22 research outputs found
An Efficient Iteration Method for Toeplitz-Plus-Band Triangular Systems Generated from Fractional Ordinary Differential Equation
It is time consuming to numerically solve fractional differential equations. The fractional ordinary differential equations may produce Toeplitz-plus-band triangular systems. An efficient iteration method for Toeplitz-plus-band triangular systems is presented with OMlogM computational complexity and OM memory complexity in this paper, compared with the regular solution with OM2 computational complexity and OM2 memory complexity. M is the discrete grid points. Some methods such as matrix splitting, FFT, compress memory storage and adjustable matrix bandwidth are used in the presented solution. The experimental results show that the presented method compares well with the exact solution and is 4.25 times faster than the regular solution
Approximate solutions for solving nonlinear variable-order fractional Riccati differential equations
In this manuscript, we introduce a spectral technique for approximating the variable-order fractional Riccati equation (VO-FRDEs). Firstly, the solution and its space fractional derivatives is expanded as shifted Chebyshev polynomials series. Then we determine the expansion coefficients by reducing the VO-FRDEs and its conditions to a system of algebraic equations. We show the accuracy and applicability of our numerical approach through four numerical examples.
 
Optimal Collocation Nodes for Fractional Derivative Operators
Spectral discretizations of fractional derivative operators are examined,
where the approximation basis is related to the set of Jacobi polynomials. The
pseudo-spectral method is implemented by assuming that the grid, used to
represent the function to be differentiated, may not be coincident with the
collocation grid. The new option opens the way to the analysis of alternative
techniques and the search of optimal distributions of collocation nodes, based
on the operator to be approximated. Once the initial representation grid has
been chosen, indications on how to recover the collocation grid are provided,
with the aim of enlarging the dimension of the approximation space. As a
results of this process, performances are improved. Applications to fractional
type advection-diffusion equations, and comparisons in terms of accuracy and
efficiency are made. As shown in the analysis, special choices of the nodes can
also suggest tricks to speed up computations