Manuscript Topics

Partial differential equations (PDEs) on graphs and networked domains provide a rigorous mathematical framework for modeling dynamics on heterogeneous structures, where continuous processes evolve along network components and interact through coupling conditions at junctions or interfaces. Such formulations naturally arise in traffic and transport networks, biological and neural systems, energy and infrastructure networks, and multi-physics settings involving components of different spatial dimensions.

Compared with classical PDEs posed on smooth continuous domains, PDEs on graphs explicitly incorporate network topology by defining differential operators on edges together with coupling conditions at vertices. This setting enables the modeling of hybrid continuous–discrete phenomena, where time and state variables remain continuous while spatial structures are represented by networks, thereby bridging graph-based approaches and continuum PDE theory.

Recent advances have highlighted the central role of graph Laplacians and related discrete differential operators in the analysis of diffusion, transport, and wave propagation on networks. Spectral methods, eigenvalue problems, and discrete analogues of geometric Laplacians provide powerful tools to characterize network topology, analyze well-posedness and stability, and design efficient numerical schemes.

Despite significant progress, many challenges remain, including the formulation and analysis of elliptic and hyperbolic PDEs on graphs, interface and coupling conditions, control and optimization of networked PDE systems, and the development of structure-preserving numerical methods.

This Special Issue aims to bring together recent contributions on the analysis, numerical approximation, control, and applications of PDEs on graphs and hybrid continuous–discrete networked systems, fostering a unified perspective that integrates theory, computation, and modeling of complex and heterogeneous media in Networks and Heterogeneous Media.

Main Entry Points for Submission

This Special Issue particularly welcomes submissions in the following three directions:
• Diffusion and transport processes on graph-based networks
• Control, stabilization, and optimization of PDE systems on networked graph structures
• Hybrid continuous–discrete models and coupled PDE–ODE systems on graphs and networks

These three directions are intended to provide accessible entry points for both theoretical and application-driven contributions while preserving the original mathematical focus of PDEs on graphs and networked systems.

Topics of Interest

Topics include, but are not limited to:
• PDEs on graphs, metric graphs, and quantum graphs
• Graph Laplacian–based diffusion, transport, and reaction models
• Hybrid PDE–ODE systems and coupled network dynamics
• Spectral analysis, stability, and control of networked PDE systems
• Numerical methods and structure-preserving schemes for PDEs on networks
• Applications to traffic, biological and neural networks, and energy systems
• Data-informed or inverse problems for PDEs on graphs and networks

Instructions for authors
https://www.aimspress.com/nhm/news/solo-detail/instructionsforauthors
Please submit your manuscript to online submission system
https://aimspress.jams.pub/