zbMATH — the first resource for mathematics

Multi-order fractional differential equations and their numerical solution. (English) Zbl 1060.65070
The paper presents the numerical solution of multi-order fractional differential equations of the general (possibly nonlinear) form \[ y^{(\alpha )}(t)=f(t, y(t), y^{(\beta_1)}(t), y^{(\beta_2)}(t),\dots, y^{(\beta_n)}(t)) \] with \(\alpha>\beta_n>\beta_{n-1}>...>\beta_1\) and \(\alpha -\beta_n\leq1\), \(\beta_j -\beta_{j-1}\leq 1\) for all \(j\) and \(0<\beta_1\leq 1\). Its linear case is \(y^{(\alpha )}(t)=\lambda_oy(t)+ \sum_{j=1}^n\lambda_jy^{(\beta_j)}(t)+f(t)\). The initial conditions have the form \(y^{k}(t)=y_o^{(k)}\), \(k=0,1,\dots,\lceil\alpha\rceil -1\). The derivatives are understood in the Caputo sense.
A generalization of an approach employed in the solution of ordinary differential equations of order two or higher converting such equation to a system of equations of order one is used. It uses the fact that any real number can be approximated arbitrarily closely by a rational number. Thereby, the assumption on the commensuracy for fractional order equations can be ensured by an appropriate order approximation. A simple generalization of the theorem on the equivalence of a nonlinear system and the linear systems theory is presented first. Then, the nonlinear problem includes two Gronwall-type results for a two-term equation, the general existence-uniqueness as well as the structural stability results. A convergent and stable Adams-type numerical method is proposed including a specific numerical example.

65L05 Numerical methods for initial value problems
65L20 Stability and convergence of numerical methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems, general theory
26A33 Fractional derivatives and integrals
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
Full Text: DOI
[1] C.T.H. Baker, M.S. Derakhshan, Stability barriers to the construction of ρ,σ-reducible and fractional quadrature rules, in: H. Braß, G. Hämmerlin (Eds.), Numerical Integration III, Vol. 85 of Internat. Ser. Numer. Math., Birkhäuser, Basel, 1988, pp. 1-15 · Zbl 0652.65014
[2] L. Blank, Numerical treatment of differential equations of fractional order, Numerical Analysis Report 287, Manchester Centre for Computational Mathematics, 1996
[3] Diethelm, K, An algorithm for the numerical solution of differential equations of fractional order, Elect. transact. numer. anal., 5, 1-6, (1997) · Zbl 0890.65071
[4] Lubich, C, Fractional linear multistep methods for abel – volterra integral equations of the second kind, Math. comp., 45, 463-469, (1985) · Zbl 0584.65090
[5] Brunner, H; van der Houwen, P.J, The numerical solution of Volterra equations, (1986), North Holland Amsterdam · Zbl 0611.65092
[6] Diethelm, K; Ford, N.J, Numerical solution of the bagley – torvik equation, Bit, 42, 490-507, (2002) · Zbl 1035.65067
[7] K. Diethelm, Y. Luchko, Numerical solution of linear multi-term differential equations of fractional order, J. Comput. Anal. Appl., in press · Zbl 1083.65064
[8] A.R. Nkamnang, Diskretisierung von mehrgliedrigen Abelschen Integralgleichungen und gewöhnlichen Differentialgleichungen gebrochener Ordnung, Ph.D. thesis, Freie Universität, Berlin, available electronically from http://darwin.inf.fu-berlin.de/1999/23, 1999
[9] Podlubny, I, Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[10] Diethelm, K; Freed, A.D, On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity, (), 217-224
[11] Oldham, K.B; Spanier, J, The fractional calculus, vol. 111 of mathematics in science and engineering, (1974), Academic Press New York, London
[12] Samko, S.G; Kilbas, A.A; Marichev, O.I, Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach Yverdon · Zbl 0818.26003
[13] Gaul, L; Klein, P; Kempfle, S, Damping description involving fractional operators, Mech. syst. signal process., 5, 81-88, (1991)
[14] Glöckle, W.G; Nonnenmacher, T.F, A fractional calculus approach to self-similar protein dynamics, Biophys. J., 68, 1, 46-53, (1995)
[15] Mainardi, F, Fractional calculus: some basic problems in continuum and statistical mechanics, (), 291-348 · Zbl 0917.73004
[16] 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)
[17] Caputo, M, Linear models of dissipation whose Q is almost frequency independent, II, Geophys. J. royal astronom. soc., 13, 529-539, (1967)
[18] Edwards, J.T; Ford, N.J; Simpson, A.C, The numerical solution of linear multi-term fractional differential equations: systems of equations, J. comput. appl. math., 148, 401-418, (2002) · Zbl 1019.65048
[19] Diethelm, K; Ford, N.J, Analysis of fractional differential equations, J. math. anal. appl., 265, 229-248, (2002) · Zbl 1014.34003
[20] K. Diethelm, A.D. Freed, The FracPECE subroutine for the numerical solution of differential equations of fractional order, in: S. Heinzel, T. Plesser (Eds.), Forschung und wissenschaftliches Rechnen 1998, no. 52 in GWDG-Berichte, Gesellschaft für wissenschaftliche Datenverarbeitung, Göttingen, 1999, pp. 57-71
[21] Diethelm, K; Freed, A.D; Ford, N.J, A predictor – corrector approach for the numerical solution of fractional differential equations, Nonlinear dyn., 29, 3-22, (2002) · Zbl 1009.65049
[22] K. Diethelm, A.D. Freed, N.J. Ford, Detailed convergence analysis for a fractional Adams method, Berichte der Mathematischen Institute der TU Braunschweig 02/02, http://www.tu-bs.de/ diethelm/publications/adams.ps · Zbl 1055.65098
[23] Ford, N.J; Simpson, A.C, The numerical solution of fractional differential equations: speed versus accuracy, Numer. algorithms, 26, 333-346, (2001) · Zbl 0976.65062
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.