Matrix Mittag‑Leffler function in fractional systems and its computation

• Autorzy: Matychyn I. , Onyshchenko V.

Identyfikatory

DOI

10.24425/124266

• Języki publikacji: EN

Abstrakty

EN

Matrix Mittag‑Leffler functions play a key role in numerous applications related to systems with fractional dynamics. That is why the methods for computing the matrix Mittag‑Leffler function are so important. The matrix Mittag‑Leffler function is a generalization of matrix exponential function. This implies that some of numerous existing methods for computing the matrix exponential can be adapted for matrix Mittag‑Leffler functions as well. Unfortunately, the technique of scaling and squaring, widely used in computing of the matrix exponential, cannot be applied to matrix Mittag‑Leffler functions, as the latter do not possess the semigroup property. Here we describe a method of computing the matrix Mittag‑Leffler function based on the Jordan canonical form representation. This method is implemented with Matlab code [1].

Słowa kluczowe

EN

matrix Mittag‑Leffler function; Jordan canonical form; fractional calculus; fractional differential equation

PL

równanie różniczkowe cząstkowe; funkcja Mittag-Lefflera; rozkład Jordana

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2018
• Tom: Vol. 66, nr 4
• Strony: 495--500
• Opis fizyczny: Bibliogr. 20 poz., wykr.

Twórcy

autor

Matychyn I.

• University of Warmia and Mazury, Faculty of Mathematics and Computer Science, 54 Słoneczna St., 10-710 Olsztyn, Poland

autor

Onyshchenko V.

• State University of Telecommunications, 7 Solomyanska St., 03110 Kyiv, Ukraine

Bibliografia

• [1] I. Matychyn, “Matrix Mittag-Leffler function”, MATLAB Central File Exchange, File ID: 62790 (2017).
• [2] B. Datsko, Y. Luchko, and V. Gafiychuk, “Pattern formation in fractional reaction-diffusion systems with multiple homogeneous states”, International Journal of Bifurcation and Chaos, 22 (4), 1250087 (10) (2012).
• [3] Y. Luchko and M. Yamamoto, “General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems”, Fractional Calculus and Applied Analysis, 19 (3), 676‒695 (2016).
• [4] Y. Povstenko, Linear Fractional Diffusion-Wave Equation for Scientists and Engineers, Birkhäuser, New York, 2015.
• [5] A.B. Malinowska and D.F. Torres, Introduction to the Fractional Calculus of Variations, Imperial College Press, London, 2012.
• [6] N.J. Higham, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, 2008.
• [7] C. Moler and C. Van Loan, “Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later”, SIAM review, 45 (1), 3‒49 (2003).
• [8] R.L. Bagley and P.J. Torvik, “On the appearance of the fractional derivative in the behavior of real materials”, J. Appl. Mech., 51 (2), 294‒298 (1984).
• [9] A. Chikrii and S. Eidelman, “Generalized Mittag-Leffler matrix functions in game problems for evolutionary equations of fractional order”, Cybern. Syst. Analysis, 36 (3), 315‒338 (2000).
• [10] S. Samko, A. Kilbas, and O. Marichev, Fractional Integrals and Derivatives, Gordon & Breach, Amsterdam, 1993.
• [11] A. Kilbas, H. Srivastava, and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
• [12] A. Chikrii and I. Matichin, “Presentation of solutions of linear systems with fractional derivatives in the sense of Riemann–Liouville, Caputo, and Miller–Ross”, J. Autom. Inf. Sci., 40 (6), 1‒11 (2008).
• [13] T. Kaczorek, “Fractional positive continuous-time linear systems and their reachability”, Int. J. Appl. Math. Comput. Sci., 18 (2), 223‒228 (2008).
• [14] I. Podlubny, Fractional Differential Equations, Acad. Press, San Diego, 1999.
• [15] K. Diethelm and J. Ford, “Numerical solution of the Bagley-Torvik equation”, BIT Numerical Mathematics, 42 (3), 490‒507 (2002).
• [16] K. Balachandran, J. Park, and J. Trujillo, “Controllability of nonlinear fractional dynamical systems”, Nonlinear Analysis: Theory, Methods & Applications, 75 (4), 1919‒1926 (2012).
• [17] I. Matychyn and V. Onyshchenko, “Time-optimal control of fractional-order linear systems”, Fractional Calculus and Applied Analysis, 18 (3), 687‒696 (2015).
• [18] F.R. Gantmacher, The Theory of Matrices, AMS Chelsea Publishing, New York, 1959.
• [19] R. Garrappa, “Numerical evaluation of two and three parameter Mittag-Leffler functions”, SIAM J. Numer. Anal., 53 (3), 1350-1369 (2015).
• [20] I. Podlubny, “Matrix approach to discrete fractional calculus”, Fractional Calculus and Applied Analysis, 3 (4), 359‒386 (2000).

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).