Fractional differential equations are one of the important contents of advanced mathematics courses. The article uses fractional differential equations to describe the effects of children’s innovative education courses. Through the qualitative analysis of the basic model, several conditions to ensure the effect of children’s innovative education courses are obtained. At the same time, combined with practical experience, the teaching curriculum case design analyzes the specific application of the fractional differential equation in the effect of children’s innovative education curriculum. Research has found that the fractional differential equation algorithm improves the efficiency of innovation.

#### Keywords

- Children’s innovative education
- Educational fees
- Differential equation model
- Educational cost-sharing

#### MSC 2010

- 34A08

Higher education fees are a major matter related to the interests of the country, higher education institutions, educated persons, and their families. Over the years, several educational theory scholars have conducted useful discussions on this from academic economics, sociology, and law perspectives and put forward some corresponding countermeasures [1]. In this article, our main purpose is to put forward two differential equation models for higher education fees. At the same time, we will further study the macro-control issues of higher education fees and enrollment scale.

We set the number of enrollment of colleges and universities N as a continuous variable, and the education charge of colleges and universities is

Where

Assuming that

In the same way, we can also obtain (2) the corresponding differential equation model (IV) describing the issue of college education charges

If

In this section, we mainly use qualitative analysis (2) and (4) to seek solutions to the problem of college education fees.

This shows that a sufficient and necessary condition for universities to keep their education fees unchanged is to balance the per capita education costs shared by individuals and the unpaid expenses _{0}, so that the number of college enrollment _{0} and the per capita education cost _{0} = _{0}) shared by individuals should be determined [5]. Therefore, under the condition of expanding the enrollment scale of colleges and universities, increasing government investment in education funds from policy measures, preferential incentives, and attracting social donations to run schools can solve the problem. At the same time, student loan scholarship, work-study, and other forms to solve the ability of poor students to pay are all solutions to achieve a balance.

(1) When _{0}) > 0,
_{0}) = 0, _{0}) > 0 colleges and universities reach a minimum. This shows that under the condition that the per capita education cost _{0}) of colleges and universities can be kept unchanged.

(2) When _{0}) < 0,
_{0}) = 0, _{0}) > 0 college education fees that reach a minimum. This shows that if the growth rate of the per capita unpaid expenses of poor students exceeds a degree
_{0}) of colleges and universities remain unchanged.

(3) When _{0}) > 0,
_{0}) = 0, _{0}) < 0 colleges and universities reach a maximum value. This shows that when the per capita education cost _{0}) unchanged.

(4) Under the condition _{0}) > 0,

This requires (6) there is a positive root _{0}, and there is also _{0} = _{0}). Because _{0} is the equilibrium point of (4), we have the following stability conclusions about education fees _{0} according to the stability judgment method.

(5) When
_{0}) > 0, _{0} is the stable equilibrium point of the model (4). This shows that under the condition that the per capita education cost _{0} of colleges and universities remain stable.

(6) When
_{0}) < 0, _{0} is also the stable equilibrium point of the model (4). This shows that if the growth rate of unpaid expenses per capita of poor students exceeds a degree
_{0} of colleges and universities remain stable.

(7) The school’s education fees _{0} are unstable under other conditions.

In free education (_{M}

Under the condition that individuals’ per capita education cost increases with the increase in the number of enrolled students, we set _{0}(

At _{0}. Because of 0 < _{M}

It can be seen that the adjustment of education fees according to (8) will be higher. If _{0}^{2} is changed to a quadratic function in a model (2), then it is easy to know that the scale of enrollment of colleges and universities whose education fees are positive is

It can be seen that when the government stipulates that the per capita education cost to be shared is lower (_{M}_{0}^{1+T} is a power function. Where _{0}, the per capita education cost-shared by individuals is _{0}. Then we have _{0}d^{1+T} ^{1+T}. Where

Model (10) has an unstable equilibrium point _{1} = 0 and a stable equilibrium point
_{2} > _{0} when
_{2} > _{0}. This shows that when individuals’ per capita education cost-shared exceeds one degree, the phenomenon of some students in arrears in tuition fees will reduce the stable education fees of colleges and universities.

We assume in the model (2) that
_{0}

Points (11) The relationship between education fees and enrollment is logarithmic

Because of
_{M}_{0}^{2} is assumed in model (2), then it is easy to know that the number of students enrolled in universities to maintain an increase in education fees is

And the education fees of colleges and universities are

When

The formula (15) shows that the enrollment of colleges and universities must reach a certain scale to avoid the result of school losses [9].

We assume
_{0}

Points (16) The relationship between education fees and enrollment is logarithmic

If _{0}^{2} is assumed in model (2), then it is easy to know that when _{0}

Therefore, under condition _{0}

Finally, we assume
_{0}^{2} in the model (4). If the education fee set by the government is _{0}, the per capita education cost-shared by individuals is _{0}, then we have _{1}

When _{0}_{1}, equation (19) has unstable education charges

According to different enrollment scales, colleges and universities consider adopting corresponding education charging models to mix two or more models [10]. We adjust the charging method of a college education according to the number of people

Universities in different (regions) all adopt the same charging model, and there is a certain mutual influence between them [11]. Then we can integrate more than two models. Consider the logistic model of second-order nonlinear coupling:

Due to the imbalance of economic and social development, different colleges and universities adopt different charging models, which leads to the unfair phenomenon of education fees and the scale of enrollment [12]. If the control variable

This article mainly proposes two basic differential equation models (2) and (4) to describe the issue of college education fees. By analyzing several specialcoun models (2) and (4) in three situations, some new conclusions and suggestions on macro-control of university education fees and enrollment scale are given.

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