Numerical analysis for distributed-order differential equations

Journal of Computational and Applied Mathematics

Volumn 225, Issue 1, 2009, Pages 96-104

  Diethelm, Kai ^a   Ford, Neville J ^b  

^a GNS Gesellschaft für Numerische Simulation mbH Am Gaußberg 2   (Germany)

^b UNIVERSITY OF CHESTER   (United Kingdom)

Author keywords

Convergence; Distributed order differential equation; Multi term fractional differential equation; Numerical solution

Indexed keywords

DIFFERENTIAL EQUATIONS; NUMBER THEORY; NUMERICAL ANALYSIS; PROGRAMMING THEORY;

CONVERGENCE; CONVERGENCE THEORY; DISTRIBUTED-ORDER DIFFERENTIAL EQUATION; MULTI-TERM FRACTIONAL DIFFERENTIAL EQUATION; NUMERICAL SOLUTION; POSITIVE REAL;

CONVERGENCE OF NUMERICAL METHODS;

EID: 58549120748     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2008.07.018     Document Type: Article

References (28)

1. Atanackovic T.M., Budincevic M., and Pilipovic S. On a distributed-order oscillator. J. Phys. A 38 (2005) 6703-6713
2. Bagley R.L., and Torvik P.J. On the existence of the order domain and the solution of distributed order equations-Part I. Int. J. Appl. Math. 2 (2000) 865-882
3. Bagley R.L., and Torvik P.J. On the existence of the order domain and the solution of distributed order equations-Part II. Int. J. Appl. Math. 2 (2000) 965-987
4. Brass H. Quadraturverfahren (1977), Vandenhoeck & Ruprecht, Göttingen
5. Caputo M. Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. Roy. Astronom. Soc. 13 (1967) 529-539
6. Caputo M. Elasticitá e Dissipazione (1969), Zanichelli, Bologna
7. M. Caputo, 1995. Mean fractional-order-derivatives: Differential equations and filters. Ann. Univ. Ferrara Sez. VII (N. S.) XLI, 73-84
8. Caputo M. Distributed order differential equations modelling dielectric induction and diffusion. Fract. Calc. Appl. Anal. 4 (2001) 421-442
9. Caputo M. Diffusion with space memory modelled with distributed order space fractional differential equations. Ann. Geophys. 46 (2003) 223-234
10. Diethelm K. An algorithm for the numerical solution of differential equations of fractional order. Electron. Trans. Numer. Anal. 5 (1997) 1-6
11. m E methods. Computing 71 (2003) 305-319
12. K. Diethelm, 2008. Multi-term fractional differential equations, multi-order fractional differential systems and their numerical solution, J. Eur. Syst. Autom. 42 (in press)
13. Diethelm K., Ford J.M., Ford N.J., and Weilbeer M. Pitfalls in fast numerical solution of fractional differential equations. J. Comput. Appl. Math. 186 (2006) 482-503
14. Diethelm K., and Ford N.J. Numerical solution methods for distributed order differential equations. Fract. Calc. Appl. Anal. 4 (2001) 531-542
15. Diethelm K., and Ford N.J. Analysis of fractional differential equations. J. Math. Anal. Appl. 265 (2002) 229-248
16. Diethelm K., and Ford N.J. Numerical solution of the Bagley-Torvik equation. BIT 42 (2002) 490-507
17. Diethelm K., and Ford N.J. Multi-order fractional differential equations and their numerical solution. Appl. Math. Comp. 154 (2004) 621-640
18. Diethelm K., Ford N.J., and Freed A.D. Detailed error analysis for a fractional Adams method. Numer. Algorithms 36 (2004) 31-52
19. Diethelm K., and Luchko Y. Numerical solution of linear multi-term initial value problems of fractional order. J. Comput. Anal. Appl. 6 (2004) 243-263
20. Edwards J.T., Ford N.J., and Simpson A.C. The numerical solution of multi-term fractional differential equations: Systems of equations. J. Comput. Appl. Math. 148 (2002) 401-418
21. T.T. Hartley, C.F. Lorenzo, 1999. Fractional system identification: An approach using continuous order-distributions, NASA Technical Memorandum 1999-209640, NASA Glenn Research Center, Cleveland OH
22. Lubich C. Fractional linear multistep methods for Abel-Volterra integral equations of the second kind. Math. Comp. 45 (1985) 463-469
23. Lubich C. A stability analysis of convolution quadratures for Abel-Volterra integral equations. IMA J. Numer. Anal. 6 (1986) 87-101
24. Lubich C. Convolution quadrature revisited. BIT 44 (2004) 503-514
25. Luchko Y., and Gorenflo R. An operational method for solving fractional differential equations with the Caputo derivatives. Acta. Math. Vietnam. 24 (1999) 207-233
26. A.R. Nkamnang, 1998. Diskretisierung von mehrgliedrigen Abelschen Integralgleichungen und gewöhnlichen Differentialgleichungen gebrochener Ordnung, Ph.D. Thesis, Freie Universität Berlin
27. Sokolov I.M., Chechkin A.V., and Klafter J. Distributed order fractional kinetics. Acta Phys. Polon. B 35 (2004) 1323-1341
28. Umarov S., and Gorenflo R. Cauchy and non-local multi-point problems for distributed order pseudo-differential equations, Part I. J. Anal. Appl. 24 (2005) 449-466