Numerical solution for a class of multi-order fractional differential equations with error correction and convergence analysis

Abstract

In this article, we investigate numerical solution of a class of multi-order fractional differential equations with error correction and convergence analysis. According to fractional differential definition in Caputo's sense, fractional differential operator matrix is deduced. The problem is reduced to a set of algebraic equations, and we apply MATLAB to solve the equation. In order to improve the precision of numerical solution, the process of error correction for multi-order fractional differential equation is introduced. By constructing the multi-order fractional differential equation of the error function, the approximate error function is obtained so that the numerical solution is corrected. Then, we analyze the convergence of the shifted Chebyshev polynomials approximation function. Numerical experiments are given to demonstrate the applicability of the method and the validity of error correction.

Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration

“…From these results, we can conclude that m = 4 and m = 6 give larger absolute error, while m = 7 gives smaller absolute error (10 −16 ) and more precise numerical solution. These comparisons also shows that the results obtained by proposed method is closer to the exact solution than the results of [59]. In Figure 4, we show the graphical representation of absolute errors obtained by using proposed method and the method of [59] with m, n = 6.…”

“…These comparisons also shows that the results obtained by proposed method is closer to the exact solution than the results of [59]. In Figure 4, we show the graphical representation of absolute errors obtained by using proposed method and the method of [59] with m, n = 6. From Figure 4, we can conclude that the absolute error obtained by our method is remaining smaller and stable while the absolute error of other method is increasing in the interval [0, 1].…”

“…Table 1, compares the obtained results for absolute error with m = 4, 6, 7. We observe from Table 1 that, the absolute errors for presented method are smaller and the numerical solution is more accurate for the same size of m. In Figures 1-3, we present the graphical representation of comparison between exact solution and the numerical solutions obtained by proposed method and the method of [59] for the problem (19) with m = 4, 6, 7 respectively. From these results, we can conclude that m = 4 and m = 6 give larger absolute error, while m = 7 gives smaller absolute error (10 −16 ) and more precise numerical solution.…”

“…To show the efficiency, we compared the numerical results with the method given in [59]. Table 1, compares the obtained results for absolute error with m = 4, 6, 7.…”

A fast numerical method for fractional partial differential equations

“…The spectral methods using Chebyshev polynomials are well known for ordinary and partial differential equations with rapid convergence property [30][31][32][33][34][35][36][37][38][39][40]. An important advantage of these methods over finite-difference methods is that computing the coefficient of the approximation completely determines the solution at any point of the desired interval.…”