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The Homework Model of Screening English Teaching Courses Based on Fractional Differential Equations

Data publikacji: 15 Jul 2022
Tom & Zeszyt: AHEAD OF PRINT
Zakres stron: -
Otrzymano: 19 Feb 2022
Przyjęty: 23 Apr 2022
Informacje o czasopiśmie
License
Format
Czasopismo
eISSN
2444-8656
Pierwsze wydanie
01 Jan 2016
Częstotliwość wydawania
2 razy w roku
Języki
Angielski
Introduction

Informatization changes the learning mode of students and changes the teaching mode of teachers. The communication content between teachers and students includes discussions, courseware, assignments, etc. Most of this work is done online. The convenience of obtaining network resources brings certain benefits to teaching work and increases the difficulty for teachers to evaluate students' learning effects. Teachers are increasingly relying on information to solve the problems encountered in the traditional teaching process, such as plagiarism checks and automatic evaluation of English teaching coursework [1]. The network collects students' learning behavior data and conducts statistical analysis through a reasonable model to provide a reasonable basis for plagiarism inspection and automatic evaluation of English teaching coursework. This method has become an important subject of teaching management in colleges and universities.

Plagiarism detection technology is widely used in the field of intellectual property protection. Howie's academic misconduct detection system relies on its huge literature database to detect plagiarism. The ROST anti-plagiarism system of Wuhan University provides the service of judging the similarity between the papers under investigation and the existing literature. Open source plagiarism detection platforms such as Plagiarisma abroad provide plagiarism detection services through the support of search engines.

In the study of the online learning environment by Underwood et al., we found that the sources of student plagiarism are mainly local and global. If all or part of a student's English coursework or dissertation is taken from outside the learning community, this type of plagiarism is called global plagiarism [2]. In global plagiarism, the sources of plagiarism are relatively wide. The content includes the Internet, academic monographs, paper publications, P2P networks, etc. Many commercial and non-commercial systems provide global plagiarism detection services. One of the most famous systems is the Turnitin system developed by the American IPA plagiarism company. The current international mainstream e-learning system. Blackboard, WebCT, ANGEL, and Moodle, etc., are all directly using the Turnitin system of the plug-ins provided by it.

According to the time axis relationship, most of the existing plagiarism detection systems determine the plagiarism relationship [3]. The plagiarist between two papers with a high degree of similarity must be a later publication date. However, there is no necessary relationship between the time sequence of students submitting assignments and the plagiarism of students' assignments, so the relationship of the time axis cannot judge the plagiarism relationship of students' assignments. The homework plagiarism detection module of LMS2.0 can only judge the set of homework that may have a plagiarism relationship. The system cannot determine who is a plagiarist and who is a plagiarized person in this collection. Some scholars have analyzed thesis-type assignments and found a certain relationship between students' usual study habits and their attitude toward assignments. Students who study seriously also take homework relatively seriously. In this study, various learning behavior data of students were recorded in LMS3.0 utilizing the fractional differential equation method [4]. We designed an English online homework-assisted evaluation model based on fractional differential equations based on the original plagiarism detection module. We use this model to solve the problem of determining the plagiarism relationship of students' homework.

Review the design of the model

The homework-assisted evaluation and correction model based on LMS3.0 is shown in Figure 1. We divide the determination of plagiarism detection into three stages. The first stage is job similarity determination. We use the similarity detection fractional differentiation algorithm to find assignments whose similarity threshold reaches a certain level from assignments submitted by students [5]. The second stage is through the hierarchical clustering method. We cluster assignments with a threshold of similarity to a certain degree according to the source of plagiarism. Output a list of assignments from the same plagiarized source and their corresponding student lists. The third stage assigns different weights to the students' learning behavior and learning result data collected by LMS3.0 according to their reliability and importance. In this way, a learning evaluation weight vector is formed [6]. We use the analytic hierarchy process to calculate the evaluation value of the students in the student list output in the previous step on each evaluation index. This results in a composite score sheet for all student learning using the same plagiarism source. The calculation results provide a basis for judging plagiarists.

Figure 1

Architecture diagram of the homework-assisted assessment model

Fractional differential equation model establishment

This paper uses variational methods to study the structure of the following fractional differential equations with instantaneous and non-instantaneous impulses: {ddt(120Dtβu(t)+12tDt1βu(t))=f1(t,u(t)),t(0,t1]ddt(12s1Dtβu(t)+12tDTβu(t))=f2(t,u(t)),t(s1,T]12(tDs1βu)|t=t1+12(0Dtβu)|t=t1=I(u(t1))t1Dtβu(t)+tDs1βu(t)=(tDs1βu)|t=t1+,t(t1,s1](t1Dtβu|t=s1=(t1DTβu|t=s1+au(0)b2(tDt1βu)|t=0=0,cu(T)+d2(s1Dtβu)|t=T=0 \left\{ {\matrix{ { - {d \over {dt}}\left( {{1 \over 2}0D_t^{ - \beta }u^\prime \left( t \right) + {1 \over 2}tD_{{t_1}}^{ - \beta }u^\prime \left( t \right)} \right) = {f_1}\left( {t,u\left( t \right)} \right),t \in (0,{t_1}]} \hfill \cr { - {d \over {dt}}\left( {{1 \over 2}{s_1}D_t^{ - \beta }u^\prime \left( t \right) + {1 \over 2}tD_T^{ - \beta }u^\prime \left( t \right)} \right) = {f_2}\left( {t,u\left( t \right)} \right),t \in ({s_1},T]} \hfill \cr {{1 \over 2}\left( {_tD_{{s_1}}^{ - \beta }u^\prime } \right){{\left| {_{t = t_1^ + } - {1 \over 2}\left( {_0D_t^{ - \beta }u^\prime } \right)} \right|}_{t = t_1^ - }} = I\left( {u\left( {{t_1}} \right)} \right)} \hfill \cr {{{\left. {{t_1}D_t^{ - \beta }u^\prime \left( t \right){ + _t}D_{{s_1}}^{ - \beta }u^\prime \left( t \right) = \left( {_tD_{{s_1}}^{ - \beta }u^\prime } \right)} \right|}_{t = t_1^ + }},\,t \in ({t_1},{s_1}]} \hfill \cr {\left( {_{{t_1}}D_t^{ - \beta }u^\prime } \right.{{\left| {_{t = s_1^ - } = \left( {_{{t_1}}D_T^{ - \beta }u^\prime } \right.} \right|}_{t = s_1^ + }}} \hfill \cr {au\left( 0 \right) - {b \over 2}\left( {_tD_{{t_1}}^{ - \beta }u^\prime } \right){{\left| {_{t = 0} = 0,\,cu\left( T \right) + {d \over 2}\left( {_{{s_1}}D_t^{ - \beta }u^\prime } \right)} \right|}_{t = T}} = 0} \hfill \cr } } \right. Where 0Dtβ _0D_t^{ - \beta } , t1Dtβ _{{t_1}}D_t^{ - \beta } and s1Dtβ _{{s_1}}D_t^{ - \beta } are left Riemann-Liouville type fractional integrals. tDt1β _tD_{{t_1}}^{ - \beta } , tDs1β _tD_{{s_1}}^{ - \beta } and tDTβ _tD_T^{ - \beta } are fractional integrals of the right Riemann-Liouville type. 0 ≤ β < 1, a, c > 0, b, d ≥ 0, b, d ≥ 0, b, d ≥ 0, IC(R, R), f1C((0, t1] × R, R), f2C((s1, T] × R, R). If β = 0, b = d = 0, problem (1) degenerates into the following standard second-order differential equation boundary value problem with transient impulses, non-instantaneous impulses, and Dirichlet boundary conditions: {u(t)=f1(t,u(t)),t(0,t1]u(t)=f2(t,u(t)),t(s1,T]u(t)=u(t1+),t(t1,s1]u(s1)=u(s1+)u(0)=u(T)=0 \left\{ {\matrix{ { - u^{''}\left( t \right)\, = {f_1}\left( {t,\,u\left( t \right)} \right),\,t\, \in (0,\,{t_1}]} \hfill \cr { - u^{''}\left( t \right)\, = {f_2}\left( {t,\,u\left( t \right)} \right),\,t\, \in ({s_1},T]} \hfill \cr {u^\prime \left( t \right) = u^\prime \left( {t_1^ + } \right),\,t \in ({t_1},\,{s_1}]} \hfill \cr {u^\prime \left( {s_1^ - } \right) = u^\prime \left( {s_1^ + } \right)} \hfill \cr {u\left( 0 \right) = u\left( T \right) = 0} \hfill \cr } } \right.

Definition 2.1 (left and right Riemann-Liouville type fractional integrals)

Suppose f is a function defined on [a, b] and 0 < γ < 1. aDtγ _aD_t^{ - \gamma } is. represents the left Riemann-Liouville type fractional integral. Defined as follows: aDtγf(t)=1Γ(γ)at(ts)γ1f(s)ds,t[a,b] _aD_t^{ - \gamma }f\left( t \right) = {1 \over {\Gamma \left( \gamma \right)}}\int_a^t {{{\left( {t - s} \right)}^{\gamma - 1}}f\left( s \right)ds,t \in \left[ {a,\,b} \right]}

The Riemann-Liouville type fractional integral is denoted by tDbγ _tD_b^{ - \gamma } . It is defined as follows: tDbγf(t)=1Γ(γ)tb(st)γ1f(s)ds,t[a,b] _tD_b^{ - \gamma }f\left( t \right) = {1 \over {\Gamma \left( \gamma \right)}}\int_t^b {{{\left( {s - t} \right)}^{\gamma - 1}}f\left( s \right)ds,t \in \left[ {a,\,b} \right]} Where Γ > 0 is the classical gamma function.

Definition 2.2 (Left and Right Riemann-Liouville Type Fractional Differentiation)

We assume that f is a function defined on [a, b] and 0 < γ < 1. aDtγ _aD_t^{ - \gamma } . represents the left Riemann-Liouville type fractional derivative. It is defined as follows: aDtγf(t)=ddtaDtγ1f(t)=1Γ(1γ)ddt(at(ts)γf(s)ds),t[a,b] _aD_t^{ - \gamma }f\left( t \right) = {d \over {dt}}{\;_a}D_t^{ - \gamma - 1}f\left( t \right) = {1 \over {\Gamma \left( {1 - \gamma } \right)}}{d \over {dt}}\left( {\int_a^t {{{\left( {t - s} \right)}^\gamma }f\left( s \right)} ds} \right),t \in \left[ {a,\,b} \right] tDbγf(t)=ddttDbγ1f(t)=1Γ(1γ)ddt(tb(st)γf(s)ds),t[a,b] _tD_b^\gamma f\left( t \right) = - {d \over {dt}}{\;_t}D_b^{\gamma - 1}f\left( t \right) = - {1 \over {\Gamma \left( {1 - \gamma } \right)}}{d \over {dt}}\left( {\int_t^b {{{\left( {s - t} \right)}^{ - \gamma }}f\left( s \right)} ds} \right),t \in \left[ {a,\,b} \right] acDtγf(t)=aDtγ1f(t)=1Γ(1γ)(at(ts)γf(t)ds),t[a,b] _a^cD_t^\gamma f\left( t \right) = {\;_a}D_t^{\gamma - 1}f^\prime \left( t \right) = {1 \over {\Gamma \left( {1 - \gamma } \right)}}\left( {\int_a^t {{{\left( {t - s} \right)}^{ - \gamma }}f^\prime \left( t \right)} ds} \right),t \in \left[ {a,\,b} \right]

The right Caputo-type fractional derivative is defined as follows: tcDbγf(t)=tDbγ1f(t)=1Γ(1γ)(tb(st)γf(t)ds),t[a,b] _t^cD_b^\gamma f\left( t \right) = { - _t}D_b^{\gamma - 1}f^\prime \left( t \right) = {1 \over {\Gamma \left( {1 - \gamma } \right)}}\left( {\int_t^b {{{\left( {s - t} \right)}^{ - \gamma }}f^\prime \left( t \right)} ds} \right),t \in \left[ {a,\,b} \right]

Job similarity determination

This paper follows the similarity algorithm in the homework plagiarism detection module of LMS2.0. In the homework plagiarism detection module of LMS2.0, we use asymmetric similarity to judge the similarity between students' homework [7]. We are given two documents a and b. This ratio is also called the inclusion ratio of document a by b. which is expressed as Cab=|AB|/|A| {C_{ab}} = \left| {A \cap B} \right|/\left| A \right| Where A and B are the sample fingerprint sets of documents a and b, respectively. The similarity of documents is an asymmetric matrix, CabCba. Asymmetric similarity can clearly tell the degree of inclusion and inclusion between tasks [8]. This paper focuses on the closeness of the correlation between documents. Therefore, we need to use the correlation coefficient ρab to represent the correlation between jobs, ρab=|AB|/(|A||B|) {\rho _{ab}} = \left| {A \cap B} \right|/\left( {\sqrt {\left| A \right|} \sqrt {\left| B \right|} } \right) The relationship between the correlation coefficient and asymmetric similarity can be derived from the following equation CabCba=(|AB|/|A|)(|AB|/|B|=|AB|2/(|A||B|)ρab2ρab=CabCba {C_{ab}}{C_{ba}} = \left( {\left| {A \cap B} \right|/\left| A \right|} \right)\,\left( {\left| {A \cap B} \right|/\left| B \right| = {{\left| {A \cap B} \right|}^2}/\left( {\left| A \right|\left| B \right|} \right)\rho _{ab}^2 \Rightarrow {\rho _{ab}} = \sqrt {{C_{ab}}{C_{ba}}} } \right. It can be known from the above formula that 0 ≤ ρab ≤ 1. The ρab size represents the degree of linear correlation between the two document vectors. The closer ρab is to 1, the closer the linear relationship between the two document vectors is. On the contrary, it means that the linear correlation between the two vectors is weaker. We use 1 − ρab to denote the distance of the document vector. The smaller the 1 − ρab value, the higher the similarity between the two documents.

Hierarchical Clustering

The purpose of this stage is to cluster assignments that may be plagiarized from each other. This facilitates subsequent determination of who is a plagiarist in each cluster. We adopt the agglomerative hierarchical clustering method to analyze the target documents bottom-up according to the distance between the jobs obtained in the previous stage. Documents with the smallest distances are merged into one class first [9]. And so on until all documents are merged into one class or a certain threshold is reached. We ended up combining all the assignments into one class in this article.

Before clustering, we first treat each document vector as a class. The inter-class distance is the distance 1 − ρab of the document vector. In the clustering process, if k documents i1, i2, ⋯ ik are merged into a class i, the distance between this class and other classes j = {j1, i2, ⋯ jm} is defined as follows dij=min(1ρisjt)(s=1,2,,k;t=1,2,,m) \matrix{ {{d_{ij}} = \min \left( {1 - {\rho _{{i_s}jt}}} \right)} \hfill \cr {\left( {s = 1,\,2,\, \cdots ,\,k;\,t = 1,\,2,\, \cdots ,\,m} \right)} \hfill \cr } Where k and m are the number of document vectors in the two classes, respectively. dij is the distance between the two classes. This distance is the minimum distance between all documents in the two classes.

The hierarchical clustering method is an “Overview of Computer Technology Development” assignment for plagiarism correlation analysis. The results are shown in Figure 2. In Figure 2, the ordinate is the relative distance between the assignments, and the abscissa is the number of the assignments handed in by the students [10]. Hierarchical clustering means that the smaller the correlation distance, the higher the similarity between two documents. Suppose we take 0.1 as the similar job determination threshold. Eight plagiarism sources from the same source were found through hierarchical clustering. More than 90% of the homework plagiarism groups are similar to the plagiarism source.

Figure 2

Job correlation pedigree analysis diagram

Plagiarism determination

Judging a plagiarist among several similar assignments is a relatively subjective act. We conducted a detailed analysis of students with plagiarism in LMS2.0 and found a certain correlation between students' daily learning behavior and learning effect and assignment plagiarism [11]. Students with good study habits and learning effects are less likely to plagiarize. Based on this phenomenon, this paper proposes the Analytic Hierarchy Process. We collected five learning evaluation indicators, including unit test result (X1), course visit frequency (X2), learning progress (X3), in-class communication participation (X4) and attendance result (X5) collected through LMS3.0. We build a hierarchical structure model, as shown in Figure 3.

Figure 3

Hierarchical structure model diagram

Suppose X = {X1, X2, ⋯, X5} is the influencing factor of decision target. Preset its impact factor to get a pairwise comparison matrix. We normalize it to get the learning evaluation weight vector W = [ω1, ω2, …, ω5]. {Y1, Y2, …, Yn} is the list of plagiarism-related students obtained through hierarchical clustering. We calculate the pairwise comparison matrix of each student on each indicator in D and standardize it to get the corresponding evaluation matrix R=[r11r12r1nr21r22r2nr51r52r5n] R = \left[ {\matrix{ {{r_{11}}} & {{r_{12}}} & \cdots & {{r_{1n}}} \cr {{r_{21}}} & {{r_{22}}} & \cdots & {{r_{2n}}} \cr \cdots & {} & {} & {} \cr {{r_{51}}} & {{r_{52}}} & \cdots & {{r_{5n}}} \cr } } \right]

In the formula rij is the evaluation score of the j student on the i index. We perform the following operations on the learning evaluation weight vector and the evaluation matrix to obtain the student's total learning evaluation score. Students with lower scores were more likely to plagiarize other people's work.

Y=WR=[ω1ω2ω5]T[r11r12r1nr21r22r2nr51r52r5n]=[y1y2yn] Y = W\,R = {\left[ {\matrix{ {{\omega _1}} \cr {{\omega _2}} \cr \cdots \cr {{\omega _5}} \cr } } \right]^T}\left[ {\matrix{ {{r_{11}}} & {{r_{12}}} & \cdots & {{r_{1n}}} \cr {{r_{21}}} & {{r_{22}}} & \cdots & {{r_{2n}}} \cr \cdots & {} & {} & {} \cr {{r_{51}}} & {{r_{52}}} & \cdots & {{r_{5n}}} \cr } } \right] = \left[ {\matrix{ {{y_1}} \cr {{y_2}} \cr \cdots \cr {{y_n}} \cr } } \right]

We assume a(12,1],p[1,+) a \in ({1 \over 2},1],p \in [1, + \infty ) , the fractional differential space X = Ea,2 = {u : [0, T] → R.

Assuming α=1β2,b,d>0,) \alpha = 1 - {\beta \over 2},b,d > 0,) the new energy functional φ : XR is defined as follows: φ(u)=120t1(0cDtau,tcDt1au)dt12t1s1(t1cDtau,tcDs1au)dt12s1T(s1cDtau,tcDTau)dt+a2b(u(0))2+c2d(u(T))20t1F1(t,u(t))dts1TF2(t,u(t))dt+0u(t1)I(s)ds \matrix{ {\varphi \left( u \right) = - {1 \over 2}\int_0^{{t_1}} {\left( {_0^cD_t^au,\,_t^cD_{{t_1}}^au} \right)dt - {1 \over 2}} \int_{{t_1}}^{{s_1}} {\left( {_{{t_1}}^cD_t^au,\,_t^cD_{{s_1}}^au} \right)dt - {1 \over 2}} \int_{{s_1}}^T {\left( {_{{s_1}}^cD_t^au,\,_t^cD_T^au} \right)dt} } \hfill \cr { + {a \over {2b}}{{\left( {u\left( 0 \right)} \right)}^2} + {c \over {2d}}{{\left( {u\left( T \right)} \right)}^2} - \int_0^{{t_1}} {{F_1}\left( {t,u\left( t \right)} \right)dt - \int_{{s_1}}^T {{F_2}\left( {t,u\left( t \right)} \right)dt + \int_0^{u\left( {{t_1}} \right)} {I\left( s \right)ds} } } } \hfill \cr } Where Fi(t,u)=0ufi(t,s)ds(i=1,2) {F_i}\left( {t,u} \right) = \int_0^u {{f_i}\left( {t,s} \right)ds\left( {i = 1,\,2} \right)} . Since f1, f2 and I are continuous functions, we can get φC1 (X, R) and φ(u),v=120t1(0Dtβu+tDt1βu,v)dt+12t1s1(t1Dtβu+tDs1βu,v)dt \left\langle {\varphi ^\prime \left( u \right),v} \right\rangle = {1 \over 2}\int_0^{{t_1}} {\left( {_0D_t^{ - \beta }u^\prime { + _t}D_{{t_1}}^{ - \beta }u^\prime ,v^\prime } \right)dt + {1 \over 2}} \int_{{t_1}}^{{s_1}} {\left( {_{{t_1}}D_t^{ - \beta }u^\prime { + _t}D_{{s_1}}^{ - \beta }u^\prime ,v^\prime } \right)dt} u, vX: φ(u),v=120t1(0cDtau,tcDt1av)+(tcDt1au,0cDtav)dt12t1s1(t1cDtau,tcDs1av)+(tcDs1au,t1cDtav)dt12s1T(s1cDtau,tcDTav)+(tcDTau,s1cDtav)dt+abu(0)v(0)+cdu(T)v(T)0t1f1(t,u(t))v(t)dts1Tf2(t,u(t))v(t)dt+I(u(t1))v(t1) \matrix{ {\left\langle {\varphi ^\prime \left( u \right),v} \right\rangle = {1 \over 2}\int_0^{{t_1}} {\left( {_0^cD_t^au,\,_t^cD_{{t_1}}^av} \right) + \left( {_t^cD_{{t_1}}^au,\,_0^cD_t^av} \right)dt - {1 \over 2}} \int_{{t_1}}^{{s_1}} {\left( {_{{t_1}}^cD_t^au,\,_t^cD_{{s_1}}^av} \right)} } \hfill \cr { + \left( {_t^cD_{{s_1}}^au,\,_{{t_1}}^cD_t^av} \right)dt - {1 \over 2}\int_{{s_1}}^T {\left( {_{{s_1}}^cD_t^au,\,_t^cD_T^av} \right) + \left( {_t^cD_T^au,\,_{{s_1}}^cD_t^av} \right)dt} } \hfill \cr { + {a \over b}u\left( 0 \right)v\left( 0 \right) + {c \over d}u\left( T \right)v\left( T \right) - \int_0^{{t_1}} {{f_1}\left( {t,\,u\,\left( t \right)} \right)v\left( t \right)dt} } \hfill \cr { - \int_{{s_1}}^T {{f_2}\left( {t,u\left( t \right)} \right)} v\left( t \right)dt + I\left( {u\left( {{t_1}} \right)} \right)v\left( {{t_1}} \right)} \hfill \cr }

Combining the above definitions, we can get 120t1(0cDtau,tcDt1av)+(tcDt1au,0cDtav)dt=120t1(0cDtau,tDt1a1v)+(tcDt1au,0Dta1v)dt=120t1(0Dta1(0cDtau),v)+(tDt1a1(tcDt1au),v)dt=120t1(0Dta1(0Dta1u),v)+(tDt1a1(tDt1a1u),v)dt \matrix{ { - {1 \over 2}\int_0^{{t_1}} {\left( {_0^cD_t^au,\,_t^cD_{{t_1}}^av} \right) + \left( {_t^cD_{{t_1}}^au,\,_0^cD_t^av} \right)dt} } \hfill \cr { = - {1 \over 2}\int_0^{{t_1}} {\left( {_0^cD_t^au,{ - _t}D_{{t_1}}^{a - 1}v^\prime } \right) + \left( {_t^cD_{{t_1}}^au{,_0}D_t^{a - 1}v^\prime } \right)dt} } \hfill \cr { = - {1 \over 2}\int_0^{{t_1}} {\left( {_0D_t^{a - 1}\left( {_0^cD_t^au} \right), - v^\prime } \right) + \left( {_tD_{{t_1}}^{a - 1}\left( {_t^cD_{{t_1}}^au} \right),v^\prime } \right)dt} } \hfill \cr { = - {1 \over 2}\int_0^{{t_1}} {\left( {_0D_t^{a - 1}\left( {_0D_t^{a - 1}u^\prime } \right), - v^\prime } \right) + \left( {_tD_{{t_1}}^{a - 1}\left( {{ - _t}D_{{t_1}}^{a - 1}u^\prime } \right),v^\prime } \right)dt} } \hfill \cr }

Similarly, you can also get 12t1s1(t1cDtau,tcDs1av)+(tcDs1au,t1cDtav)dt=12t1s1(t1Dtβu+tDs1βu,v)dt - {1 \over 2}\int_{{t_1}}^{{s_1}} {\left( {_{{t_1}}^cD_t^au,\,_t^cD_{{s_1}}^av} \right) + \left( {_t^cD_{{s_1}}^au,_{{t_1}}^cD_t^av} \right)dt = {1 \over 2}\int_{{t_1}}^{{s_1}} {\left( {_{{t_1}}D_t^{ - \beta }u^\prime { + _t}D_{{s_1}}^{ - \beta }u^\prime ,v^\prime } \right)dt} }

As well as 12s1T(s1cDtau,tcDTav)+(tcDTau,s1cDtav)dt=12s1T(s1Dtβu+tDTβu,v)dt - {1 \over 2}\int_{{s_1}}^T {\left( {_{{s_1}}^cD_t^au,\,_t^cD_T^av} \right) + \left( {_t^cD_T^au,_{{s_1}}^cD_t^av} \right)dt = {1 \over 2}\int_{{s_1}}^T {\left( {_{{s_1}}D_t^{ - \beta }u^\prime { + _t}D_T^{ - \beta }u^\prime ,v^\prime } \right)dt} }

If the function u makes 0Dtβu _0D_t^{ - \beta }u^\prime , tDt1βuC1(0,t1] _tD_{{t_1}}^{ - \beta }u^\prime \in {C^1}\left( {0,{t_1}} \right] , s1Dtβu _{{s_1}}D_t^{ - \beta }u^\prime , tDTβuC1(s1,T] _tD_T^{ - \beta }u^\prime \in {C^1}\left( {{s_1},T} \right] and satisfies the transient impulse, non-transient impulse, and Sturm-Liouville boundary conditions in the problem.

Experiments and Analysis

The homework-assisted evaluation model proposed in this paper has been applied to the teaching platform of college English majors. In the experiment, we tracked and collected the learning behavior data of about 300 students in 5 courses in the teaching platform. We use the platform's plagiarism detection module to detect 8 of these course papers [12]. We found a total of 47 homework plagiarism groups with a similarity of more than 80%. Table 1 shows the accuracy of the homework-assisted evaluation and correction model for judging plagiarists, including the number of people in the homework plagiarism group. n2 is the number of homework plagiarism groups. η is the judgment accuracy.

The relationship between the size of the plagiarism group and the accuracy of plagiarism judgment

N1 n2 η/%
2 23 96
3 12 83
4 7 43
5 5 20

When the homework plagiarism group contains a small number of people (2 to 3 people), the judgment accuracy of the auxiliary evaluation model is higher. This means that those with higher learning total evaluation scores are more likely to be non-plagiarism. When the homework plagiarism group contains many people, the judgment accuracy of the auxiliary evaluation model is not ideal. After inspection, it is found that when the same homework plagiarism group contains many people, the learning total evaluation score depends on the latter, the plagiarism probability is higher. However, the basis for judging whether the person with the highest total score in the learning evaluation is a non-plagiarism is not necessarily established.

Conclusion

The English homework-aided assessment model proposed in this paper based on fractional differential equations is ideal for determining homework plagiarism. However, there is a certain error in the judgment of non-plagiarists when the plagiarism group contains a large number of people.

Figure 1

Architecture diagram of the homework-assisted assessment model
Architecture diagram of the homework-assisted assessment model

Figure 2

Job correlation pedigree analysis diagram
Job correlation pedigree analysis diagram

Figure 3

Hierarchical structure model diagram
Hierarchical structure model diagram

The relationship between the size of the plagiarism group and the accuracy of plagiarism judgment

N1 n2 η/%
2 23 96
3 12 83
4 7 43
5 5 20

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