Singularly perturbed reaction-diffusion problems on a k-star graph

Singularly perturbed reaction-diffusion equations on a star graph (having k + 1 nodes and k edges) resulting in a system with k individual partial differential equations along the edges with coupling conditions at the common junction are presented. In the singular limit, as the diffusion parameter tends to zero, possibly individually along each edge, boundary layers may occur at the multiple nodes as well as at the simple nodes. Numerically, the proposed equations are solved using central finite difference schemes on properly extended Shishkin meshes. Error estimates are discussed and validated by solving a test problem on a graph with three edges (tripod). A more general graph problem with eight edges and three connecting nodes has also been solved numerically.