A collocation methods based on the quadratic quadrature technique for fractional differential equations

Abstract

<abstract><p>In this paper, we introduce a mixed numerical technique for solving fractional differential equations (FDEs) by combining Chebyshev collocation methods and a piecewise quadratic quadrature rule. For getting solutions at each integration step, the fractional integration is calculated in two intervals-all previous time intervals and the current time integration step. The solution at the current integration step is calculated by using Chebyshev interpolating polynomials. To remove a singularity which belongs originally to the FDEs, Lagrangian interpolating technique is considered since the Chebyshev interpolating polynomial can be rewritten as a Lagrangian interpolating form. Moreover, for calculating the fractional integral on the whole previous time intervals, a piecewise quadratic quadrature technique is applied to get higher accuracy. Several numerical experiments demonstrate the efficiency of the proposed method and show numerically convergence orders for both linear and nonlinear cases.</p></abstract>

A highly accurate artificial neural networks scheme for solving higher multi‐order fractal‐fractional differential equations based on generalized Caputo derivative

“…Figure 22 shows the convergence behavior of the calculated weight parameters. When we compare our results to the result obtained in Reference 62 for $$$ \alpha =1.5 $$$, we find that Reference 62's absolute error is approximately $$$ 7.9944\times 1{0}^{-6} $$$ while in the proposed method's result is $$$ 8.0704\times 1{0}^{-9} $$$.…”

Error estimation using neural network technique for solving ordinary differential equations

“…For this reason, several numerical techniques for solving ODEs have been developed during last few decades. Also, the numerical methods can be broadly classified into the following categories: the first class consists of one-step multistage techniques such as Runge-Kutta-type methods [5,13,17], the second includes BDF-type multistep methods [6], and the last is a group of deferred or error correction methods [4,7,18,19] such as spectral deferred correction (SDC) methods [8,11], etc.…”