Oldham,
K. B.
, and
Spanier,
J.
, 1974, The Fractional Calculus,
Academic Press,
New York.

Miller,
K. S.
, and
Ross,
B.
, 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations,
Wiley,
New York.

Bagley,
R. L.
, and
Torvik,
P. J.
, 1985, “
Fractional Calculus in the Transient Analysis of Viscoelastically Damped Structures,” AIAA J.,
23, pp. 918–925.

[CrossRef]
Mainardi,
F.
, 1997, “
Fractional Calculus, Some Basic Problems in Continuum and Statistical Mechanics,” Fractals and Fractional Calculus in Continuum Mechanics,
A. Carpinteri
, and
F. Mainardi
, eds.,
Springer Verlag,
New York, pp. 291–348.

Rossikhin,
Y. A.
, and
Shitikova,
M. V.
, 1997, “
Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids,” ASME Appl. Mech. Rev.,
50, pp. 15–67.

[CrossRef]
Oldham,
K. B.
, 2010, “
Fractional Differential Equations in Electrochemistry,” Adv. Eng. Soft., **20**, pp. 9–12.

Krishnasamy,
V. S.
, and
Razzaghi,
M.
, 2016, “
The Numerical Solution of the Bagley-Torvik Equation With Fractional Taylor Method,” ASME J. Comput. Nonlinear Dyn.,
11, p. 051010.

[CrossRef]
Zak,
M. A.
, and
Tenreiro Machado,
J. A.
, 2017, “
On the Formulation and Numerical Simulation of Distributed Order Fractional Optimal Control,” Commun. Nonlinear Sci. Numer. Simul.,
52, pp. 177–189.

[CrossRef]
Li,
Y.
,
Sheng,
H.
, and
Chen,
Y. Q.
, 2011, “
On Distributed Order Integrator/Differentiator,” Signal Process,
91, pp. 1079–1084.

[CrossRef]
Kharazmi,
E.
,
Zayernouri,
M.
, and
Karniadakis,
G. E.
, 2017, “
Petrov-Galerkin and Spectral Collocation Methods for Distributed Order Differential Equations,” Signal Process.,
91, pp. 1079–1084.

Lorenzo,
C. F.
, and
Hartley,
T. T.
, 2002, “
Variable Order and Distributed Order Fractional Operators,” Nonlinear Dyn.,
29, pp. 57–98.

[CrossRef]
Chechkin,
A. V.
,
Klafter,
J.
, and
Sokolov,
I. M.
, 2003, “
Fractional Fokker-Planck Equation for Ultraslow Kinetics,” Europhys. Lett.,
63, pp. 326–332.

[CrossRef]
Naber,
M.
, 2004, “
Distributed Order Fractional Sub-Diffusion,” Fractals,
12, pp. 23–32.

[CrossRef]
Kochubei,
A. N.
, 2008, “
Distributed Order Calculus and Equations of Ultraslow Diffusion,” J. Math. Anal. Appl.,
340, pp. 252–281.

[CrossRef]
Diethelm,
K.
, and
Ford,
N. J.
, 2009, “
Numerical Analysis for Distributed Order Differential Equations,” J. Comput. Appl. Math.,
225, pp. 96–104.

[CrossRef]
Caputo,
M.
, 1995, “
Mean Fractional-Order-Derivative Differential Equation and Filters,” Annali dell'Università di Ferrara,
41(1) pp. 73–84.

Chechkin,
A. V.
,
Gorenflo,
R.
,
Sokolov,
I. M.
, and
Gonchar,
V. Y.
, 2003, “
Distributed Order Time Fractional Diffusion Equation,” Fract. Calc. Appl. Anal.,
6, pp. 259–279.

Bagley,
R. L.
, and
Torvik,
P. J.
, 2000, “
On the Existence of the Order Domain and the Solution of Distributed Order Equations-Part I,” Int. J. Appl. Math.,
2, pp. 865–882.

Bagley,
R. L.
, and
Torvik,
P. J.
, 2000, “
On the Existence of the Order Domain and the Solution of Distributed Order Equations-Part II,” Int. J. Appl. Math.,
2, pp. 965–988.

Jiao,
Z.
,
Chen,
Y. Q.
, and
Podlubny,
I.
, 2012, Distributed Order Dynamic System Stability, Simulation and Perspective,
Springer,
London.

Li,
J.
,
Liu,
F.
,
Feng,
L.
, and
Turner,
I.
, 2017, “
A Novel Finite Volume Method for the Riesz Space Distributed-Order Diffusion Equation,” Comput. Math. Appl.,
74, pp. 772–783.

[CrossRef]
Sokolov,
I. M.
,
Chechkin,
A. V.
, and
Klafter,
J.
, 2004, “
Distributed-Order Fractional Kinetics,” Acta Phys. Polo.,
35, pp. 1323–1341.

Hartley,
T. T.
, and
Lorenzo,
C. F.
, 2003, “
Fractional-Order System Identification Based on Continuous Order-Distributions,” Signal Process.,
83(1), pp. 2287–2300.

[CrossRef]
Zhou,
F.
,
Zhao,
Y.
,
Li,
Y.
, and
Chen,
Y. Q.
, 2013, “
Design, Implementation and Application of Distributed Order PI Control,” ISA Trans.,
52, pp. 429–437.

[CrossRef] [PubMed]
Lazovic,
G.
,
Vosika,
Z.
,
Lazarevi,
M.
,
Simic-Krsti,
J.
, and
Koruga,
D.
, 2014, “
Modeling of Bioimpedance for Human Skin Based on Fractional Distributed-Order Modified Cole Model,” FME Trans.,
42, pp. 74–81.

[CrossRef]
Su,
N.
,
Nelson,
P. N.
, and
Connor,
S.
, 2015, “
The Distributed-Order Fractional Diffusion-Wave Equation of Groundwater Flow: Theory and Application to Pumping and Slug Tests,” J. Hydrol.,
529, pp. 1262–1273.

[CrossRef]
Atanackovic,
T. M.
, 2002, “
A Generalized Model for the Uniaxial Isothermal Deformation of a Viscoelastic Body,” Acta Mech.,
159, pp. 77–86.

[CrossRef]
Atanackovic,
T. M.
,
Budincevic,
M.
, and
Pilipovic,
S.
, 2005, “
On a Fractional Distributed Order Oscillator,” J. Phys. A, Math. Gen.,
38(30), pp. 6703–6713.

[CrossRef]
Atanackovic,
T. M.
,
Pilipovic,
S.
, and
Zorica,
D.
, 2011, “
Distributed Order Fractional Wave Equation on a Finite Domain.Stress Relaxation in a Rod,” Int. J. Eng. Sci,
49(2), pp. 175–190.

[CrossRef]
Aminikhah,
H.
,
Sheikhani,
A. H. R.
,
Houlari,
T.
, and
Rezazadeh,
H.
, 2017, “
Numerical Solution of the Distributed-Order Fractional Bagley-Torvik Equation,” J. Autom. Sin., epub.

Katsikadelis,
J. T.
, 2014, “
Numerical Solution of Distributed Order Fractional Differential Equations,” J. Comput. Phys.,
259, pp. 11–22.

[CrossRef]
Mashayekhi,
S.
, and
Razzaghi,
M.
, 2016, “
Numerical Solution of Distributed Order Fractional Differential Equations by Hybrid Functions,” J. Comput. Phys.,
315, pp. 169–181.

[CrossRef]
Trung Duong,
P. L.
,
Kwok,
E.
, and
Lee,
M.
, 2016, “
Deterministic Analysis of Distributed Order Systems Using Operational Matrix,” Appl. Math. Model.,
40(3), pp. 1929–1940.

[CrossRef]
Mashoof,
M.
, and
Refahi Sheikhani,
A. H.
, 2017, “
Simulating the Solution of the Distributed Order Fractional Differential Equations by Block-Pulse Wavelets,” UPB Sci. Bull., Ser. A: Appl. Math. Phys.,
79, pp. 193–206.

Semary,
M. S.
,
Hassan,
H. N.
, and
Radwan,
A. G.
, 2018, “
Modified Methods for Solving Two Classes of Distributed Order Linear Fractional Differential Equations,” Appl. Math. Comput.,
323, pp. 106–119.

Zhu,
L.
, and
Fan,
Q.
, 2012, “
Solving Fractional Nonlinear Fredholm Integro-Differential Equations by the Second Kind Chebyshev Wavelet,” Commun. Nonlinear Sci. Num. Simul.,
17(6), pp. 2333–2341.

[CrossRef]
Heydari,
M. H.
,
Hooshmandasl,
M. R.
, and
Mohammadi,
F.
, 2014, “
Legendre Wavelets Method for Solving Fractional Partial Differential Equations With Dirichlet Boundary Conditions,” Appl. Math. Comput.,
276, pp. 267–276.

Saeedi,
H.
,
Mohseni Moghadam,
M.
,
Mollahasani,
N.
, and
Chuev,
G. N.
, 2011, “
A CAS Wavelet Method for Solving Nonlinear Fredholm Integro-Differential Equations of Fractional Order,” Commun. Nonlinear Sci. Num. Simul.,
16(3), pp. 1154–1163.

[CrossRef]
Li,
Y.
, and
Zhao,
W.
, 2010, “
Haar Wavelet Operational Matrix of Fractional Order Integration and Its Applications in Solving the Fractional Order Differential Equations,” Appl. Math. Comput.,
216(8), pp. 2276–2285.

Wang,
X. T.
, and
Li,
Y. M.
, 2009, “
Numerical Solutions of Integro Differential Systems by Hybrid of General Block-Pulse Functions and the Second Chebyshev Polynomials,” Appl. Math. Comput.,
209(2), pp. 266–272.

Singh,
V. K.
,
Pandey,
R. K.
, and
Singh,
S.
, 2010, “
A Stable Algorithm for Hankel Transforms Using Hybrid of Block-Pulse and Legendre Polynomials,” Comput. Phys. Commun.,
181(1), pp. 1–10.

[CrossRef]
Marzban,
H. R.
, and
Razzaghi,
M.
, 2005, “
Analysis of Time-Delay Systems Via Hybrid of Block-Pulse Functions and Taylor Series,” J. Vib. Control,
11(12), pp. 1455–1486.

Mashayekhi,
S.
,
Ordokhani,
Y.
, and
Razzaghi,
M.
, 2012, “
Hybrid Functions Approach for Nonlinear Constrained Optimal Control Problems,” Commun. Nonlinear Sci. Numer. Simul.,
17(4), pp. 1831–1843.

[CrossRef]
Marzban,
H. R.
, 2016, “
Parameter Identification of Linear Multi-Delay Systems Via a Hybrid of Block-Pulse Functions and Taylors Polynomials,” Int. J. Control,
90(3), pp. 504–518.

[CrossRef]
Mashayekhi,
S.
, and
Razzaghi,
M.
, 2015, “
Numerical Solution of Nonlinear Fractional Integro-Differential Equations by Hybrid Functions,” Eng. Anal. Bound. Elem.,
56, pp. 81–89.

[CrossRef]
Diethelm,
K.
, and
Ford,
N. J.
, 2001, “
Numerical Solution Methods for Distributed Order Differential Equations,” Fract. Calc. Appl. Anal.,
4, pp. 531–542.

Canuto,
C.
,
Hussaini,
M. Y.
,
Quarteroni,
A.
, and
Zang,
T. A.
, 2006, Spectral Methods: Fundamentals in Single Domains,
Springer,
New York.

Mashayekhi,
S.
,
Ordokhani,
Y.
, and
Razzaghi,
M.
, 2013, “
A Hybrid Functions Approach for the Duffing Equation,” Phys. Scr.,
888(2), pp. 1–8.

Mashayekhi,
S.
,
Razzaghi,
M.
, and
Wattanataweekul,
M.
, 2016, “
Analysis of Multi-Delay and Piecewise Constant Delay Systems by Hybrid Functions Approximation,” Differ. Equations Dyn. Syst.,
24(1), pp. 1–20.

[CrossRef]
Mainardi,
F.
,
Mura,
A.
,
Goreno,
R.
, and
Stojanovic,
M.
, 2007, “
The Two Form of Fractional Relaxation of Distributed Order,” J. Vib. Control,
13(9–10), pp. 1249–1268.

[CrossRef]
Podlubny,
I.
,
Skovranek,
T.
,
Vinagre Jara,
B. M.
,
Petras,
I.
,
Verbitsky,
V.
, and
Chen,
Y. Q.
, 2013, “
Matrix Approach to Discrete Fractional Calculus—III: Non-Equidistant Grids, Variable Step Length and Distributed Orders,” Phil. Trans. R. Soc. A,
371(1990), p. 20120153.

[CrossRef]
Katsikadelis, J. T.
, 2012, “
The Fractional Distributed Order Oscillator. A Numerical Solution,” J. Serb. Soc. Comput. Mech.,
6(1), pp. 148–159.