# zbMATH — the first resource for mathematics

Distributed order equations as boundary value problems. (English) Zbl 1268.45005
Summary: There has been a recent increase in interest in the use of distributed order differential equations (particularly in the case where the derivatives are given in the Caputo sense) to model various phenomena. Recent papers have provided insights into the numerical approximation of the solution, and some results on existence and uniqueness have been proved. In each case, the representation of the solution depends, among other parameters, on Caputo-type initial conditions. In this paper we discuss the existence and uniqueness of solutions and we propose a numerical method for their approximation in the case where the initial conditions are not known and, instead, some Caputo-type conditions are given away from the origin.

##### MSC:
 45J05 Integro-ordinary differential equations 65R20 Numerical methods for integral equations 34A08 Fractional ordinary differential equations and fractional differential inclusions
Full Text:
##### 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] Caputo, M., Elasticité e dissipazione, (1969), Zanichelli Bologna [3] Caputo, M., Distributed order differential equations modelling dielectric induction and diffusion, Fract. calc. appl. anal., 4, 421-442, (2001) · Zbl 1042.34028 [4] Bagley, R.L.; Torvik, P.J., On the existence of the order domain and the solution of distributed order equations – part I, Int. J. appl. math., 2, 865-882, (2000) · Zbl 1100.34006 [5] Bagley, R.L.; Torvik, P.J., On the existence of the order domain and the solution of distributed order equations – part II, Int. J. appl. math., 2, 965-987, (2000) · Zbl 1142.34301 [6] Hartley, T.T.; Lorenzo, C.F., Fractional system identification: an approach using continuous order-distributions, () · Zbl 1229.93142 [7] Lorenzo, C.F.; Hartley, T.T., Variable order and distributed order fractional operators, Nonlinear dyn., 29, 57-98, (2002) · Zbl 1018.93007 [8] Chechkin, A.V.; Gorenflo, R.; Sokolov, I.M.; Gonchar, V. Yu., Distributed order time fractional diffusion equation, Fract. calc. appl. anal., 6, 259-279, (2003) · Zbl 1089.60046 [9] Caputo, M., Mean fractional-order-derivatives: differential equations and filters, Ann. univ. ferrara sez. VII (NS), XLI, 73-84, (1995) · Zbl 0882.34007 [10] Diethelm, K.; Ford, N.J., Numerical analysis for distributed order differential equations, J. comp. appl. math., 225, 96-104, (2009) · Zbl 1159.65103 [11] Diethelm, K.; Ford, N.J., Numerical solution methods for distributed order differential equations, Fract. calc. appl. anal., 4, 531-542, (2001) · Zbl 1032.65070 [12] Ford, N.J.; Morgado, M.L., Fractional boundary value problems: analysis and numerical methods, Fract. calc. appl. anal., 14, 554-567, (2011) · Zbl 1273.65098 [13] K. Diethelm, N.J. Ford, Volterra integral equations and fractional calculus: do neighbouring solutions intersect? J. Integral Equ. Appl. (in press). · Zbl 1238.45003 [14] N.J. Ford, M.L. Morgado, Numerical methods for multi-term fractional boundary value problems, Springer Proceedings in Mathematics, (submitted for publication). · Zbl 1320.34008 [15] Diethelm, K.; Ford, N.J., Numerical solution of the bagley – torvik equation, Bit, 42, 490-507, (2002) · Zbl 1035.65067 [16] Diethelm, K.; Ford, N.J.; Freed, A.D., A predictor – corrector approach for the numerical solution of fractional differential equations, Nonlinear dyn., 29, 3-22, (2002) · Zbl 1009.65049 [17] Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, Electr. trans. numer. anal., 5, 1-6, (1997) · Zbl 0890.65071 [18] Diethelm, K.; Ford, N.J., Multi-order fractional differential equations and their numerical solution, Appl. math. comp., 154, 621-640, (2004) · Zbl 1060.65070
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.