On reflecting boundary conditions for space-fractional equations on a finite interval: proof of the matrix transfer technique - Cusimano, N; Burrage, K; Turner, I; Kay, D (Elsevier, 2016)
Even in the one-dimensional case, dealing with the analysis of space-fractional differential equations on finite domains is a difficult issue. On a finite interval coupled with zero ux boundary conditions different approaches have been proposed to define a space-fractional differential operator and to compute the solution to the corresponding fractional problem, but to the best of our knowledge, a clear relationship between these strategies is yet to be established. Here, by using the theory of α-stable symmetric Lévy flights and the master equation, we derive a discrete representation of the non-local operator embedding in its definition the concept of re ecting boundary conditions. We refer to this discrete operator as the re ection matrix and provide (and prove) a theorem on the analytic expression of its eigenvalues and eigenvectors. We then use this result to compare the re ection matrix to the discrete operator defined via the matrix transfer technique, and establish the validity of the latter technique in producing the correct solution to a space-fractional differential equation on a finite interval with re ecting boundary conditions. We finally discuss and emphasize the challenges in the generalisation of the proposed result to more than one spatial dimension.