Abstract

Fractional calculus provides a rigorous mathematical framework to describe anomalous stochastic processes by generalizing the notion of classical differential equations to their fractional-order counterparts. By introducing the fractional orders as uncertain variables, we develop an operator-based uncertainty quantification framework in the context of stochastic fractional partial differential equations (SFPDEs), subject to additive random noise. We characterize different sources of uncertainty and then, propagate their associated randomness to the system response by employing a probabilistic collocation method (PCM). We develop a fast, stable, and convergent Petrov–Galerkin spectral method in the physical domain in order to formulate the forward solver in simulating each realization of random variables in the sampling procedure.

Issue Section:
Research Papers
Topics:
Partial differential equations, Space, Uncertainty, Uncertainty quantification, Boundary-value problems, Dimensions, Stochastic processes, Noise (Sound), Differential equations, Stability

References

1.
Zhang
,
Y.
,
Sun
,
H.
,
Stowell
,
H. H.
,
Zayernouri
,
M.
, and
Hansen
,
S. E.
,
2017
, “
A Review of Applications of Fractional Calculus in Earth System Dynamics
,”
Chaos, Solitons Fractals
,
102
, pp.
29
–
46
.10.1016/j.chaos.2017.03.051
2.
Jaishankar
,
A.
, and
McKinley
,
G. H.
,
2014
, “
A Fractional K-BKZ Constitutive Formulation for Describing the Nonlinear Rheology of Multiscale Complex Fluids
,”
J. Rheol.
,
58
(
6
), pp.
1751
–
1788
.10.1122/1.4892114
3.
Jha
,
R.
,
Kaw
,
P. K.
,
Kulkarni
,
D. R.
,
Parikh
,
J. C.
, and
ADITYA Team
,
2003
, “
Evidence of Lévy Stable Process in Tokamak Edge Turbulence
,”
Phys. Plasmas
,
10
(
3
), pp.
699
–
704
.10.1063/1.1541607
4.
del Castillo-Negrete
,
D.
,
Carreras
,
B. A.
, and
Lynch
,
V. E.
,
2004
, “
Fractional Diffusion in Plasma Turbulence
,”
Phys. Plasmas
,
11
(
8
), pp.
3854
–
3864
.10.1063/1.1767097
5.
Jaishankar
,
A.
, and
McKinley
,
G. H.
,
2013
, “
Power-Law Rheology in the Bulk and at the Interface: Quasi-Properties and Fractional Constitutive Equations
,”
Proc. R. Soc. A
,
469
(
2149
), p.
20120284
.10.1098/rspa.2012.0284
6.
Naghibolhosseini
,
M.
,
2015
, “
Estimation of Outer-Middle Ear Transmission Using DPOAEs and Fractional-Order Modeling of Human Middle Ear
,” Ph.D. thesis, City University of New York, New York.
7.
Naghibolhosseini
,
M.
, and
Long
,
G. R.
,
2017
, “
Fractional-Order Modelling and Simulation of Human Ear
,”
Int. J. Comput. Math.
,
95
(
6–7
), pp.
1257
–
1273
.10.1080/00207160.2017.1404038
8.
Samiee
,
M.
,
Akhavan-Safaei
,
A.
, and
Zayernouri
,
M.
,
2019
, “
A Fractional Subgrid-Scale Model for Turbulent Flows: Theoretical Formulation and a Priori Study
,” preprint arXiv: 1909.09943.
9.
Meral
,
F. C.
,
Royston
,
T. J.
, and
Magin
,
R.
,
2010
, “
Fractional Calculus in Viscoelasticity: An Experimental Study
,”
Commun. Nonlinear Sci. Numer. Simul.
,
15
(
4
), pp.
939
–
945
.10.1016/j.cnsns.2009.05.004
10.
Cullen
,
A. C.
, and
Frey
,
H. C.
,
1999
,
Probabilistic Techniques in Exposure Assessment: A Handbook for Dealing With Variability and Uncertainty in Models and Inputs
,
Springer Science & Business Media
, New York, p.
336
.
11.
Roy
,
C. J.
, and
Oberkampf
,
W. L.
,
2011
, “
A Comprehensive Framework for Verification, Validation, and Uncertainty Quantification in Scientific Computing
,”
Comput. Methods Appl. Mech. Eng.
,
200
(
25–28
), pp.
2131
–
2144
.10.1016/j.cma.2011.03.016
12.
Mullins
,
J.
, and
Mahadevan
,
S.
,
2016
, “
Bayesian Uncertainty Integration for Model Calibration, Validation, and Prediction
,”
ASME J. Verif., Validation Uncertainty Quantif.
,
1
(
1
), p.
011006
.10.1115/1.4032371
13.
Der Kiureghian
,
A.
, and
Ditlevsen
,
O.
,
2009
, “
Aleatory or Epistemic? Does It Matter?
,”
Struct. Saf.
,
31
(
2
), pp.
105
–
112
.10.1016/j.strusafe.2008.06.020
14.
Benson
,
D. A.
,
Wheatcraft
,
S. W.
, and
Meerschaert
,
M. M.
,
2000
, “
Application of a Fractional Advection-Dispersion Equation
,”
Water Resour. Res.
,
36
(
6
), pp.
1403
–
1412
.10.1029/2000WR900031
15.
Baeumer
,
B.
,
Benson
,
D. A.
,
Meerschaert
,
M. M.
, and
Wheatcraft
,
S. W.
,
2001
, “
Subordinated Advection-Dispersion Equation for Contaminant Transport
,”
Water Resour. Res.
,
37
(
6
), pp.
1543
–
1550
.
16.
Fishman
,
G. S.
,
1996
,
Monte Carlo: Concepts, Algorithms, and Applications
, Springer, New York.
17.
Del Moral
,
P.
,
Doucet
,
A.
, and
Jasra
,
A.
,
2006
, “
Sequential Monte Carlo Samplers
,”
J. R. Stat. Soc.: Ser. B
,
68
(
3
), pp.
411
–
436
.10.1111/j.1467-9868.2006.00553.x
18.
Beskos
,
A.
,
Jasra
,
A.
,
Law
,
K.
,
Tempone
,
R.
, and
Zhou
,
Y.
,
2017
, “
Multilevel Sequential Monte Carlo Samplers
,”
Stochastic Processes Their Appl.
,
127
(
5
), pp.
1417
–
1440
.10.1016/j.spa.2016.08.004
19.
Jasra
,
A.
,
Law
,
K. J.
, and
Zhou
,
Y.
,
2016
, “
Forward and Inverse Uncertainty Quantification Using Multilevel Monte Carlo Algorithms for an Elliptic Nonlocal Equation
,”
Int. J. Uncertainty Quantif.
,
6
(
6
), pp.
501
–
514
.10.1615/Int.J.UncertaintyQuantification.2016018661
20.
Owen
,
N. E.
,
Challenor
,
P.
,
Menon
,
P. P.
, and
Bennani
,
S.
,
2017
, “
Comparison of Surrogate-Based Uncertainty Quantification Methods for Computationally Expensive Simulators
,”
SIAM/ASA J. Uncertainty Quantif.
,
5
(
1
), pp.
403
–
435
.10.1137/15M1046812
21.
Ghanem
,
R. G.
, and
Spanos
,
P. D.
,
2003
,
Stochastic Finite Elements: A Spectral Approach
,
Courier Corporation
, Springer, New York.
22.
Xiu
,
D.
, and
Karniadakis
,
G. E.
,
2002
, “
Modeling Uncertainty in Steady State Diffusion Problems Via Generalized Polynomial Chaos
,”
Comput. Methods Appl. Mech. Eng.
,
191
(
43
), pp.
4927
–
4948
.10.1016/S0045-7825(02)00421-8
23.
Xiu
,
D.
, and
Karniadakis
,
G. E.
,
2002
, “
The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations
,”
SIAM J. Sci. Comput.
,
24
(
2
), pp.
619
–
644
.10.1137/S1064827501387826
24.
Knio
,
O. M.
, and
Le Maître
,
O. P.
,
2006
, “
Uncertainty Propagation in CFD Using Polynomial Chaos Decomposition
,”
Fluid Dyn. Res.
,
38
(
9
), pp.
616
–
640
.10.1016/j.fluiddyn.2005.12.003
25.
Najm
,
H. N.
,
2009
, “
Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics
,”
Annu. Rev. Fluid Mech.
,
41
(
1
), pp.
35
–
52
.10.1146/annurev.fluid.010908.165248
26.
Van Dam
,
N.
, and
Rutland
,
C.
,
2016
, “
Uncertainty Quantification of Large-Eddy Spray Simulations
,”
ASME J. Verif., Validation Uncertainty Quantif.
,
1
(
2
), p.
021006
.10.1115/1.4032196
27.
Babuska
,
I.
,
Tempone
,
R.
, and
Zouraris
,
G. E.
,
2004
, “
Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations
,”
SIAM J. Numer. Anal.
,
42
(
2
), pp.
800
–
825
.10.1137/S0036142902418680
28.
Babuška
,
I.
,
Tempone
,
R.
, and
Zouraris
,
G. E.
,
2005
, “
Solving Elliptic Boundary Value Problems With Uncertain Coefficients by the Finite Element Method: The Stochastic Formulation
,”
Comput. Methods Appl. Mech. Eng.
,
194
(
12–16
), pp.
1251
–
1294
.10.1016/j.cma.2004.02.026
29.
Le Maître
,
O. P.
,
Knio
,
O. M.
,
Najm
,
H. N.
, and
Ghanem
,
R. G.
,
2004
, “
Uncertainty Propagation Using Wiener–Haar Expansions
,”
J. Comput. Phys.
,
197
(
1
), pp.
28
–
57
.10.1016/j.jcp.2003.11.033
30.
Le Maître
,
O. P.
,
Najm
,
H. N.
,
Ghanem
,
R. G.
, and
Knio
,
O. M.
,
2004
, “
Multi-Resolution Analysis of Wiener-Type Uncertainty Propagation Schemes
,”
J. Comput. Phys.
,
197
(
2
), pp.
502
–
531
.10.1016/j.jcp.2003.12.020
31.
Schuss
,
Z.
,
1980
, “
Singular Perturbation Methods in Stochastic Differential Equations of Mathematical Physics
,”
SIAM Rev.
,
22
(
2
), pp.
119
–
155
.10.1137/1022024
32.
Babuška
,
I.
, and
Chatzipantelidis
,
P.
,
2002
, “
On Solving Elliptic Stochastic Partial Differential Equations
,”
Comput. Methods Appl. Mech. Eng.
,
191
(
37–38
), pp.
4093
–
4122
.10.1016/S0045-7825(02)00354-7
33.
Todor
,
R. A.
,
2005
, “
Sparse Perturbation Algorithms for Elliptic PDE's With Stochastic Data
,” Ph.D. thesis, ETH Zurich, Zurich, Switzerland.
34.
Winter
,
C. L.
, and
Tartakovsky
,
D. M.
,
2002
, “
Groundwater Flow in Heterogeneous Composite Aquifers
,”
Water Resour. Res.
,
38
(
8
), pp.
23
–
1
–
23-11
.10.1029/2001WR000450
35.
Liu
,
W. K.
,
Belytschko
,
T.
, and
Mani
,
A.
,
1986
, “
Probabilistic Finite Elements for Nonlinear Structural Dynamics
,”
Comput. Methods Appl. Mech. Eng.
,
56
(
1
), pp.
61
–
81
.10.1016/0045-7825(86)90136-2
36.
Liu
,
W. K.
,
Belytschko
,
T.
, and
Mani
,
A.
,
1986
, “
Random Field Finite Elements
,”
Int. J. Numer. Methods Eng.
,
23
(
10
), pp.
1831
–
1845
.10.1002/nme.1620231004
37.
Xiu
,
D.
, and
Hesthaven
,
J. S.
,
2005
, “
High-Order Collocation Methods for Differential Equations With Random Inputs
,”
SIAM J. Sci. Comput.
,
27
(
3
), pp.
1118
–
1139
.10.1137/040615201
38.
Babuška
,
I.
,
Nobile
,
F.
, and
Tempone
,
R.
,
2007
, “
A Stochastic Collocation Method for Elliptic Partial Differential Equations With Random Input Data
,”
SIAM J. Numer. Anal.
,
45
(
3
), pp.
1005
–
1034
.10.1137/050645142
39.
Nobile
,
F.
,
Tempone
,
R.
, and
Webster
,
C. G.
,
2008
, “
A Sparse Grid Stochastic Collocation Method for Partial Differential Equations With Random Input Data
,”
SIAM J. Numer. Anal.
,
46
(
5
), pp.
2309
–
2345
.10.1137/060663660
40.
Smolyak
,
S.
,
1963
, “
Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions
,”
Sov. Math. Dokl.
,
4
(
5
), pp.
240
–
243
.http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=dan&paperid=27586&option_lang=eng
41.
Gorenflo
,
R.
,
Mainardi
,
F.
,
Moretti
,
D.
, and
Paradisi
,
P.
,
2002
, “
Time Fractional Diffusion: A Discrete Random Walk Approach
,”
Nonlinear Dyn.
,
29
(
1–4
), pp.
129
–
143
.10.1023/A:1016547232119
42.
Sun
,
Z.
, and
Wu
,
X.
,
2006
, “
A Fully Discrete Difference Scheme for a Diffusion-Wave System
,”
Appl. Numer. Math.
,
56
(
2
), pp.
193
–
209
.10.1016/j.apnum.2005.03.003
43.
Lin
,
Y.
, and
Xu
,
C.
,
2007
, “
Finite Difference/Spectral Approximations for the Time-Fractional Diffusion Equation
,”
J. Comput. Phys.
,
225
(
2
), pp.
1533
–
1552
.10.1016/j.jcp.2007.02.001
44.
Wang
,
H.
,
Wang
,
K.
, and
Sircar
,
T.
,
2010
, “
A Direct O(N log2N) Finite Difference Method for Fractional Diffusion Equations
,”
J. Comput. Phys.
,
229
(
21
), pp.
8095
–
8104
.10.1016/j.jcp.2010.07.011
45.
Wang
,
K.
, and
Wang
,
H.
,
2011
, “
A Fast Characteristic Finite Difference Method for Fractional Advection–Diffusion Equations
,”
Adv. Water Resour.
,
34
(
7
), pp.
810
–
816
.10.1016/j.advwatres.2010.11.003
46.
Cao
,
J.
, and
Xu
,
C.
,
2013
, “
A High Order Schema for the Numerical Solution of the Fractional Ordinary Differential Equations
,”
J. Comput. Phys.
,
238
(
1
), pp.
154
–
168
.10.1016/j.jcp.2012.12.013
47.
Zeng
,
F.
,
Li
,
C.
,
Liu
,
F.
, and
Turner
,
I.
,
2015
, “
Numerical Algorithms for Time-Fractional Subdiffusion Equation With Second-Order Accuracy
,”
SIAM J. Sci. Comput.
,
37
(
1
), pp.
A55
–
A78
.10.1137/14096390X
48.
Zayernouri
,
M.
, and
Matzavinos
,
A.
,
2016
, “
Fractional Adams-Bashforth/Moulton Methods: An Application to the Fractional Keller–Segel Chemotaxis System
,”
J. Comput. Phys.
,
317
, pp.
1
–
14
.10.1016/j.jcp.2016.04.041
49.
Karniadakis
,
G.
, and
Sherwin
,
S.
,
2013
,
Spectral/hp Element Methods for Computational Fluid Dynamics
,
Oxford University Press
, New York.
50.
Canuto
,
C.
,
Hussaini
,
M. Y.
,
Quarteroni
,
A.
, and
Zang
,
T. A.
,
2006
,
Spectral Methods
,
Springer
, New York.
51.
Rawashdeh
,
E.
,
2006
, “
Numerical Solution of Fractional Integro-Differential Equations by Collocation Method
,”
Appl. Math. Comput.
,
176
(
1
), pp.
1
–
6
.10.1016/j.amc.2005.09.059
52.
Khader
,
M.
,
2011
, “
On the Numerical Solutions for the Fractional Diffusion Equation
,”
Commun. Nonlinear Sci. Numer. Simul.
,
16
(
6
), pp.
2535
–
2542
.10.1016/j.cnsns.2010.09.007
53.
Khader
,
M.
, and
Hendy
,
A.
,
2012
, “
The Approximate and Exact Solutions of the Fractional-Order Delay Differential Equations Using Legendre Pseudospectral Method
,”
Int. J. Pure Appl. Math.
,
74
(
3
), pp.
287
–
297
.https://ijpam.eu/contents/2012-74-3/1/
54.
Li
,
X.
, and
Xu
,
C.
,
2009
, “
A Space-Time Spectral Method for the Time Fractional Diffusion Equation
,”
SIAM J. Numer. Anal.
,
47
(
3
), pp.
2108
–
2131
.10.1137/080718942
55.
Li
,
X.
, and
Xu
,
C.
,
2010
, “
Existence and Uniqueness of the Weak Solution of the Space-Time Fractional Diffusion Equation and a Spectral Method Approximation
,”
Commun. Comput. Phys.
,
8
(
5
), pp.
1016
–
1051
.10.4208/cicp.020709.221209a
56.
Chen
,
S.
,
Shen
,
J.
, and
Wang
,
L.
,
2014
, “
Generalized Jacobi Functions and Their Applications to Fractional Differential Equations
,”
Math. Comput.
85
, pp.
1603
–
1638
.https://www.ams.org/journals/mcom/2016-85-300/S0025-5718-2015-03035-X/home.html
57.
Wang
,
H.
, and
Zhang
,
X.
,
2015
, “
A High-Accuracy Preserving Spectral Galerkin Method for the Dirichlet Boundary-Value Problem of Variable-Coefficient Conservative Fractional Diffusion Equations
,”
J. Comput. Phys.
,
281
, pp.
67
–
81
.10.1016/j.jcp.2014.10.018
58.
Bhrawy
,
A. H.
,
Doha
,
E. H.
,
Baleanu
,
D.
, and
Ezz-Eldien
,
S. S.
,
2015
, “
A Spectral Tau Algorithm Based on Jacobi Operational Matrix for Numerical Solution of Time Fractional Diffusion-Wave Equations
,”
J. Comput. Phys.
,
293
, pp.
142
–
156
.10.1016/j.jcp.2014.03.039
59.
Zayernouri
,
M.
, and
Karniadakis
,
G. E.
,
2013
, “
Fractional Sturm–Liouville Eigen-Problems: Theory and Numerical Approximations
,”
J. Comput. Phys
.,
252
, pp.
495
–
517
.https://www.sciencedirect.com/science/article/pii/S0021999113004610
60.
Zayernouri
,
M.
,
Ainsworth
,
M.
, and
Karniadakis
,
G. E.
,
2015
, “
Tempered Fractional Sturm–Liouville Eigenproblems
,”
SIAM J. Sci. Comput.
,
37
(
4
), pp.
A1777
–
A1800
.10.1137/140985536
61.
Lischke
,
A.
,
Zayernouri
,
M.
, and
Zhang
,
Z.
, “
Spectral and Spectral Element Methods for Fractional Advection–Diffusion–Reaction Equations
,”
Handbook of Fractional Calculus With Applications, Numerical Methods
, Walter de Gruyter GmbH & Co KG, Berlin.
62.
Lischke
,
A.
,
Zayernouri
,
M.
, and
Karniadakis
,
G. E.
,
2017
, “
A Petrov–Galerkin Spectral Method of Linear Complexity for Fractional Multiterm ODEs on the Half Line
,”
SIAM J. Sci. Comput.
,
39
(
3
), pp.
A922
–
A946
.10.1137/17M1113060
63.
Suzuki
,
J. L.
,
Zayernouri
,
M.
,
Bittencourt
,
M. L.
, and
Karniadakis
,
G. E.
,
2016
, “
Fractional-Order Uniaxial Visco-Elasto-Plastic Models for Structural Analysis
,”
Comput. Methods Appl. Mech. Eng.
,
308
, pp.
443
–
467
.10.1016/j.cma.2016.05.030
64.
Suzuki
,
J. L.
, and
Zayernouri
,
M.
,
2018
, “
An Automated Singularity-Capturing Scheme for Fractional Differential Equations
,” preprint arXiv:1810.12219.
65.
Samiee
,
M.
,
Zayernouri
,
M.
, and
Meerschaert
,
M. M.
,
2019
, “
A Unified Spectral Method for PDEs With Two-Sided Derivatives; Part I: A Fast Solver
,”
J. Comput. Phys.
,
385
, pp.
225
–
243
.10.1016/j.jcp.2018.02.014
66.
Samiee
,
M.
,
Kharazmi
,
E.
, and
Zayernouri
,
M.
,
2017
, “
Fast Spectral Methods for Temporally-Distributed Fractional PDEs
,” Spectral and High Order Methods for Partial Differential Equations (ICOSAHOM 2016) (Lecture Notes in Computational Science and Engineering, Vol. 119), Springer, Cham, Switzerland, pp.
651
–
667
.
67.
Samiee
,
M.
,
Zayernouri
,
M.
, and
Meerschaert
,
M. M.
,
2019
, “
A Unified Spectral Method for FPDEs With Two-Sided Derivatives—Part II: Stability, and Error Analysis
,”
J. Comput. Phys.
,
385
, pp.
244
–
261
.10.1016/j.jcp.2018.07.041
68.
Samiee
,
M.
,
Kharazmi
,
E.
,
Zayernouri
,
M.
, and
Meerschaert
,
M. M.
,
2018
, “
Petrov–Galerkin Method for Fully Distributed-Order Fractional Partial Differential Equations
,” preprint arXiv:1805.08242.
69.
Kharazmi
,
E.
,
Zayernouri
,
M.
, and
Karniadakis
,
G. E.
,
2017
, “
Petrov–Galerkin and Spectral Collocation Methods for Distributed Order Differential Equations
,”
SIAM J. Sci. Comput.
,
39
(
3
), pp.
A1003
–
A1037
.10.1137/16M1073121
70.
Kharazmi
,
E.
,
Zayernouri
,
M.
, and
Karniadakis
,
G. E.
, “
A Petrov–Galerkin Spectral Element Method for Fractional Elliptic Problems
,”
Comput. Methods Appl. Mech. Eng.
,
324
, pp.
512
–
536
.10.1016/j.cma.2017.06.006
71.
Kharazmi
,
E.
, and
Zayernouri
,
M.
,
2018
, “
Fractional Pseudo-Spectral Methods for Distributed-Order Fractional PDEs
,”
Int. J. Comput. Math.
,
95
(
6–7
), pp.
1340
–
1361
.10.1080/00207160.2017.1421949
72.
Kharazmi
,
E.
, and
Zayernouri
,
M.
,
2019
, “
Fractional Sensitivity Equation Method: Application to Fractional Model Construction
,”
J. Sci. Comput.
,
80
(
1
), pp.
110
–
140
.10.1007/s10915-019-00935-0
73.
Rizzi
,
F.
,
Najm
,
H. N.
,
Debusschere
,
B. J.
,
Sargsyan
,
K.
,
Salloum
,
M.
,
Adalsteinsson
,
H.
, and
Knio
,
O. M.
,
2012
, “
Uncertainty Quantification in MD Simulations. Part I: Forward Propagation
,”
Multiscale Model. Simul.
,
10
(
4
), pp.
1428
–
1459
.10.1137/110853169
74.
Miller
,
K. S.
, and
Ross
,
B.
,
1993
,
An Introduction to the Fractional Calculus and Fractional Differential Equations
,
Wiley
,
New York
.
75.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
San Diego, CA
.
76.
Atanackovic
,
T. M.
,
Pilipovic
,
S.
,
Stankovic
,
B.
, and
Zorica
,
D.
,
2014
,
Fractional Calculus With Applications in Mechanics: Vibrations and Diffusion Processes
,
Wiley
, Hoboken, NJ.
77.
Afzali
,
F.
,
Kapucu
,
O.
, and
Feeny
,
B. F.
,
2016
, “
Vibrational Analysis of Vertical-Axis Wind-Turbine Blades
,”
ASME
Paper No: DETC2016-60374.10.1115/DETC2016-60374
78.
Afzali
,
F.
,
Acar
,
G. D.
, and
Feeny
,
B. F.
,
2017
, “
Analysis of the Periodic Damping Coefficient Equation Based on Floquet Theory
,”
ASME
Paper No. DETC2017-68450
.10.1115/DETC2017-68450
79.
de Moraes
,
E. A. B.
,
Zayernouri
,
M.
, and
Meerschaert
,
M. M.
, “
An Integrated Sensitivity-Uncertainty Quantification Framework for Stochastic Phase-Field Modeling of Material Damage
,”
Int. J. Numer. Methods Eng.
(submitted).
80.
Zamani
,
V.
,
Kharazmi
,
E.
, and
Mukherjee
,
R.
,
2015
, “
Asymmetric Post-Flutter Oscillations of a Cantilever Due to a Dynamic Follower Force
,”
J. Sound Vib.
,
340
, pp.
253
–
266
.10.1016/j.jsv.2014.11.020
81.
Varghaei
,
P.
,
Kharazmi
,
E.
,
Suzuki
,
J. L.
, and
Zayernouri
,
M.
, “
Nonlinear Vibration of Fractional Viscoelastic Cantilever Beam: Application to Structural Health Monitoring
,”
ASME J. Vib. Acoust.
(submitted).
82.
Ari
,
M.
,
Faal
,
R. T.
, and
Zayernouri
,
M.
,
2019
, “
Vibrations Suppression of Fractionally Damped Plates Using Multiple Optimal Dynamic Vibration Absorbers
,”
Int. J. Comput. Math.
, pp.
1
–
24
.10.1080/00207160.2019.1594792
83.
Sapmaz
,
A.
,
Acar
,
G. D.
, and
Feeny
,
B. F.
,
2019
, “
Approximate General Responses of Tuned and Mistuned 4-Degree-of-Freedom Systems With Parametric Stiffness
,”
Topics in Modal Analysis & Testing
, Vol.
9
, Springer, Cham, Switzerland, pp.
315
–
324
.: https://link.springer.com/chapter/10.1007/978-3-319-74700-2_35
84.
Loéve
,
M.
,
1977
,
Probability Theory
,
Springer-Verlag
,
New York
.
85.
Su
,
C. H.
, and
Lucor
,
D.
,
2006
, “
Covariance Kernel Representations of Multidimensional Second-Order Stochastic Processes
,”
J. Comput. Phys.
,
217
(
1
), pp.
82
–
99
.10.1016/j.jcp.2006.02.006
86.
Oksendal
,
B.
,
1998
,
Stochastic Differential Equations
,
Springer-Verlag
, New York.
87.
Foo
,
J.
,
Wan
,
X.
, and
Karniadakis
,
G. E.
,
2008
, “
The Multi-Element Probabilistic Collocation Method (ME-PCM): Error Analysis and Applications
,”
J. Comput. Phys.
,
227
(
22
), pp.
9572
–
9595
.10.1016/j.jcp.2008.07.009
88.
Barthelmann
,
V.
,
Novak
,
E.
, and
Ritter
,
K.
,
2000
, “
High Dimensional Polynomial Interpolation on Sparse Grids
,”
Adv. Comput. Math.
,
12
(
4
), pp.
273
–
288
.10.1023/A:1018977404843
89.
Novak
,
E.
, and
Ritter
,
K.
,
1999
, “
Simple Cubature Formulas With High Polynomial Exactness
,”
Constr. Approximation
,
15
(
4
), pp.
499
–
522
.10.1007/s003659900119
90.
Novak
,
E.
, and
Ritter
,
K.
,
1996
, “
High Dimensional Integration of Smooth Functions Over Cubes
,”
Numer. Math.
,
75
(
1
), pp.
79
–
97
.10.1007/s002110050231
91.
Ervin
,
V. J.
, and
Roop
,
J. P.
,
2007
, “
Variational Solution of Fractional Advection Dispersion Equations on Bounded Domains in Rd
,”
Numer. Methods Partial Differ. Equations
,
23
(
2
), pp.
256
–
281
.10.1002/num.20169
Copyright © 2019 by ASME
You do not currently have access to this content.