Speaker: Kai Diethelm, Technical University of Applied Sciences Wurzburg-Schweinfur

Time : 15:00 - 16.00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

Zoom : A link will appear here, an hour before the talk.

Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: Riemann-Liouville integral, numerical method, fast solution, diffusive representation, sum-of-exponentials method, kernel compression.

Abstract: Because of the inherent hereditary properties of fractional operators, the numerical evaluation of the Riemann-Liouville integral of some given vector-valued function at a large number of grid points is a computationally expensive and memory-intesive task when classical algorithms are used [3, 4, 8]. Applying FFT-based summation techniques to the classical methods significantly reduces the run time but not the memory demands [5, 6]. In recent years, we have thus witnessed the development of a growing number of algorithms aimed at reducing both run time and memory requirements. These methods typically use kernel compression techniques, sum-of-exponentials approximations to the kernel function, or nonclassical representations of the fractional integral operator, cf., e.g.,[1, 2, 7, 9, 10, 11, 12]. In this talk, we present a common framework, based on the concept of diffusive representations of fractional operators, under which we can subsume many seemingly different types of such novel methods. This more abstract perspective allows to obtain additional insight and a better understanding of how the algorithms behave. This is particularly significant because these methods are at the core of efficient schemes for solving fractional differential equations

Biography: Kai Diethelm is Professor for Mathematics and Applied Computer Science at the Technical University of Applied Sciences Wurzburg-Schweinfurt, Germany, where he leads the Scientific Computing Laboratory. He holds a Diploma in Mathematics from TU Braunschweig and a PhD in Computer Science from the University of Hildesheim. His main research interests are approximation theory and its applications, analytical and numerical aspects of fractional calculus (especially, fractional ordinary differential equations), and high performance computing

Bibliography

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[2] K. Diethelm. “Diffusive representations for the numerical evaluation of fractional integrals”. In: 2023 International Conference on Fractional Differentiation and Its Applications (ICFDA). IEEE. 2023, pp. 1–6.

[3] K. Diethelm, N. J. Ford, and A. D. Freed. “A predictor-corrector approach for the numerical solution of fractional differential equations”. In: Nonlinear Dynam. 29 (2002), pp. 3–22.

[4] K. Diethelm, N. J. Ford, and A. D. Freed. “Detailed error analysis for a fractional Adams method”. In: Numer. Algorithms 36 (2004), pp. 31–52.

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[7] J.-R. Li. “A fast time stepping method for evaluating fractional integrals”. In: SIAM J. Sci. Comput. 31 (2010), pp. 4696–4714.

[8] C. Lubich. “Fractional linear multistep methods for Abel-Volterra integral equations of the second kind”. In: Math. Comput. 45 (1985), pp. 463–469.

[9] W. McLean. “Exponential sum approximations for t−β ”. In: Contemporary computational mathematics—a celebration of the 80th birthday of Ian Sloan. Cham: Springer, 2018, pp. 911–930.

[10] S. J. Singh and A. Chatterjee. “Galerkin projections and finite elements for fractional order derivatives”. In: Nonlinear Dynam. 45 (2006), pp. 183–206.

[11] L. Yuan and O. P. Agrawal. “A numerical scheme for dynamic systems containing fractional derivatives”. In: J. Vibration Acoust. 124 (2002), pp. 321–324.

[12] W. Zhang et al. “An efficient and accurate method for modeling nonlinear fractional viscoelastic biomaterials”. In: Comput. Methods Appl. Mech. Eng. 362, 112834 (2020)