A comparison between numerical solutions to fractional differential equations: Adams-type predictor-corrector and multi-step generalized differential transform method

In this note, two numerical methods of solving fractional differential equations (FDEs) are briefly described, namely predictor-corrector approach of Adams-Bashforth-Moulton type and multi-step generalized differential transform method (MSGDTM), and then a demonstrating example is given to compare the results of the methods. It is shown that the MSGDTM, which is an enhancement of the generalized differential transform method, neglects the effect of non-local structure of fractional differentiation operators and fails to accurately solve the FDEs over large domains.Comment: 12 pages, 2 figure

A M\"untz-Collocation spectral method for weakly singular volterra integral equations

In this paper we propose and analyze a fractional Jacobi-collocation spectral method for the second kind Volterra integral equations (VIEs) with weakly singular kernel $(x-s)^{-\mu},0<\mu<1$. First we develop a family of fractional Jacobi polynomials, along with basic approximation results for some weighted projection and interpolation operators defined in suitable weighted Sobolev spaces. Then we construct an efficient fractional Jacobi-collocation spectral method for the VIEs using the zeros of the new developed fractional Jacobi polynomial. A detailed convergence analysis is carried out to derive error estimates of the numerical solution in both $L^{\infty}$- and weighted $L^{2}$-norms. The main novelty of the paper is that the proposed method is highly efficient for typical solutions that VIEs usually possess. Precisely, it is proved that the exponential convergence rate can be achieved for solutions which are smooth after the variable change $x\rightarrow x^{1/\lambda}$ for a suitable real number $\lambda$. Finally a series of numerical examples are presented to demonstrate the efficiency of the method