Diffusion Equation with Riemann–Liouville Derivative in the Angular Domain

This paper is concerned with a boundary value problem for a fractional-order diffusion equation involving the Riemann–Liouville fractional derivative with respect to time in an angular domain. The distinctive feature of the formulation is that the lower limit of the fractional operator coincides with the spatial variable. For fractional orders strictly between zero and one and for boundary and source data satisfying the entire-function and exponential-type assumptions, we prove that the problem has exactly one regular solution in a class of functions of at most exponential growth. To solve the problem, the Laplace transform is used with respect to the time variable with a variable lower limit of integration. The solution is given by an explicit representation, which combines inverse Laplace transforms with Wright function potentials. The resulting formula is specified for the model case of a constant source and constant boundary data, which demonstrates the possibility of using the result as an analytical test for problems of anomalous diffusion in domains with a moving boundary.