Step-by-step integration for fractional operators. (English) Zbl 1510.65177
Summary: In this paper, an approach based on the definition of the Riemann-Liouville fractional operators is proposed in order to provide a different discretisation technique as alternative to the Grünwald-Letnikov operators. The proposed Riemann-Liouville discretisation consists of performing step-by-step integration based upon the discretisation of the function \(f(t)\). It has been shown that, as \(f(t)\) is discretised as stepwise or piecewise function, the Riemann-Liouville fractional integral and derivative are governing by operators very similar to the Grünwald-Letnikov operators.
In order to show the accuracy and capabilities of the proposed Riemann-Liouville discretisation technique and the Grünwald-Letnikov discrete operators, both techniques have been applied to: unit step functions, exponential functions and sample functions of white noise.
In order to show the accuracy and capabilities of the proposed Riemann-Liouville discretisation technique and the Grünwald-Letnikov discrete operators, both techniques have been applied to: unit step functions, exponential functions and sample functions of white noise.
MSC:
| 65L99 | Numerical methods for ordinary differential equations |
| 26A33 | Fractional derivatives and integrals |
Software:
ma2dfcReferences:
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