×

zbMATH — the first resource for mathematics

Analysis of fractional differential equations. (English) Zbl 1014.34003
The authors discuss the existence, uniqueness and structural stability of solutions to nonlinear differential equations of fractional order. They take the differential operators in the Riemann-Liouville sense and the initial conditions are specified according to Caputo’s suggestion, in order to allow for an interpretation in a physically meaningful way.
They also investigate the dependence of the solution on the order of the differential equation and on the initial condition, and they relate their results to the selection of an appropriate numerical scheme for solving fractional differential equations.

MSC:
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
26A33 Fractional derivatives and integrals
34A45 Theoretical approximation of solutions to ordinary differential equations
65L05 Numerical methods for initial value problems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Caputo, M.: Linear models of dissipation whose Q is almost frequency independent, II. Geophys. J. Roy. astronom. Soc. 13, 529-539 (1967)
[2] Corduneanu, C.: Principles of differential and integral equations. (1971) · Zbl 0208.10701
[3] Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order. Electron. trans. Numer. anal. 5, 1-6 (1997) · Zbl 0890.65071
[4] Diethelm, K.: Generalized compound quadrature formulae for finite-part integrals. IMA J. Numer. anal. 17, 479-493 (1997) · Zbl 0871.41021
[5] Diethelm, K.; Freed, A. D.: On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity. Scientific computing in chemical engineering II–computational fluid dynamics, reaction engineering and molecular properties, 217-224 (1999)
[6] Diethelm, K.; Walz, G.: Numerical solution of fractional order differential equations by extrapolation. Numer. algorithms 16, 231-253 (1997) · Zbl 0926.65070
[7] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G.: Higher transcendental functions. (1955) · Zbl 0064.06302
[8] Gaul, L.; Klein, P.; Kempfle, S.: Damping description involving fractional operators. Mech. systems signal processing 5, 81-88 (1991)
[9] Glöckle, W. G.; Nonnenmacher, T. F.: A fractional calculus approach to self-similar protein dynamics. Biophys. J. 68, 46-53 (1995)
[10] Gorenflo, R.; Mainardi, F.: Fractional oscillations and Mittag–Leffler functions. (1996) · Zbl 0916.34011
[11] Gorenflo, R.; Rutman, R.: On ultraslow and intermediate processes. Transform methods and special functions, sofia 1994, 61-81 (1995) · Zbl 0923.34005
[12] Linz, P.: Analytical and numerical methods for Volterra equations. (1985) · Zbl 0566.65094
[13] Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. Fractals and fractional calculus in continuum mechanics, 291-348 (1997) · Zbl 0917.73004
[14] Metzler, R.; Schick, W.; Kilian, H. -G.; Nonnenmacher, T. F.: Relaxation in filled polymers: A fractional calculus approach. J. chem. Phys. 103, 7180-7186 (1995)
[15] Oldham, K. B.; Spanier, J.: The fractional calculus. Mathematics in science and engineering 111 (1974) · Zbl 0292.26011
[16] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications. (1993) · Zbl 0818.26003
[17] Weissinger, J.: Zur theorie und anwendung des iterationsverfahrens. Math. nachr. 8, 193-212 (1952) · Zbl 0046.34105
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.