A numerical study of fractional relaxation-oscillation equations involving -Caputo fractional derivative

We provide a numerical method to solve a certain class of fractional differential equations involving -Caputo fractional derivative. The considered class includes as particular case fractional relaxation-oscillation equations. Our approach is based on operational matrix of fractional integration of a new type of orthogonal polynomials. More precisely, we introduce -shifted Legendre polynomial basis, and we derive an explicit formula for the -fractional integral of -shifted Legendre polynomials. Next, via an orthogonal projection on this polynomial basis, the problem is reduced to an algebraic equation that can be easily solved. The convergence of the method is justified rigorously and confirmed by some numerical experiments.