Numerical differentiation of fractional order derivatives based on inverse source problems of hyperbolic equations

In this paper, we mainly study the numerical differentiation problem of computing the fractional order derivatives from noise data of a single variable function. Firstly, the numerical differentiation problem is reformulated into an inverse source problem of first order hyperbolic equation, and the corresponding solvability and the conditional stability are provided under suitable conditions. Then, four regularization methods are proposed to reconstruct the unknown source of hyperbolic equation which is the numerical derivative, and they are implemented by utilizing the finite dimensional expansion of source function and the superposition principle of hyperbolic equation. Finally, Numerical experiments are presented to show effectiveness of the proposed methods. It can be conclude that the proposed methods are very effective for small noise levels, and they are simpler and easier to be implemented than the previous PDEs-based numerical differentiation method based on direct and inverse problems of parabolic equations.