Singularities in solutions of fractional-derivative problems are unavoidable
By means of simple examples we show that weak singularities are present in typical solutions of fractional-derivative problems with Caputo or Riemann-Liouville derivatives. The presence of these singularities significantly complicates the numerical analysis of these problems, and affects the rates of convergence attained by numerical methods. If one removes the singularity by replacing the singular kernel in the fractional-derivative operator with a smoother kernel (as has been suggested by some authors), this leads to an unreasonable restriction on the class of problems that can be studied.
obtained his PhD from Oregon State University, USA, in 1977. He worked most of his career at University College, Cork, Ireland (the city where he attended high school and undergraduate university). He has been at Beijing CSRC since 2014, where he is a Chair Professor under China’s “1000 Talents” program. He is currently on the editorial boards of the journals Advances in Computational Mathematics, Applied Numerical Mathematics, and Computational Methods in Applied Mathematics. For more information see orcid.org, where his ORCID ID is 0000-0003-2085-7354.