Mathematical Modeling of AI Adoption in SMEs via Fractional Calculus Techniques

The paper proposes a system of Caputo-Katugampola fractional partial differential equations to model the spatiotemporal dynamics of the adoption of artificial intelligence (AI) in small and medium-sized enterprises (SMEs). The model embeds the Technology-Organization-Environment (TOE) framework into four coupled equations for AI readiness (R), technological infrastructure (T), organizational readiness (O) and cultural acceptance (C). The Caputo–Katugampola derivative of order α ∈ 2 (0, 1] with scaling parameter ρ > 0 encapsulates memory and non-local effects. The system is numerically solved using a space discretization based on the Legendre-Gauss-Lobatto spectral collocation method and a time discretization by a finite difference method. Lower α delays early adoption but speeds up later diffusion, AI readiness acts as a moderator for the TOE pathways, and the cultural acceptance of AI is moderated by the ethical interaction term η2, as revealed from numerical experiments applied to a representative domain of Indian SMEs. The findings provide quantitative managerial advice on the trade-off between the rate of adoption and ethical protections, and show that fractional models can be useful in application of the study of technology diffusion in management.