and there is a certain mutual influence between them, then we can integrate more than two models, such as the Logistic model that mainly considers the second-order nonlinear coupling. Study the synchronisation of the scale of enrolment and charging standards in different (regions) universities. (22) {R1′=δd1R1−σg0d12R22R2′=δd1R2−σg0d12R12R1(N0)=R10,R2(N0)=R20, \left\{ {\matrix{ {R_1^\prime = \delta {d_1}{R_1} - \sigma {g_0}d_1^2R_2^2} \hfill \cr {R_2^\prime = \delta {d_1}{R_2} - \sigma {g_0}d_1^2R_1^2} \hfill \cr {{R_1}\left( {{N_0}} \right) = {R_{10}},{R_2}\left( {{N_0}} \right) = {R_{20}},} \hfill \cr } } \right.

Due to the imbalance of economic and social development, different colleges and universities adopt different charging models, which has led to the unfair phenomenon of education fees and the scale of enrolment. To promote education fairness and build a harmonious society, the government needs to be If we introduce the control variable a(N) that changes with the number of students enrolled, and consider the linear feedback control between two different models, then we can get a three-dimensional differential equation model. For example, (23) {R1′=δd1R1−σg0aR1′=δd2R2−σg0d22R12a′=e1R1+e2R2R1(N0)=R10,R2(N0)=R20,a(N0)=a0 \left\{ {\matrix{ {R_1^\prime = \delta {d_1}{R_1} - \sigma {g_0}a} \hfill \cr {R_1^\prime = \delta {d_2}{R_2} - \sigma {g_0}d_2^2R_1^2} \hfill \cr {{a^\prime} = {e_1}{R_1} + {e_2}{R_2}} \hfill \cr {{R_1}\left( {{N_0}} \right) = {R_{10}},{R_2}\left( {{N_0}} \right) = {R_{20}},a\left( {{N_0}} \right) = {a_0}} \hfill \cr } } \right. In turn, we can in-depth study the dynamics of the bifurcation and chaos of different (regional) colleges and universities’ education fees and enrolment scale.