Abstract
In this chapter we present a general approach to numerical solution to discretization of distributed-order derivatives and integrals, and to numerical solution of ordinary and partial differential equations of distributed order.
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Notes
- 1.
Here due to the use of the descending numbering of discretization nodes the roles of the matrices \(B_n^{(\alpha )}\) (originally for backward fractional differences) and \(F_n^{(\alpha )}\) (originally for forward fractional differences) are swapped in comparison with Podlubny (2000), where these matrices were introduced for the first time. However, we would like to preserve the notation \(B_n^{(\alpha )}\) for the case of the backward fractional differences approximation and \(F_n^{(\alpha )}\) for the case of the forward fractional differences approximation.
References
Bulgakov BV (1954) Kolebaniya (Vibrations). Gostekhizdat, Moscow
Gantmakher FR (1988) Theory of matrices. Nauka, Moscow
Loan CFV (2000) The ubiquitous Kronecker product. J Comput Appl Math 123:85–100
Ortigueira MD (2006) Riesz potential operators and inverses via fractional centred derivatives. Int J Math Math Sci 48391:1–12
Ortigueira MD, Batista AG (2008) On the relation between the fractional Brownian motion and the fractional derivatives. Phys Lett A 372:958–968
Podlubny I (2000) Matrix approach to discrete fractional calculus. Fract Cal Appl Anal 3(4):359–386
Podlubny I (2011) Matrix approach to distributed-order ODEs and PDEs. http://www.mathworks.com/matlabcentral/fileexchange/
Podlubny I, Skovranek T, Vinagre BM (2008) Matrix approach to discretization of ODEs and PDEs of arbitrary real order. http://www.mathworks.com/matlabcentral/fileexchange/22071
Podlubny I, Skovranek T, Vinagre BM (2009a) Matrix approach to discretization of ordinary and partial differential equations of arbitrary real order: the matlab toolbox. In: Proceedings of the ASME 2009 international design engineering technical conferences and computers and information in engineering conference IDETC/CIE 2009 August 30–September 2, 2009, San Diego, USA, article DETC2009-86944
Podlubny I, Chechkin A, Skovranek T, Chen YQ, Vinarge BM (2009b) Matrix approach to discrete fractional calculus ii: partial fractional differential equations. J Comput Phys 228(8):3137–3153
Podlubny I, Skovranek T, Verbickij V, Vinagre BM, Chen YQ, Petras I (2011) Discrete fractional calculus: non-equidistant grids and variable step length. In: Proceedings of the ASME 2011 international design engineering technical conferences and computers and information in engineering conference IDETC/CIE 2011 August 28–31, 2011, Washington, DC, USA, article DETC2011-47623
Pskhu AV (2005) Differencial’nye uravneniya drobnogo poryadka (Partial differential equations of fractional order). Nauka, Moscow (in Russian)
Skovranek T, Verbickij V, Tarte Y, Podlubny I (2010) Discretization of fractional-order operators and fractional differential equations on a non-equidistant mesh. In: Proceedings of the 4TH IFAC workshop on fractional differentiation and its applications, Badajoz, Spain, October 18–20, 2010, article FDA10-157
Suprunenko DA, Tyshkevich RI (1966) Commutative matrices. Nauka i Tekhnika, Minsk
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Jiao, Z., Chen, Y., Podlubny, I. (2012). Numerical Solution of Differential Equations of Distributed Order. In: Distributed-Order Dynamic Systems. SpringerBriefs in Electrical and Computer Engineering(). Springer, London. https://doi.org/10.1007/978-1-4471-2852-6_5
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DOI: https://doi.org/10.1007/978-1-4471-2852-6_5
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