We consider numerical solutions of nonlinear multiterm fractional integrodifferential equations, where the order of the highest derivative is fractional and positive but is otherwise arbitrary. Here, we extend and unify our previous work, where a Galerkin method was developed for efficiently approximating fractional order operators and where elements of the present differential algebraic equation (DAE) formulation were introduced. The DAE system developed here for arbitrary orders of the fractional derivative includes an added block of equations for each fractional order operator, as well as forcing terms arising from nonzero initial conditions. We motivate and explain the structure of the DAE in detail. We explain how nonzero initial conditions should be incorporated within the approximation. We point out that our approach approximates the system and not a specific solution. Consequently, some questions not easily accessible to solvers of initial value problems, such as stability analyses, can be tackled using our approach. Numerical examples show excellent accuracy.

1.
Bagley
,
R. L.
, and
Torvik
,
P. J.
, 1983, “
A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity
,”
J. Rheol.
0148-6055,
27
(
3
), pp.
201
210
.
2.
Torvik
,
P. J.
, and
Bagley
,
R. L.
, 1984, “
On the Appearance of the Fractional Derivative in the Behavior of Real Materials
,”
ASME J. Appl. Mech.
0021-8936,
51
, pp.
294
298
.
3.
Podlubny
,
I.
, 1999, “
Fractional-Order Systems and PIλDμ-Controllers
,”
IEEE Trans. Autom. Control
0018-9286,
44
(
1
), pp.
208
214
.
4.
Moreau
,
X.
,
Ramus-Serment
,
C.
, and
Oustaloup
,
A.
, 2002, “
Fractional Differentiation in Passive Vibration Control
,”
Nonlinear Dyn.
0924-090X,
29
(
1–4
), pp.
343
362
.
5.
Agrawal
,
O. P.
, 2004, “
A General Formulation and Solution Scheme for Fractional Optimal Control Problems
,”
Nonlinear Dyn.
0924-090X,
38
, pp.
323
337
.
6.
Mainardi
,
F.
, 1996, “
Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena
,”
Chaos, Solitons Fractals
0960-0779,
7
(
9
), pp.
1461
1477
.
7.
Mainardi
,
F.
,
Pironi
,
P.
, and
Tampieri
,
F.
, 1995, “
On a Generalization of Basset Problem via Fractional Calculus
,”
Proceedings of the CANCAM 95
.
8.
Koh
,
C. G.
, and
Kelly
,
J. M.
, 1990, “
Application of Fractional Derivative to Seismic Analysis of Base-Isolated Models
,”
Earthquake Eng. Struct. Dyn.
0098-8847,
19
, pp.
229
241
.
9.
Grigolini
,
P.
,
Rocco
,
A.
, and
West
,
B. J.
, 1999, “
Fractional Calculus as a Macroscopic Manifestation of Randomness
,”
Phys. Rev. E
1063-651X,
59
(
3
), pp.
2603
2613
.
10.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
, 2006,
Theory and Applications of Fractional Differential Equations
,
Elsevier
,
New York
.
11.
Samko
,
S. G.
,
Kilbas
,
A. A.
, and
Marichev
,
O. I.
, 1993,
Fractional Integrals and Derivatives: Theory and Applications
,
Gordon and Breach
,
Amsterdam
.
12.
Momani
,
S.
, 2006, “
A Numerical Scheme for the Solution of Multi-Order Fractional Differential Equations
,”
Appl. Math. Comput.
0096-3003,
182
, pp.
761
770
.
13.
Diethelm
,
K.
,
Ford
,
N. J.
, and
Freed
,
A. D.
, 2002, “
A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations
,”
Nonlinear Dyn.
0924-090X,
29
, pp.
3
22
.
14.
Diethelm
,
K.
, and
Ford
,
N. J.
, 2004, “
Multi-Order Fractional Differential Equations and Their Numerical Solution
,”
Appl. Math. Comput.
0096-3003,
154
, pp.
621
640
.
15.
Chen
,
J. -H.
, and
Chen
,
W. -C.
, 2008, “
Chaotic Dynamics of the Fractionally Damped van der Pol Equation
,”
Chaos, Solitons Fractals
0960-0779,
35
, pp.
188
198
.
16.
Diethelm
,
K.
, 2009, “
An Improvement of a Nonclassical Numerical Method for the Computation of Fractional Derivatives
,”
ASME J. Vibr. Acoust.
0739-3717,
131
(
1
), p.
014502
.
17.
Barbosa
,
R. S.
,
Machado
,
J. A. T.
,
Vinagre
,
B. M.
, and
Claderón
,
A. J.
, 2007, “
Analysis of the van der Pol Oscillator Containing Derivatives of Fractional Order
,”
J. Vib. Control
1077-5463,
13
(
9–10
), pp.
1291
1301
.
18.
Ge
,
Z. -M.
, and
Hsu
,
M. -Y.
, 2007, “
Chaos in a Generalized van der Pol System and in Its Fractional Order System
,”
Chaos, Solitons Fractals
0960-0779,
33
, pp.
1711
1745
.
19.
Yuan
,
L.
, and
Agrawal
,
O. P.
, 2002, “
Numerical Scheme for Dynamic Systems Containing Fractional Derivatives
,”
ASME J. Vibr. Acoust.
0739-3717,
124
, pp.
321
324
.
20.
Agrawal
,
O. P.
, 2009, “
A Numerical Scheme for Initial Compliance and Creep Response of a System
,”
Mech. Res. Commun.
0093-6413,
36
(
4
), pp.
444
451
.
21.
Singh
,
S. J.
, and
Chatterjee
,
A.
, 2006, “
Galerkin Projections and Finite Elements for Fractional Order Derivatives
,”
Nonlinear Dyn.
0924-090X,
45
(
1–2
), pp.
183
206
.
22.
Caputo
,
M.
, 1967, “
Linear Models of Dissipation Whose Q is Almost Frequency Independent, Part II
,”
Geophys. J. R. Astron. Soc.
0016-8009,
13
, pp.
529
539
.
23.
Singh
,
S. J.
, and
Chatterjee
,
A.
, 2008, “
DAE-Based Solution of Nonlinear Multiterm Fractional Integrodifferential Equations
,”
Journal Européen des Systèmes Automatisés
,
42
(
6–8
), pp.
677
688
.
24.
Singh
,
S. J.
, and
Chatterjee
,
A.
, 2010, “
Beyond Fractional Derivatives: Local Approximation of Other Convolution Integrals
,”
Proc. R. Soc. London, Ser. A
0950-1207,
466
, pp.
563
581
.
25.
Doetsch
,
G.
, 1974,
Introduction to the Theory and Application of the Laplace Transformation
,
Springer-Verlag
,
New York
.
26.
Hairer
,
E.
, and
Wanner
,
G.
, 1991,
Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems
,
Springer
,
Berlin
.
27.
Singh
,
S. J.
, 2007, “
New Solution Methods for Fractional Order Systems
,” Ph.D. thesis, Indian Institute of Science, Bangalore, India.
28.
Baleanu
,
D.
, 2009, “
Fractional Variational Principles in Action
,”
Phys. Scr.
0031-8949,
T136
, p.
014006
.
29.
Magin
,
R.
,
Feng
,
X.
, and
Baleanu
,
D.
, 2009, “
Solving the Fractional Order Bloch Equation
,”
Concepts in Magnetic Resonance Part A
,
34A
(
1
), pp.
16
23
.
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