Singularity preserving spectral collocation method for nonlinear systems of fractional differential equations with the right-sided Caputo fractional derivative

Abstract

The numerical treatment of fractional differential equations in an accurate way is more difficult to tackle than the standard integer-order counterpart, and occasionally non-specialists are unaware of the specific difficulties. In this paper, we consider nonlinear systems of fractional differential equations involving the right-sided Caputo fractional derivatives of order α∈(0,1). The solutions to these equations have low regularity in the usual Sobolev space even for smooth inputs, requiring regularization techniques to control arbitrary error amplification and to get adequate solutions. Since singularities of the solution usually reflect important features of practical problems, numerical methods preserving the singularities of the solution are preferable. Thus, a singularity preserving regularization method is discussed in detail. The convergence analysis is carried out and the optimal error estimates are obtained.