TITLE:
A Reaction-Diffusion Model for the Propagation of Learning and the Impact of Pedagogical Methods
AUTHORS:
Aliou Sonko, Maurice Leno, Aladji Babacar Niang, Ibrahima Sory Mamikouny Camara
KEYWORDS:
Reaction-Diffusion Equation, Knowledge Propagation, Pedagogical Methods, Logistic Growth, Hill Function, Social Facilitation, Ebbinghaus Forgetting, Stability Analysis, MATLAB Simulation
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.6,
June
12,
2026
ABSTRACT: Understanding how knowledge spreads within a learner population and how pedagogical methods shape this propagation is a fundamental challenge in educational science. In this paper, we propose a reaction-diffusion partial differential equation (PDE) framework to model the spatiotemporal dynamics of knowledge acquisition in a structured population. The state variable
u(
x,t
)∈[
0,1 ]
represents the local density of mastered competence at position
x
in an abstract learner space and at time
t≥0
. The diffusion term models the social contagion of knowledge through peer interactions, modulated by a set of pedagogical methods
(
M
i
)
whose individual diffusion efficiencies are captured by coefficients
(
δ
i
)
. The reaction term combines a logistic growth law, a Hill-type threshold function accounting for the triggering effect in learning, a social facilitation integral operator, and a linear forgetting term consistent with the Ebbinghaus forgetting curve. The learning rate
r(
x,t
)
incorporates both the positive contributions of pedagogical methods through coefficients
(
α
i
)
and the inhibitory effect of cognitive stress through a parameter
β
. We establish the mathematical well-posedness of the model, derive sufficient conditions for the existence of nontrivial steady states, and perform a local stability analysis. A fictitious numerical simulation using MATLAB is conducted to validate the qualitative behavior of the model and to illustrate the comparative impact of five pedagogical methods on the long-term knowledge distribution. The extended two-compartment system coupling cognitive and non-cognitive competences is also discussed.