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A novel derivation of a second-order accurate Grünwald-Letnikov-type approximation to the fractional derivative of a function is presented. This scheme is shown to be second-order accurate under certain modifications to account for poor accuracy in approximating the asymptotic behavior near the lower limit of differentiation. Some example functions are chosen and numerical results are presented to illustrate the efficacy of this new method over some other popular choices for discretizing fractional derivatives.

Fractional differential equations and fractional order derivatives have become increasingly popular in recent years. The use of fractional order derivatives has become abundant in modeling techniques as well as computational methods for the numerical solution of these models. Zeng et al. [

Meerschaert and Tadjeran [

Tadjeran and Meerschaert [

The present work derives a difference approximation in a novel way that produces a form similar to the known result presented by Oldham and Spanier in [

Another important technique used in the numerical application of fractional differential equations is the “short-memory” principle [

The following section presents some preliminary results used in the remainder of the paper. Section

This section presents some preliminary results that are put to use in subsequent sections. However, in the interest of brevity, we omit some useful properties but direct the interested reader to the work by Li and Deng [

The Grünwald-Letnikov derivative of order

However the limit definition is only practically useful for finite-difference implementations; hence the following definition is used.

The Grünwald-Letnikov derivative of order

Definition

The Riemann-Liouville derivative of fractional order

The right-shifted Grünwald-Letnikov approximation introduced by Meerschaert and Tadjeran [

The right-shifted Grünwald-Letnikov approximation of a function

Oldham and Spanier obtain a second-order Grünwald-Letnikov approximation simply by noting that the Riemann sum present in the derivative would converge faster if a simple modification was made.

The Grünwald-Letnikov derivative of order

The primary difference between Definition

In much the same way the Grünwald-Letnikov derivative is generalized from the first-order backward difference scheme, as described by Podlubny in [

As described by Oldham and Spanier in [

The error of

Let

Using Theorem

This section presents some numerical results for different functions. Each example shows the absolute error over the domain of the function, the consistency of each derivative as

Let

Let

Let

Let

Let

The above results indicate that the present method enjoys a high accuracy. The nature of the fractional derivative near the lower limit of differentiation results in a high gradient. Difference schemes with fixed interval steps are unable to accurately capture very steep gradients, similar to those at the asymptote of the fractional derivative. The use of the constant correction allows one to avoid the asymptotic behavior near

Figures

Absolute error over time domain for four different methods.

Absolute error over varying values of

Absolute error for increasing

Absolute error over varying values of

Absolute error over time domain for four different methods.

Absolute error over varying values of

Absolute error for increasing

Absolute error over varying values of

Absolute error over time domain for four different methods.

Absolute error over varying values of

Absolute error for increasing

Absolute error over varying values of

Absolute error over time domain for four different methods.

Absolute error over varying values of

Absolute error for increasing

Absolute error over varying values of

Absolute error over time domain for four different methods.

Absolute error over varying values of

Absolute error for increasing

Absolute error over varying values of

This work has shown a novel derivation of a second-order accurate scheme for approximating the fractional order derivative of a function. To the best of the author’s knowledge performing a linear transformation on the function to avoid asymptotic behavior near the origin is also novel. Section

Using a higher-order accurate approximation leads to the potential for higher-order accurate numerical schemes for fractional differential equations and fractional partial differential equations. Tadjeran and Meerschaert [

The accuracy of the method has been proven to be at least of

The opinions expressed and conclusions arrived at herein are those of the author and are not necessarily to be attributed to the CoE-MaSS.

The author declares that there is no conflict of interests regarding the publication of this paper.

The author would like to acknowledge and thank Professor Charis Harley for her helpful suggestions and useful insight. The DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) is acknowledged for funding under Grant no. 94005. Thanks are due to the reviewers for their comments which helped to elevate the level of this work.