Cauchy Problem for the Time-Fractional Airy-Type Equation on a Star Graph with Edge-Dependent Order of Derivative

We study the Cauchy problem for a time-fractional Airy-type equation on an infinite metric star graph with edge-dependent Caputo derivatives. By constructing layer potentials generated by fractional Airy fundamental solutions, the problem is reduced to a generalized Abel integral equation. Using Pskhu’s solvability results, we establish existence, uniqueness, and an explicit integral representation of the solution. The obtained framework extends previous results where identical fractional orders were assumed on all edges.