Boundary Value Problem for the Fractional Diffusion Equation with the Right-Sided Liouville Operator in a Triangular Domain

We study the first Dirichlet boundary-value problem for a time-fractional diffusion equation with the right-sided Liouville derivative of order strictly between zero and one, posed in a degenerate triangular domain with a moving boundary. A fundamental solution in the half-space is derived using the sine Fourier transform and the Laplace transform, yielding an explicit Green function written through the Wright and Mittag-Leffler special functions. The solution inside the domain is represented by fractional-diffusion layer potentials with unknown densities, which are determined from a system of Volterra integral equations of the second kind with weakly singular kernels. We establish jump relations for the double-layer potential on both the fixed and the slanted boundaries, and we obtain kernel estimates based on the asymptotics of the Wright function, thereby justifying passage to the limit under the integral sign. For continuous, bounded data we prove existence and uniqueness of a solution and show that the boundary densities are computable by the method of successive approximations. Explicit integral representations are provided, along with conditions ensuring the applicability of the derived estimates.