Nonlinear Dissipative System Mathematical Equations in the Multi-regression Model of Information-based Teaching

Dissipative structure theory was first used to study non-equilibrium thermodynamics. Then, non-equilibrium statistical physics is used to reveal the relevant characteristics of complex systems with the in-depth study of its scientific and philosophical significance by scholars of dissipative system theory. The theory of dissipative system has gradually developed into a theoretical system of universal significance and applied to various fields of natural and social research. An essential point of this theory is the self-organising nature of the system [2]. The entire system may undergo abrupt changes through fluctuations. This new system structure needs to continuously exchange material and energy with the outside world to maintain its orderly structure and, at the same time, have a particular ability to resist external interference. The structure with this property is named ‘dissipative structure’ by Prigogine.

The dissipative stochastic nonlinear Schrodinger equation is as follows: Consider the following one-dimensional dissipative stochastic nonlinear Schrodinger equation with additive noise (1) {du−i(Δu+iau+λ|u|2u)dt=εdWu(t,0)=u(t,1)=0, t≥0, x∈[0,1]⊂Ru(0,x)=u0(x) \left\{ {\matrix{ {du - i(\Delta u + iau + \lambda |u{|^2}u)dt = \varepsilon dW} \hfill \cr {u(t,0) = u(t,1) = 0,\quad t \ge 0,\quad x \in [0,1] \subset Ru(0,x) = {u_0}(x)} \hfill \cr } } \right. Here λ = ±1, the dissipation factor a > 0,ɛ is the noise scale. The complex-valued Wiener process W = W₁ + iW₂ is defined in the filter probability space (Ω,F,{F_t}_t≥0,P) and W has the following Karhunen-Loève expansion (2) W(t,x)=∑k=0∞Qek(x)βk(t)=∑k=0∞ηkek(x)βk(t), t≥0, x∈[0,1] W(t,x) = \sum\limits_{k = 0}^\infty Q{e_k}(x){\beta _k}(t) = \sum\limits_{k = 0}^\infty \sqrt {{\eta _k}} {e_k}(x){\beta _k}(t),\quad t \ge 0,\quad x \in [0,1] Among them Q is the linear positive definite operator on L² = L²(0,1) and it is commutative with the Δ operator. It satisfies that Qek=ηkek,{ek}k≥1 Q{e_k} = \sqrt {{\eta _k}} {e_k},\{ {e_k}{\} _{k \ge 1}} is the eigenvector of the homogeneous Δ operator. βk=βk1+iβk2 {\beta _k} = \beta _k^1 + i\beta _k^2 and {βki}k≥1,i=1,2 {\{ \beta _k^i\} _{k \ge 1,i = 1,2}} is a family of independent and identically distributed real-valued standard Brownian motions. We modify Equation (1) as follows. Define space-time white noise χ˙=dWdt \dot \chi = {{dW} \over {dt}} , set u = p + iq, χ˙=χ˙1+iχ˙2 \dot \chi = {\dot \chi _1} + i{\dot \chi _2} , and p, q, χ˙1=dW1dt {\dot \chi _1} = {{d{W_1}} \over {dt}} and χ˙2=dW2dt {\dot \chi _2} = {{d{W_2}} \over {dt}} are real-valued random processes, then Equation (1) can be rewritten as (3) {pt+qxx+ap+λ(p2+q2)q=εχ˙1−qt+pxx−aq+λ(p2+q2)p=εχ˙2 \left\{ {\matrix{ {{p_t} + {q_{xx}} + ap + \lambda ({p^2} + {q^2})q = \varepsilon {{\dot \chi }_1}} \hfill \cr { - {q_t} + {p_{xx}} - aq + \lambda ({p^2} + {q^2})p = \varepsilon {{\dot \chi }_2}} \hfill \cr } } \right. Then using v = p_x, w = q_x, z = (p,q,v,w)^T the above equation can be transformed into a compact form (4) Mdtz+K∂xzdt=−aMzdt+∇S0(z)dt+∇S1(z)∘dW1+∇S2(z)∘dW2 M{d_t}z + K{\partial _x}zdt = - aMzdt + \nabla {S_0}(z)dt + \nabla {S_1}(z) \circ d{W_1} + \nabla {S_2}(z) \circ d{W_2} Among them (5) M=(0−100100000000000), K=(00100001−10000−100) M = \left( {\matrix{ 0 & { - 1} & 0 & 0 \cr 1 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 \cr } } \right),\quad K = \left( {\matrix{ 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 1 \cr { - 1} & 0 & 0 & 0 \cr 0 & { - 1} & 0 & 0 \cr } } \right) (6) S0(z)=−λ4(p2+q2)2−12(v2+w2), S1(z)=εq, S2(z)=−εp {S_0}(z) = - {\lambda \over 4}{({p^2} + {q^2})^2} - {1 \over 2}({v^2} + {w^2}),\quad {S_1}(z) = \varepsilon q,\quad {S_2}(z) = - \varepsilon p S_i(z) means that the equation is established under the meaning of Stratonovich-type stochastic integral.

Informatisation teaching ecology is defined as the sum of the processes [3]. The research process of information-based teaching ecology and the development process of teaching application under an information technology environment reciprocate to each other. To develop teaching ecology with the help of information technology, researchers study the renewal and iteration of information teaching ecology from the physical environment and humanistic environment of classroom teaching. According to the change of teaching equipment, four stages of the development of the teaching environment are proposed: the first stage represented by blackboard or writing whiteboard, the second stage is the computer combined with projection template, the third stage is the computer combined with interactive electronic whiteboard and the fourth is the computer combined with multiple Tap of the touch screen. Scholars took the star C teaching application to state the transformation of teaching resources, teaching processes and teaching methods under the support of the fourth-generation teaching environment.

Informatisation classroom is the venue for the development of informatisation teaching. It is the principal place for teacher-student interaction and student-student interaction. In the information classroom space, teachers use students as the main object to transform the object. The cultivation of students in the information classroom is an obvious function of the information teaching ecology. Its essence is a process of objectification of the subject. In the classroom teaching ecological layer, teachers transform the material flow, energy flow and information flow in the internal and external environment into the knowledge flow, ability flow and value flow of students through multiple teaching activities [5]. The teacher community teaching is based on a particular organisational environment and organisational activities and comprises teachers with different knowledge structures, teaching abilities, thinking styles and behaviour styles. Community is the group resonance of teachers. This is an effective form of professional learning, communication, reflection and a way for teachers’ collective emotions to exist.

The intelligent classroom and the teacher community are mutually interdependent and affect each other. Moreover, the dual development of subject and object in the process of ecological practice is realised through the dual integration of ‘teaching’ and ‘research’. A smart classroom is the basis of activities of the teacher community. Teachers mainly use teaching activities to understand and promote students’ cognition, thinking and physical and mental development. Informatised classroom is the primary manifestation of teachers’ professional values, teaching ideas and teaching skills, and it is also the source of teachers’ professional reflections [6]. The teacher community outside the classroom provides teachers with a flow of information and energy for professional growth.

On the other hand, teachers in the classroom provide a steady stream of negative entropy for the growth and development of students. Students are the main targets of teachers’ practical activities and provide primary problem streams for teachers’ professional development and provide spiritual support for professional growth. It can be seen that the cultivation of students and the development of teachers are mutually the input and output, forming a dual power system.

Each organisation unit represents different professional forces in the cooperation and division of labour to form a community to construct an informatisation teaching ecology. As the supporting system of the informatisation teaching ecology, the external ecological environment has a complex and overlapping influence on the development and evolution of the ecology [7]. This is considered as the information technology material environment, information teaching operation and maintenance environment, policy and system environment for ecological development and the professional information and energy for the ecological input of teaching behaviour evaluation and teaching research through professional exchanges. Thus, the internal and external environments of the information-based teaching ecology form a broader material and humanistic ecological circle and professionals from various systems integrate to form a mutually beneficial and symbiotic ecological relationship.

The positive entropy index system of information teaching ecology mainly includes three elements of mutual influence among smart classrooms, teacher community and subsystems, as shown in Figure 2.

Fig. 2

The negative entropy index system of information teaching ecology is constructed from the three elements of information technology material environment, information teaching operation and maintenance environment and policy and institutional environment. The details are shown in Figure 3.

Fig. 3

The thesis uses the Brussels model to quantitatively analyse the dynamic evolution of the dissipative structure of the information teaching ecology. Usually, the mathematical conditions are solved to get the dynamic critical value condition for forming the dissipative structure: the negative entropy B of the system is more significant than 1 + A2. This article escapes the model based on the characteristics of the information teaching ecology [9]. We set the positive entropy of the information teaching ecology to be equal to A and the negative entropy to be equal to B. It can be confirmed that the basis for discriminating the dissipative structure of the information-based teaching ecology is: (7) |B|={<1+A2,Non−dissipative structure=1+A2,Critical structure>1+A2,Dissipative structure |B| = \left\{ {\matrix{ { < 1 + {A^2},{\rm{Non}} - {\rm{dissipative\;structure}}} \hfill \cr { = 1 + {A^2},{\rm{Critical\;structure}}} \hfill \cr { > 1 + {A^2},{\rm{Dissipative\;structure}}} \hfill \cr } } \right. Equation (7) shows that only when the negative entropy of the information-based teaching ecology is large, and the positive entropy is small, the dissipative structure will be formed to promote the continuous evolution of the information-based teaching ecology.

First, we can calculate the entropy value of an index system with m variable layers and n evaluation levels according to Boltzmann's formula. The formula for calculating the entropy value of the i evaluation index is as follows: (8) ei=k∑j=1npijlnpij {e_i} = k\sum\limits_{j = 1}^n {p_{ij}}\ln {p_{ij}} Among them, k is a constant, usually k_A = −1/ln n is used for the calculation of positive entropy and k_B = 2/ln n ○ p_ij is used for calculation of negative entropy. From formula (8), it can be concluded that the value range of e_Ai is usually [0,1]. e_Ai The closer it is to 0, the more stable the internal structure of the ecology and the opposite, the more disorderly. The value range of e_Bi is [−2,0]. When e_Bi is closer to −2, it indicates that the external ecological environment contributes to the ordering of the ecological interior and vice versa. Secondly, the entropy weight of the i evaluation index can be calculated according to the entropy value of the index. The entropy weight calculation formulas of the i evaluation index in positive and negative entropy are shown in formulas (9) and (10), respectively. (9) ωAi=1−eAim−∑i=1meAi \omega {A_i} = {{1 - {e_{Ai}}} \over {m - \sum\limits_{i = 1}^m {e_{Ai}}}} (10) ωBi=2+eBi2m+∑i=1meBi \omega {B_i} = {{2 + {e_{Bi}}} \over {2m + \sum\limits_{i = 1}^m {e_{Bi}}}} Further, the positive entropy value E_A and negative entropy value E_B of the information teaching ecology can be obtained based on the overall entropy value calculation formula (10) of the index system. (11) E=∑i=1meiωi E = \sum\limits_{i = 1}^m {e_i}{\omega _i}

This article selects 41 digital demonstration schools in H Province as samples for empirical analysis. Based on the informatisation teaching ecological indicator system, two sets of questionnaires for schools and teachers are compiled. It is further divided into five grades n_A = n_B = 5 based on the results of the questionnaire. At the same time, m_A = 12, m_B = 16 can be known from the evaluation index system. Based on formulas (8)–(11), the information teaching ecology's positive and negative entropy values are obtained. The details are shown in Tables 1 and 2.

It can be seen from the table that the total entropy of the informatisation teaching ecology of 41 digital demonstration schools in H province indicates that the overall entropy is in a relatively orderly state. Thus, the existing external environment of the information teaching ecology promotes the continuous ordering of smart classrooms, teacher communities and their influence. At the same time, |B| = 0.8811 < 1 + A² = 1.1938 can be known according to the discriminant formula of the dissipative structure. This shows that the current information teaching ecology is not in a state of the dissipative structure. That is, it has failed to cause a change in the information teaching ecology.

From the above results, it can be seen that the average entropy value of the positive entropy index of the informatisation teaching ecology is 0.4946 [10]. On the other hand, the relatively high entropy values of teaching model innovation (0.7060), teaching demonstration and training (0.7220) and online information sharing (0.6673) indicate that there is still an imbalance in the development of teaching model innovation, demonstration and guidance between schools and teachers. This requires the further improvement and promotion of teaching innovation model exploration and demonstration leading practice. This is also a vital issue in the realisation of educational informatisation from 1.0 to 2.0.

On the other hand, in the negative entropy index of the information technology teaching ecology, the absolute values of the negative entropy of the campus network construction (−0.6133), data centre management (−0.7946) and teacher-student electronic terminal (−0.7969) are low in the material environment elements of information technology, while the values of digital teaching space (−1.6168) and innovation space (−1.7051) are relatively high. This is because the development of essential equipment such as networks, data centres and terminals in each school is more consistent and has reached the requirements of digital campus construction. However, there are differences in constructing a more prosperous digital environment, and each school can make breakthroughs in the information infrastructure. This provides a particular foundation for the reform of the information teaching ecology [11]. The relatively high absolute value of negative entropy of related indicators of information-based teaching operation and maintenance environment elements can effectively inject entropy flow into the information-based teaching ecology. On the other hand, the absolute value of negative entropy of the overall indicators is low in terms of policy and institutional environmental elements, especially the guarantee and incentive system (−0.2422) indicator. At present, each unit has the same level of school management rules and regulations, teaching incentive policies and resource co-construction and sharing mechanisms. Be assured. However, formulating an interconnected system for the overall development of the school that is suitable for education informatisation 2.0 needs to be innovated and explored.

The reshaping of the teaching ecology is a comprehensive adjustment process of human resources, materials, systems and information. Note that the positioning of the value of organisational philosophy is first reflected in the organisational structure. So implement decentralisation and proper distributed management on management decision-making power. Exploratory problems often require group decision-making and the participation of all teachers can significantly atimulate and enhance the initiative and enthusiasm of teachers [12]. The transformation of management from administrative order to service management requires corresponding structural changes. Build intercommunication channels conducive to the efficient transmission of information while maintaining the flat and simplified organisational structure and enhancing the organisation's collaboration characteristics and decision-making benefits. This is conducive to the rapid deployment of school human resources and materials and promotes the flow and effective conversion of information and energy in the ecology.

Innovative organisations are organisations that can guide and encourage innovative behaviours. First of all, cultivating an organisational atmosphere for expressing different opinions and tolerating new ideas and new ideas should be done. It requires a relatively loose mechanism environment and a democratic and equal decision-making mechanism to ensure the participation of teachers in learning and research. Secondly, it takes both operational guidance and active encouragement into consideration [13]. Finally, while launching a wealth of information-based teaching and research activities, it encourages innovative ideas and behaviours of informal organisations and pays attention to the psychological achievements of teachers.

Strengthen school-to-school exchanges, promote mutual learning of advanced educational management experience and explore suitable operating mechanisms between schools to form a cooperative model that can effectively implement, evaluate, feedback and adjust. Provide opportunities for in-depth exchanges between teachers through foreign cooperation, including class management, education and teaching, teaching, research and scientific research. Provide opportunities for in-depth exchanges between students through foreign cooperation, including mutual learning and mutual exchanges. It will bring more resources and opportunities to the respective schools through foreign cooperation, bringing advantages in school management, education, teaching, campus culture, research and scientific research, exchanges, and learning.