World Journal of Mechanics, 2012, 2, 325-333
doi:10.4236/wjm.2012.26038 Published Online December 2012 (http://www.SciRP.org/journal/wjm)
A Time History Method for Analysing Operational Piping
Vibrations
Subrata Saha
Department of Piping, Reliance Ports & Terminals Ltd.—Engineering Division, Reliance Refinery, Jamnagar, India
Received October 2, 2012; revised November 4, 2012; accepted November 16, 2012
ABSTRACT
Vibration failure of piping is a serious problem and a matter of concern for safety and reliability of plant operations.
Fatigue is the main cause of such failures. Due to the complexity of the phenomenon no closed form design solutions
are available. In our study an analytical technique based on the theory of vibrations in the time domain has been pre-
sented. Using the inverse theory, the problem has been reduced to a system of Volterra Integral equations to be solved
simultaneously at every time step. The solution of the inverse problem may be used in the conventional method to cal-
culate stresses and end reactions which are important from the perspective of engineering design and condition moni-
toring. The method is robust, simple and can b e easily adopted by practicing engineers.
Keywords: Vibration; Inverse Problem; Direct Problem; Time History; Fatigue Failure; Vibration Screening Criterion;
Integral Equation; Frequency; Damping
1. Introduction
Piping witnesses various vibratory loads throughout its
life cycle. These vibrations if not controlled will lead to
fatigue failures at points of high stress intensity or could
even damage the supports. All these could lead to plant
outage or even have more severe consequences like fire
and loss of human lives [1]. Thus it is imperativ e that the
piping system is to be safeguarded against s uc h h aza rd s.
For the standpoint of engineering design adequacy
check, dynamic analysis has to be carried out for the
piping system for which the forcing function has to be
known. This is the conventional method of analysis,
which is also termed mathematically as the direct prob-
lem [2-4]. But the major difficulty in the dealing with the
vibration problems lies in the estimation of the forcing
function. If the exciting forces can be quantified pre-
cisely, the system response can be determined with great
accuracy by the existing analytical methods. Thus the
estimation of the forcing function is essential for carrying
out the dynamic analysis and subsequent engineering
design check.
Unfortunately this is not readily possible in most cases
since the vibrations in an operating pipeline are flow in-
duced. The complexity of flow patterns and the mecha-
nism of force coupling render the determination of the
forcing function extremely difficult. In such a scenario
data in the form of field vibration measurements in con-
junction with some analytical methods can provide a
basis for estimating the dynamic force and stress [5-9].
The theory of Inverse Problems [2-4] invariably forms
the basic theoretical framework for such studies. Inverse
Theory has found wide applications in the fields of engi-
neering and mathematics. It has become indispensible
where the problems are ill-posed in absence of data. In
this sense Inverse Theory has got tremendous practical
value.
The vibration problems can be studied in both the fre-
quency and the time domains. In the frequency domain
the frequency response of the system is studied for the
determination of various parameters of interest [7-10].
For a recent work in the frequency domain for a closely
related application, one may refer to [8]. In the time do-
main the response time history of different points in the
form of observations are taken as the input for the study.
Some good amount of work has been done in the ap-
plication of inverse problems in the field of hyperbolic
partial differential equations or wave equations [5,11-
14]. The determination of point sources from observa-
tions has been the main theme of their studies. The pre-
sent study may be considered as a special case of such
applications. However there are some major differences.
Concentrated forces in the form of point loads at interior
points in the domain have been considered in the previ-
ous works, whereas in our case the point sources are the
end moments at the boundaries. Also we have considered
damping in the system which simulates the real case and
thus the treatment is to a great extent different from the

S. SAHA
326
previous ones. According to us such study in the time
domain has not been done. A few references [7] can be
found wherein dynamic stresses have been computed
from displacement measurements in the time domain.
The method for computing stresses from the displace-
ment measurements has been shown. However the me t hod
of force estimation is not provided. Also the topics on
existence and uniqueness of the solution which are the
key issues for inverse problems have not been addressed.
In our paper we shall present the theory in the time
domain and also the numerical scheme for the problem.
The numerical algorithm is simple and it can be easily
built into any of the common spreadsheet programs with
the help of macros. This we be lieve will find wide appli-
cation amongst the practicing engineers.
2. Current Practice—Vibration Screening
Criteria
The current practice is the vibration screening criteria
method. In this method the vibration response parameters
like velocity or displacements are measured in situ and
compared against some acceptance criteria. These are in
the form of graphs known vibration severity charts [15].
For refinery and petrochemical industries, these charts
are being extensively used. They are normally found to
be conservative.
Another widely used criterion is of ASME OM Code
[16] a standard followed for nuclear piping. Here the
vibration velocity for a piping span between nodes is the
criterion. The limiting valu e of the ve locity is d etermined
by the empirical relationship involving coefficients which
depend on several parameters like weld arrangements,
mass lumping etc. When the peak value of velocity is
less that 12.7 mm/sec, it may be assumed that the piping
has sufficient dynamic capacity. If the vibration exceeds
this level, the guide recommends reviewing the same
with more information on the potential reasons of vibra-
tions and improving the vibration levels.
It is seen that all the above methods are conservative
and provide a cook book or a go/no-go approach. They
only tell us whether the vibrations are within the accept-
able levels or not. It is not possible to have a quantitative
estimation of the forcing function and the actual stress
levels which are essential fo r a design check. In our work
this problem has been studied on the framework of In-
verse Theory as mentioned earlier.
3. Mathematical Background
It has been shown that for a simply supported pipe [vide
Figure 1] the response at any location in the span may be
determined by the vibration measurements at two distinct
points in the span. For the straight span, the excitation
source is through moments from the adjoining segments
as there are no points of excitation by forces in the span.
A distinguishing feature of this method is that no infor-
mation is required on the natural B.C’s. This is remark-
able since in the direct formulation, the B.C’s govern the
solution, whereas in th is case they are not playing a role.
This is also significant for the fact that practically it is
impossible to measure the B.C’s.
3.1. Notations
In this section we will describe the notations used in the
sequel. Please refer to Table 1.
3.2. Problem Formulation
The basic configuration is shown in Figure 1 in which
the simply supported pipe is excited by moments
()
0
M
t
and
()
L
M
t at the ends. The length of the span is L.
Considering Bernoulli-Euler formulation and viscous
damping, the dynamic equation of motion in the time
domain [17,18] is as follows:
() ()()
4
,, ,0
cEI
u xtu xtDu xt
mm
 
++
 = 
 
  (1)
Table 1. Nomenclature.
Symbol Description
x Space variable.
T Time variable.
T Total time.
L Length of the pipe span.
 Spatial derivative operator.
f
 Time derivative of f (t).
0,
L
Υ
Υ
Shape functions.
m Mass per un it leng th .
c Viscous d ampi ng co effi cient .
E Modulus of elasticity.
I Moment of inertia.
()
,uxtTotal displacement variable.
()
,vxt Dynamic displacement variable.
0,
L
θθ
 Rotational accelerations.
0,
L
M
MEn d mo me n t s .
n
ω
Undamped natural frequency (nth mode) .
dn
ω Damped frequency (nth mode).
n
φ
Mode shape (nth mode).
n
ξ
Modal damping.
()
n
qt
Generalized modal displacement.
0, nnL
ΓΓ Modal participation factors.
[
]
0,T Closed interval between 0 to T.
[
]
0,CT Space of continuous functions in [0, T].
u
 Acceleration measurement time history.
0,
L
K
KKernel of the integral equation.
TH. Time History.

S. SAHA
327
Figure 1. Piping configuration.
Boundary Conditions (B.C’s):
()( )
;0, 0, 0ut uLt== (2)
()()( )()
22
0;0,, L
EI utMtEI uLtMt==DD (3)
Equation (1) pertains to vibrations without any exter-
nal loading in the span. It is similar to the free vibration
equation. However for our case the excitations are thro ugh
end moments. This is shown in B.C’s (3). In absence of
any forcing function in the span the sources of vibrations
are through the ends. As mentioned earlier, this is a sig-
nificant development, since in the earlier studies the
point sources of excitation forces have been dealt with.
Our study is aimed at the determination of the end mo-
ments by observation of the response of some internal
points. Then the response at any point in the span can be
determined. It also assumed that the system starts from
rest (i.e. it has zero initial conditions).
We now express the total displacement function in
terms of dynamic and quasi-static components as below.
() ()()
,,,uxtvxt gxt=+ (4)
The quasi-static part can be written in terms of the
shape or pa rticipation f un ct i ons
()
0
x
Υ
and
()
L
x
Υ
as
()() ()()()
00
,LL
g
xtx txt θθ =+
Υ
Υ
(5)
The function
()
0
x
Υ
(respectively)
()
L
x
Υ
is de-
fined as the displacement of the points in the span with a
unit positive rotation at end at 0x= and
x
L= re-
spectively. Since the system starts from rest we haves
() ()
000 00 L
θθ == (6)
() ()
000 00 L
θθ ==

(7)
The following can be easily verified.
()( )
0, 0, 0vt vLt== (8)
()( )
0, 0, 0vt vLt==DD (9)
() ()
001 01 L
==
Υ
Υ
DD (10)
() ()
44
000 L
xx==
Υ
Υ
DD (11)
() ()
,0 0,0 0vx vx==
 (12)
Using (7) to (10) we can recast (1) as follows:
() ()()
() ()
()
()
4
00
,, ,
,
LL
cEI
vxtvxtDvxt
mm
c
ttgxt
m
θθ
 
++
 
=

 

+−

−

 
 
Υ
Υ
(13)
It is customary to consider damping in terms of dy-
namic displacements only and hence the last term in (13)
may be dropped . Equation (13) represen ts a forced vi bra-
tion problem with a distributed loading for a pipe with
clamped ends. Equation (12) represents the initial condi-
tions. However this being an inverse problem, the forcing
function 0
θ
 and
L
θ
 (the rotational accelerations) be-
come the unknown quantities which are to be determined.
Once they are found out, the problem is transformed into
a direct problem and is solvable using commonly used
numerical methods. The modal superposition method
will be the basis of our study in the sequel.
In line with the modal superposition theory the dy-
namic displacement may be expressed as the sum of
modal components as below:
() ()()
1
,nn
n
vxtxq t φ
∞
=
=; (14)
Here
()
n
x
φ is the Eigen-function for the th n mode
for the clamped pipe.
()
n
x
φ satisfies the B.C’s (7) and
(8). In addition we have the orthonormal properties:
0
ord;f0
L
nm
x
nm φφ =≠
 (15)
0
d1
L
nmx
φφ =
 (16)
Further we define :
() ()
00
0
d
L
nn
x
xx φ =Γ 
Υ
(17)
() ()
0
d
L
Ln Ln
x
xx φ =Γ 
Υ
(18)
With the above properties we get modal equation from
(12) as below:
()() ()
() ()
2
00
2
nnnnnn
nnLL
qtqt qt
tt
ωξ ω
θθ
++
=Γ +Γ
 
  (19)
Equation (19) is the differential equation for the gen-
eralized modal displacement
()
n
qt
. This is a second
order differential in time variable. Two initial conditions
are required for its solution. In our case we have zero
displacement and zero velocity at time 0t=. The solu-
tion for
()
n
qt
is:

S. SAHA
328
() () () ()
()
()
()
()
()
00
0
1
expsin d
t
ndnnnLL
nn dn
qt
tt
ωθτθτ
ξ
ωτω ττ
=Γ+Γ
⋅− −−
 
(20)
Here 2
1
dn nn
ωω
ξ
=− . It is also known as the
damped natural frequency.
It is clearly seen that we need to get estimates of the
rotational accelerations to obtain
()
n
qt
. The modal ac-
celeration is obtained differentiating (20) twice and using
the below identity:
() ()()
,d, t ut utut
τ
ττ τ
=
=+

 (21)
We now de fin e the following te rms:
()() ()
00nnLL
h τθτθτ =Γ +Γ
 
(22)
()
()
() ()
22 2
1,sin
nnndndn
ft t τ
ξ
ωωω τ =− −
(23)
()( ) ()
2,2 cos nnndn dn
ft t τ
ξ
ωωω τ =−− (24)
() ()
()
() ()
()
12
,e ,,
nn
t
nnn
th ftft
ξω τ
ψ
ττττ
−−
=+ (25)
1
ndn
αω = (26)
The expression for generalized modal acceleration is
()( )()
0
,d
t
nnn
qttht αψττ =+

 (27)
()()()()
10
,,d
t
N
nnn
n
vxtxt ht φαψττ
=

=+



 (28)
The total acceleration which is a sum of dynamic and
quasi-static components can be written as
() ()()()()()
00
,, LL uxtvxtxtxt θθ =+ +
 
 
Υ
Υ
(29)
Substituting (19) in (20) for
()
,vxt we have
()( )()
() ()()()()
10
00
,d
,
t
N
nnn
n
LL
xtht
x
txtuxt
φαψττ
θθ
=

+


++ =

 
Υ
Υ
(30)
Equation (30) is the fundamental equation for our
study. It is an integral equation of the second kind [11,
19]. The right hand side (RHS) quantity represents the
acceleration which is the observation. The left hand side
(LHS) contains the unknown forcing functions in form of
rotational accelerations. Our study will focus on the
method of solution for the unknown rotational accelera-
tions.
4. Solution Method
We will now address the aspects of exis tence and un ique -
ness by means of the following propositions.
4.1. Proposition 1
For a system as defined by the governing differential
Equations (13) with B.C’s (8), (9) and initial conditions
(12), the respon se (i.e. displacement, velocity etc.) at any
location x can be obtained from the measurement of ac-
celeration time history at any two interior points.
Proof: We begin with the assumption that
()
0t
θ
 and
()
Lt
θ
 belong to the function space
()
0,CT
(i.e. the
space of continuous functions).
It is shown in Appendix A1 that (29) may be reduced
to
()()
()()()()
0
00
00
,d ,d
L
tt
LL
tt
Kt Kt
ϕϕ
τ
ϕ
ττ τ
ϕ
ττ
+
=+

 
 (31)
It is seen that (31) represents a pair of Volterra Integral
equations [11,19,20] (one for
()
0t
ϕ
 and the other for
()
Lt
ϕ
 ).
For the first part of the assertion we need to show that
the trivial solution is the only solution. It has been proved
in Appendix A2 that both the integral equations have
unique fixed points. It is also seen that the trivial solution
exists. Hence it is also the unique solution.
As the existence and uniqueness of (31) has been es-
tablished, we can solve for the individual forcing func-
tions at any time t from the below system:
()()()()()()
11 0101
,, LL uxt vxtxtxt θθ =+ +
 
 
Υ
Υ
(32)
()() ()()()()
22020 2
,, LL uxtvxtxtxt θθ =+ +
 
 
Υ
Υ
(33)
Equations 32 and 33 represent a system of integral
equations. The numerical solution method for a single
equation may be found in standard texts on numerical
methods [20]. Thus Equations (32) and (33) may be ex-
pressed in the matrix form as
() ()()
tt t
AX U= (34)
In an expanded form (34) can be written as
()
() ()
11 1211
0
21 2222
t
Lt
t
AA uh
AA uh
θ
θ
 
−

=

   −

 

 
  (35)
The elements of the matrix are as below
() ()
1
N
ijnj niji
n
A
xx φ
=
=Γ +

Υ
(36)
() ()
() ()
() () ()
0
1
12
ee
,,d
nn nn
Nttt
ininjnj
n
nnn
hx
ft ft
ξ
ωτ
ξ
ωτ
φαθτ
τξτ τ
−−−−
=
=Γ
+

(37)
t
iii
Uuh=−
 (38)
where 1, 2i= and 0,1j= and N is the number of

S. SAHA
329
modes. The solution t
X
is obtained from (30) as be-
low.
() ()
()
()
1
ttt
X
AU
−
= (39)
(
()
t
U is the RHS vector of the measurements).
From the existence of the solution we know that op-
erator
()
t
A
is invertible and hence
()
t
X
is the unique
solution.
4.2. Proposition 2
The response obtained at any point is unique and inde-
pendent of the observation points. This means that if
()
1,vxt
is the response calculated on the basis of obser-
vations for the set of points
()
12
,
x
x and
()
2,vxt
be
for the set
()
34
,
x
x we have 12 vv=.
Proof: From Proposition 1 we know that the rotational
accelerations are determined uniquely. The response is
calculated on the basis of the solution of the direct prob-
lem (12) which is also unique. The forcing function is the
same for all cases. Hence we have 12 vv=.
4.3. Numerical Method
In this section we shall describe the numerical scheme
for the calculation of acceleration forcing function. Let T
be the total time interval for our study and T
N the num-
ber of time steps and N the number of modes. The objec-
tive of the scheme is to obtain t
X
for all the time in-
stants t1, t2,···etc. up to T. For convenien ce
()
i
t
X
will be
denoted as i
X
. The steps are described below:
1) Start with
() () 00
XU= (40)
2) For any r t we define the following quantities
()
()
()
{
() () ()
}
0
12
,, e
,,
nn rk
rt
ninjn jk
k
nrknn rk
Hnij
f
tftt
ξω τ
φαθτ
τξ τ
−−
=
=Γ
+Δ

(41)
() ()
,N
j
injni
n
Cij x
φ
=+Γ

Υ
(42)
The components of the matrix A in (35) is constructed
as
()
,
ij ij
A
Ai jC== (43)
The RHS vector U is
()
1
01
,,
N
ii
jn
Uu Hnij
==
=−

 (44)
(Here i = 1 and 2)
3) Solve for r
X
from (39).
4) Repeat Steps 2 to 3 till T
rN=
4.4. Determination of Response Variables
We obtain the rotational accelerations as a solution of the
inverse problem. Now we can determine the forcing
function completely. Thus the problem is transformed
into a direct one, which may be solved using existing
methods for determining various response quantities like
displacement, velocity and stress time histories.
For example, Bending Moment, Shear Force and the
Bending Stress are calculated as below.
() ()
2
,,
M
xtEIu xt= (45)
() ()
3
,,
F
xtEIuxt= (46)
Bendin g Stress
M
Z= (47)
As a measure of structural integrity a mechanical de-
sign check against fatigue is required to be carried out
using the stress distribution. In the time domain it is cus-
tomary to apply Rain-Flow Counting Method [21] to
determine cumulative usage factor. The value of the us-
age factor should be less than unity which indicates that
the system is safe and no failures from fatigue are ex-
pected to occur in its design life. A value of the factor
greater than one is an indication of a possibility of failure
due to fatigue. However as a crude estimate we may con-
sider maximum zero to peak value of the stress and
compare it with the endurance limit. It should be less
than the endurance limit to designate a system as safe.
The time histories of velocities and the end reactions
can be computed through the direct problem. The end
reaction forces should be used for checking mechanical
design of the support structure. This will ensure integrity
of the pipe supports thereby accounting for an important
hazard of a vibrating piping system.
The velocity at a point may be compared against the
maximum permissible velocity as per common practices
as mentioned earlier. However in view of our detailed
analytical method they are not the essential parameters
and may be taken as an additional piece of inform ation.
4.5. Numerical Simulation and Validation
In order to validate the theory some numerical experi-
ments have been carried out. The problem considered is
as follows.
A simply supported pipe is excited through end mo-
ments. Two cases have been considered. In case 1 the
excitation moment is applied at only one end. In case 2
excitation moments are applied at both ends. For sim-
plicity the harmonic excitation comprising of sine and
cosine terms for a few frequencies have been considered
for the forcing functions. However any continuous time
varying function is permissible. The total time T consid-
ered is 200 seconds. The pipe material is steel, size 219

S. SAHA
330
mm outer diameter (O.D), thickness 8.18 mm and the
span is 8 m. A fixed damping ratio of 1% has been as-
sumed. Five points numbered 1 to 5 have been defined in
the span. Points 1 at x = 0 and 5 at x = L are the boundary
points. Points 2, 3 and 4 are interior points at locations
0.25 L, 0.5 L and 0.75 L respectively. These points have
been defined for the purpose of specifying the input and
output locations.
The direct problem is first solved using the forcing
function as the moments using standard software. The
dynamic analysis time history module of general purpose
Finite Element Analysis (FEA) software has been used
for the direct problem. This analysis model will be
termed as model D in the sequel. The results of the
analysis have been treated as the benchmark. The accel-
erations from model D have been considered as meas-
urements which are the inputs for our proposed method
which is based on Inverse Theory and denoted as model I
for reference. Displacements, stresses and end reactions
have been considered as the response parameters for
comparison with the benchmark.
5. Results and Discussions
The time step interval has been fixed based on the high-
est natural frequency. This is done for the purpose of
minimizing errors due to integration. For the details on
the theory one may refer to standard texts [17,18]. Five
modes have been considered for the problem.
Figures 2 and 8 show the moment time history for
Case 1 and Case 2 respectively. Graph D denotes the
input for direct problem model whereas graph I denotes
the calculated response for the Inverse Problem. It is seen
that the two graphs coincide implying unique correspon-
dence between the Inverse and Direct Problem for our
case.
The observation points are 2 and 4 where the accelera-
tion time histories are measured [see Figures 3 and 9].
The rotational accelerations are calculated from Equation
37 as per inverse theory. It is seen from Figure 4 that the
rotational accelerations are shown at point 1 only. This is
due to the fact that in Case 1 the excitations are applied
at one end. The other response quantities like end reac-
tions, displacements and stresses are shown in Figures 5-
7. In all cases there is no difference between the results
of the two models. In the sequel we shall use the abbre-
viations TH for time history, ATH for acceleration time
history and RTH for rotational time history.
The results for Case 2 are given in Figures 9-13. In
this case we have rotational accelerations for both the
ends unlike Case 1. Also a very close match between the
results of direct and inverse problem is observed similar
to Case 1. This is expected since the theoretical solution
for the two methods is essentially the same. The differ-
ence is basically due to the round off errors.
Plot of End Moments
Figure 2. TH. of end moment excitations (Case 1).
A
ccln. Measurement
Figure 3. ATH. at measurement points (Case 1).
Plot of Rotational Accln.
2
Figure 4. RTH. at end points (Case 1).
End Reaction Plot
Figure 5. TH. plot of end reactions (Case 1).
As mentioned earlier, the distinct advantage of the
method over the current ones is that quantitative estimate
of the stresses and the end reactions are obtained in this
method. This is significant from the aspect of condition
monitoring and engineering design. The reaction force

S. SAHA
331
Stress Plot
Figure 6. Stress TH. at interior points(Case 1).
Displacement Plot
Figure 7. Displ. TH. at interior points (Case 1).
Plot of End Moments
Figure 8. TH. of end moment excitation (Case 2).
Accln. Measurement
Figure 9. ATH. at measurement points (Case 2).
estimates will enable us to design the pipe supports,
whereas the stresses and displacements will be useful for
condition monitoring of the system.
Plot of Rotational Accln.
2
Figure 10. RTH. at end points (Case 2).
Plot of End Reactions
Figure 11. TH. plot of end reactions (Case 2).
Displacement Plot
Figure 12. Displ. TH. at interior points (Case 2).
Stress Plot
Figure 13. Stress TH. at interior points (Case 2).
6. Conclusion
Vibration failure in operational piping is a serious prob-
lem and there is a need for a comprehensive study and
analysis for its remedial measures. In this sense the pro-
posed study has got a tremendous practical value. A

S. SAHA
332
quantitative method with proper mathematical basis has
been provided in con trast to the cook book approach. By
this method it is possible to quantify stresses, velocities
and reaction forces. This gives us a basis for a proper
engineering design. The method being simple can be
easily adopted by engineers involved in trouble-shooting.
Several improvements in the model are in line and
planned for future work. These are like inclusion of
lumped mass in the span or pipe bends. These will widen
the range of application of the method and will be of
greater practical use.
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333
Annexure 1:
This section deals with some mathematical details re-
quired for Proposition 1. We define the following terms.
() ()
000
1
N
nn n
n
x
x βαφ
=
=Γ +

Υ
(A-1)
()()
0
1
N
LnnLn
n
x
x βαφ
=
=Γ +

Υ
(A-2)
() ()
000
tt ϕβθ =
 (A-3)
() ()
0LL
tt ϕβθ =
 (A-4)
()
()() ()
()
()
0
01 2
1
0
,
1,,
N
nnnnn n
n
Kt
xftft
τ
φ
ωατ τ
β
=

=−Γ+


 (A-5)
()
()() () ()
()
12
1
,
1,,
L
N
nnnnLn n
n
L
Kt
xftft
τ
φ
ωατ τ
β
=

=−Γ+


 (A-6)
Substituting the above in (29) we have
() ()
()()()()
0
00
00
,d ,d
L
tt
LL
tt
Kt Kt
ϕϕ
τ
ϕ
ττ τ
ϕ
ττ
+
=+

 
 (A-7)
Lemma 1:
The integral equation defined below has a unique triv-
ial solution.
()()( )
0
,d t Ktfft τττ = (A-8)
(Here f(t) is a continuous function belonging to C (0, T)
and the kernel K (t, τ) is also continuous in the domain
()()
0, 0,tX t with K (t, τ) = 0 for t τ <.)
Proof: We will provide the sketch of the proof. For
details on may refer to any standard text on functional
analysis (e.g. [19]).
It can be proved that the operator T defined as
()()()
0
,d
t
TftK tf τττ = (A-9)
is a contraction mapping. Hence it has a unique fixed
point. Thus (A-8) has a unique trivial solution.