Analytical and Numerical Solutions of Riesz Space Fractional Advection-Dispersion Equations with Delay

In this paper,we propose numerical methods for the Riesz space fractional advection-dispersion equations with delay(RFADED).We utilize the fractional backward differential formulas method of second order(FBDF2)and weighted shifted Grünwald difference(WSGD)operators to approximate the Riesz fractional derivative and present the finite difference method for the RFADED.Firstly,the FBDF2 and the shifted Grünwald methods are introduced.Secondly,based on the FBDF2 method and the WSGD operators,the finite difference method is applied to the problem.We also show that our numerical schemes are conditionally stable and convergent with the accuracy of O(+h2)and O(2+h2)respectively.Thirdly we find the analytical solution for RFDED in terms Mittag-Leffler type functions.Finally,some numerical examples are given to show the efficacy of the numerical methods and the results are found to be in complete agreement with the analytical solution.

A predictor-corrector scheme for solving nonlinear fractional differential equations with uniform and nonuniform meshes

In this paper,we develop two algorithms for solving linear and nonlinear fractional differential equations involving Caputo derivative.For designing new predictor–corrector(PC)schemes,we select the mesh points based on the two equal-height and equal-area distribution.Furthermore,the error bounds of PC schemes with uniform and equidistributing meshes are obtained.Finally,examples are constructed for illustrating the obtained PC schemes with uniform and equidistributing meshes.A comparative study is also presented.