Abstract
In this paper, we consider an inverse problem for a general class of fractional ordinary differential equations. Using the collage theorem, a consequence of Banach’s classical Fixed Point Theorem, we establish a “collage method” for solving this inverse problem under certain restrictions. We apply this method to some model fractional ordinary differential equations in which we only use solution data (perhaps adding relative noise to simulate experimental error) to recover other parameters present in the model.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Barnsley, M.F., Ervin, V., Hardin, D., Lancaster, J.: Solution of an inverse problem for fractals and other sets. Proc. Natl. Acad. Sci. USA 83, 1975–1977 (1995)
Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)
Groetsch, C.W.: Inverse Problems in the Mathematical Sciences. Vieweg, Wiesbaden (1993)
Kunze, H.E., Hicken, J.E., Vrscay, E.R.: Inverse problems for ODEs using contraction mappings and suboptimality of the ‘collage method’. Inverse Probl. 20, 977–991 (2004)
Kunze, H.E., La Torre, D., Levere, K.M., Vrscay, E.R.: Solving inverse problems for deterministic and random delay integral equations using the collage method. Int. J. Math. Stat. (IJMS), 11(1) (2012)
Kunze, H.E., Vasiliadis, S.: Using the collage method to solve ODEs inverse problems with multiple data sets. Theory Methods Appl. Nonlinear Anal. (2009)
Kunze, H.E., Vrscay, E.R.: Solving inverse problems for ordinary differential equations using the Picard contraction mapping. Inverse Probl. 15, 745–770 (1999)
Levere, K.: A Collage-Based Approach to Inverse Problems for Systems of Nonlinear Partial Differential Equations. Ph.D. thesis, University of Guelph, Guelph, Ontario (2012)
Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Switzerland (1993)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Levere, K.M., Van De Walker, B. (2018). Solving Inverse Problems for Fractional ODEs via the Collage Theorem. In: Kilgour, D., Kunze, H., Makarov, R., Melnik, R., Wang, X. (eds) Recent Advances in Mathematical and Statistical Methods . AMMCS 2017. Springer Proceedings in Mathematics & Statistics, vol 259. Springer, Cham. https://doi.org/10.1007/978-3-319-99719-3_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-99719-3_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-99718-6
Online ISBN: 978-3-319-99719-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)