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Article

Discretization of Fractional Operators: Analysis by Means of Advanced Computational Techniques

by
Jose Tenreiro Machado
1,*,†,
Alexandra M. Galhano
2,† and
Carla S. Cordeiro
2
1
Institute of Engineering, Polytechnic of Porto, Rua Dr. António Bernardino de Almeida, 431, 4249-015 Porto, Portugal
2
Faculdade de Ciências Naturais, Engenharias e Tecnologias, Universidade Lusófona do Porto, Rua Augusto Rosa 24, 4000-098 Porto, Portugal
*
Author to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Mathematics 2021, 9(19), 2429; https://doi.org/10.3390/math9192429
Submission received: 2 September 2021 / Revised: 22 September 2021 / Accepted: 24 September 2021 / Published: 30 September 2021
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Abstract

This paper studies the discretization of fractional operators by means of advanced clustering methods. The Grünwald–Letnikov fractional operator is approximated by series generated by the Euler, Tustin and generalized mean. The series for different fractional orders form the objects to be assessed. For this purpose, the several distances associated with the hierarchical clustering and multidimensional scaling computational techniques are tested. The Arc-cosine distance and the 3-dim multidimensional scaling produce good results. The visualization of the graphical representations allows a better understanding of the properties embedded in each type of approximation of the fractional operators.
Keywords: multidimensional scaling; clustering methods; series; fractional calculus; discretization multidimensional scaling; clustering methods; series; fractional calculus; discretization

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MDPI and ACS Style

Tenreiro Machado, J.; Galhano, A.M.; Cordeiro, C.S. Discretization of Fractional Operators: Analysis by Means of Advanced Computational Techniques. Mathematics 2021, 9, 2429. https://doi.org/10.3390/math9192429

AMA Style

Tenreiro Machado J, Galhano AM, Cordeiro CS. Discretization of Fractional Operators: Analysis by Means of Advanced Computational Techniques. Mathematics. 2021; 9(19):2429. https://doi.org/10.3390/math9192429

Chicago/Turabian Style

Tenreiro Machado, Jose, Alexandra M. Galhano, and Carla S. Cordeiro. 2021. "Discretization of Fractional Operators: Analysis by Means of Advanced Computational Techniques" Mathematics 9, no. 19: 2429. https://doi.org/10.3390/math9192429

APA Style

Tenreiro Machado, J., Galhano, A. M., & Cordeiro, C. S. (2021). Discretization of Fractional Operators: Analysis by Means of Advanced Computational Techniques. Mathematics, 9(19), 2429. https://doi.org/10.3390/math9192429

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