Abstract
A simple scheme is proposed for computing N times N spectral differentiation matrices of fractional order α for the case of Legendre approximation. The algorithm derived here is based upon a homogeneous three-term recurrence relation and is numerically stable. The matrices are then applied to numerically differentiate.
Highlights
In the last few decades, applied scientists have outstretched new models that include fractional derivatives, fractional differential equations (FDEs)
The classical and modern dynamical systems modeled by FDEs in physics, engineering, signal processing, fluid mechanics, and bioengineering, manufacturing, systems engineering, and project management can be observed in the recently published book [1]
5 Conclusion Based on the shifted Legendre polynomials, we proposed an easy procedure for calculating spectral integration/differentiation matrices of the arbitrary order α
Summary
In the last few decades, applied scientists have outstretched new models that include fractional derivatives, fractional differential equations (FDEs). The authors of [7,8,9,10,11,12,13] have published some new results of fractional operators and their applications. Those could approximate derivatives by differentiating trial or cardinal basis functions through collocation points. The explicit formulas for the order-integer derivatives of Legendre approximation already exist.
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