The fractional Sturm-Liouville problem-Numerical approximation and application in fractional diffusion

The numerical method of solving the fractional eigenvalue problem is derived in the case when the fractional Sturm-Liouville equation is subjected to the mixed boundary conditions. This non-integer order differential equation is discretized to the scheme with the symmetric matrix representing the action of the numerically expressed composition of the left and the right Caputo derivative. The numerical eigenvalues are thus real, and the eigenvectors associated to distinct eigenvalues are orthogonal in the respective finite-dimensional Hilbert space. The advantage of the proposed method is the formulation which allows us to construct the approximate eigenfunctions which form an orthonormal function system in the infinite-dimensional weighted Lebesgue integrable function space. The developed numerical method of calculation of the eigenvalues and eigenfunctions is then applied in construction of the approximate solution to the 1D space-time fractional diffusion problem in a bounded domain.