NILT and Prony technique for new definitions of fractional calculus for modeling very slow decay phenomena

Abstract Based on the data from the Tesla users about the degradation of Tesla Model S battery capacity, this paper introduces two new definitions of fractional integrals and derivatives to describe these very slow decay phenomena, which fill the blanks in this field. Several existing definitions of fractional calculus are reviewed at first, together with their drawbacks in describing very slow decay. Then, two new definitions governed by new kernel functions are introduced. With the aids of Numerical Inverse Laplace Transform (NILT), Prony technique and MATLAB command stmcb(), mathematical properties of these two fractional operators are investigated.