Two New Approximations for Variable-Order Fractional Derivatives

Ruilian Du,Zongqi Liang

doi:10.1155/2017/1586249

Abstract

We introduced a parameter σ(t) which was related to α(t); then two numerical schemes for variable-order Caputo fractional derivatives were derived; the second-order numerical approximation to variable-order fractional derivatives α(t)∈(0,1) and 3-α(t)-order approximation for α(t)∈(1,2) are established. For the given parameter σ(t), the error estimations of formulas were proven, which were higher than some recently derived schemes. Finally, some numerical examples with exact solutions were studied to demonstrate the theoretical analysis and verify the efficiency of the proposed methods.

Highlights

Fractional differential equations include constant-order and variable-order equations; a great quantity of natural phenomena can be modeled by variable-order fractional differential equations; the study of such problems has attracted much attention
Numerous problems in mathematical physics and engineering have been modeled by variable-order fractional differential equations, such as successful applications in mechanics [5], in the simulation of linear and nonlinear viscoelasticity oscillators [6], and in other cases where the order of the derivative varies with time [7]
Fu et al [12] adopted the method of approximate particular solutions for both constant-order and variable-order time fractional diffusion models

Summary

Introduction

Fractional differential equations include constant-order and variable-order equations; a great quantity of natural phenomena can be modeled by variable-order fractional differential equations; the study of such problems has attracted much attention. Numerous problems in mathematical physics and engineering have been modeled by variable-order fractional differential equations, such as successful applications in mechanics [5], in the simulation of linear and nonlinear viscoelasticity oscillators [6], and in other cases where the order of the derivative varies with time [7]. The difference schemes of fractional derivatives with constant and variable order are investigated in [10]. Fu et al [12] adopted the method of approximate particular solutions for both constant-order and variable-order time fractional diffusion models. We proposed two new approximation formulas of second-order and 3 − α(t) accuracy for variable-order time fractional operator with orders 0 < α(t) < 1 and 1 < α(t) < 2, respectively. We adopted the following definition of variable-order Caputo fractional derivatives: C0Dtα(t)f (t).

Approximation Formulas

Numerical Verification

Conclusion