Chebyshev Polynomial Based Numerical Inverse Laplace Transform Solutions of Linear Volterra Integral and Integro-Differential Equations

American Research Journal of Mathematics Original Article
ISSN 2378-704X Volume 1, Issue1, Feb-2015
www.arjonline.org 22
Chebyshev Polynomial Based Numerical Inverse Laplace
Transform Solutions of Linear Volterra Integral and
Integro-Differential Equations
Vinod Mishra1
, Dimple Rani
Department of Mathematics, Sant Longowal Institute of Engg. & Tech, Longowal (Punjab)
Abstract: There are enormous occasions when the methods for finding solutions of integral and integro-
differential equations lead to failure because of difficulty in inverting Laplace transform by standard technique.
Numerically inverting Laplace transform is cost effective in comparison to rather complicated technique of com-
plex analysis. In the process of numerical inversion, an odd cosine series which is ultimately based on Chebyshev
polynomial has been used. The adequacy of method is illustrated through numerical examples of convolution type
linear Volterra integral equations of second kind which include weakly singular Abel's integral equation and Vol-
terra integro-differential equation.
Keywords: Volterra integral equation; Volterra Integro-differential equation; Numerical Inversion of Laplace
transform; Gaussian quadrature; Cosine series; Chebyshev polynomial.
I. INTRODUCTION
Linear Volterra Integral and Integro-Differential equations are extensively used in the various specialties of science
and engineering. These include mathematical physics, chemical kinetic, heat conduction, seismology, fluid dynam-
ics, biological models, population dynamics, metallurgy and semi-conductors [11, 12 &13].
The Volterra Integral equation with a convolution kernel is defined by
  
x
Txdttftxkxyxf
0
,0,)()()()( (1)
while Volterra Integro-Differential equation by
00 )(,)()()()(
0
uxudxxutxkxuxu
x
x
  , (2)
There have been many extant methods for solving Volterra Integral and Integro-Differential equations with convolu-
tion kernels. Wazwaz[12] and Al-Hayani[18] discussed the Adomian polynomial based Laplace Transform methods
to solve these equations. Yang[13] proposed a method by which solution is expressed in power series and to im-
prove the convergence rate applied the Pade approximant. Babolian-Shamloo[7], Aznam-Hussin[17] and Mishra et
al.[9] used operational matrices of piecewise constant orthogonal function or Haar wavelet to solve these equations.
Homotopy perturbation method with finite difference technique was used by Raftari[10] to solve Volterra Inte-
gro-Differential equations. Zarebnia[14] solved these equations using Sinc function. A modified Taylor series
method has been applied to approximate the solution of linear Integro-Differential equations in [11].
1.1 Numerical Inverse Laplace Transform
The Laplace transform of function is defined by
  .)()()(
0



 dttfesFtfL st
(3)
1
Corresponding Author: vinodmishra.2011@rediffmail.com

American Research Journal of Mathematics, Volume 1, Issue 1, February 2015
ISSN 2378-704X
As usual, the equations are first converted into algebraic equations using Laplace transform. The inverse Laplace
transform is then applied and the numerical solution is finally expressed in terms of Chebyshev series.
Applying Laplace transforms on both sides of above equations and then using convolution property in (1) & (2),
reduce the equations in the form
  ).()( sFxfL 
The inversion leads to
)()( 1
sFLxf 
 .
Comprehensive literature consists of a number of methods for numerically inverting Laplace transform suited for
problems in particular situations. For a detailed survey of various methods for Laplace transform inversion numer-
ically refers to Cohen[2], Davies-Martin[5] and Mishra[16]. Bellman et al. (1966)[1] have outlined a method which
they derive from the consideration of Gauss-Legendre's quadrature rule. Substituting 0,  
t
ex in [1 & 2],
)(sF is transformed from the interval  ,0 to  1,0 as
,)(
1
)(
1
0
1/


 dxxgxsF s 

Where  





 xfxg log
1
)(

.
)(sF can be numerically inverted using Legendre series of 



0
2 )()(
k
kk xPxg  by considering )(xg being
even function in ]1,1[ . This method has slow convergence as the coefficients k decrease slowly due to the sin-
gularity of )(xg at 0x [1 & 2]. Erdelyi (1943), Papoulis (1956) and Lanczos (1957) have proposed Legendre's
function to find the approximate value of function )(tf [2, 5, 6, & 8]. In [3] Dubner and Abate (1968) have ex-
pressed the )(tf in terms Fourier cosine transforms. Durbin (1974) [4] has proposed the trapezoidal rule by result-
ing approximation.
1.2 Chebyshev Series based Inversion
Chebyshev polynomial is due to Panfnuty Chebyshev (b. 1821), a Russian. It is basically a class of orthogonal poly-
nomials. In sixties remarkable development to the theory lead to computation of function approximations, integrals
and solution of differential equations using Chebyshev polynomials, termed as Chebyshev series expansion of a
function. The range of problems covered includes singular problem and network synthesis [15].
Here we propose a technique parallel to Papoulis[6] by making the substitution
.0,sin  
  x
e (4)
The interval (0, ∞) transformed into  2,0 
and )(xf becomes
  ).(sinlog
1


gf 





 (5)
Now eq. (3) takes the form
  .)(cossin)(
2
0
1


 dgsF
s


 (6)
By setting ,)12(  ks k = 0, 1, 2…, we have

ISSN 2378-704X
  .)(cossin))12((
2
0
2


dgkF
k
 (7)
Here we assume that   0)0(2  fg 
. In case this does not hold then arrange it by subtracting a suitable function
from )(g . The function )(g can be expanded in  2,0 
as the odd cosine series




0
)12cos()(
k
k kg  (8)
and is valid in the interval  ., 22


Now we have to determine the coefficients k
  .
22
cossin
2
2





 





 

 

iikii
k ee
i
ee
Making substitution
i
ex  , we find that
   














































































































































































































1
2
)1(
2
)1(
1
2
1
1
2
)1(
2
)1(
2
1
12
2
)1(
2
2
)1(
1
2
1
1
2
)1(
2
)1(
2
1
12
2
2
21
2
1
1
2
0
2
2
11
2
111
2
1
cossin12
11
122
122
1122
12
12
12
12
2
22
k
k
k
k
xk
k
k
k
x
rk
k
rk
k
xr
k
r
k
x
k
k
k
k
x
kk
x
x
x
x
x
x
x
kkkk
rkrk
rk
rrrk
k
k
k
k
k
kkk

=      cos
2
1
2
1)122cos(
2
1
2
1)12cos(
11








































k
k
k
k
rk
r
k
r
k
k
kr

(9)
Substitution of (8) & (9) and using the result of the orthogonality
  ,
4
)12cos( 2
2/
0



 dk
eq. (7) gives
))12((  kF =
 
   


















































krk
rk
kk
r
k
r
k
k
k
k
k


 .
2
1
2
1
2
1
2
1
412
1 1
0
1
2
(10)

ISSN 2378-704X
 ))12((2
4 2


kFk
 
   


















































krk
rk
k
r
k
r
k
k
k
k
k
  .
2
1
2
1
2
1
2
1
1
1 1
0
1
(11)
which is the linear system in 0 , 1 , 2 ….. k …. It can be conveniently put in the matrix form
,ABC  (12)
where the matrix C and the coefficient matrix A can be obtained by putting the values of ,2,1,0k in LHS
and RHS of (11) respectively.
A








































k
k
k
k
k
)1(
2
1
2
132
011
001





,

















k
B





2
1
0
Thus k can be obtained by solving CAB 1
 and hence )(g can be obtained from eq. (7).
In general, we compute first )1( N terms of eq.(8), that is, the finite series
.)12cos()(
0


N
k
kN kg  (13)
As N , )()(  ggN  . From )(g we can determine )(xf
.)12cos()(
0




k
k kxf  (14)
Defining 


sin,
cos
cos
)(1  x
k
xUk , where )(xUk is the Chebyshev polynomial of second kind of degree k
Then
  2/12
1cos x
e 
 

And
  .)(1)(
0
2
2/12





k
x
kk
x
eUexf 

1.3 Numerical Examples
Example1
Consider the weakly singular Volterra integral equation of second kind [12&13]

ISSN 2378-704X
  


x
xdt
tx
tu
xxu
0
1,0,
)(
2)( (15)
Taking Laplace transform on both sides of eq. (15) and using convolution theorem, we obtain
  ).()( sF
ss
uL 




(16)
The necessary condition for compliance of inverse )(xu of eq.(16) is that 0)0( u . Following initial value theo-
rem,
  .0lim)(lim)0( 


 

ss
sssFu
ss
Therefore, the solution )(xu can be obtained and k can be computed using relation (12).
Table1.1. Coefficients in the Expansion of )(xu
k k
0 0.81399311
1 -0.0446193
2 0.03972445
3 0.00011582
4 0.01165101
5 0.00230399
6 0.00569674
7 0.00212046
8 0.00350064
9 0.00166594
10 0.00164977

ISSN 2378-704X
Table1.2. Computed Pade approximant [4/4] and Approximation Solution for Example1
X Pade approximant [4/4] method Present method Absolute Error
0 0 5.14E-17 5.13800E-17
0.1 0.41411018 0.4141034 6.78000E-06
0.2 0.50848304 0.5087519 2.68860E-04
0.3 0.56452274 0.5637984 7.24340E-04
0.4 0.60364034 0.6033569 2.83440E-04
0.5 0.63323515 0.6338924 6.57250E-04
0.6 0.65675942 0.655994 7.65420E-04
0.7 0.67610075 0.6740862 2.01455E-03
0.8 0.69240064 0.6911992 1.20144E-03
0.9 0.70639982 0.7070156 6.15780E-04
1 0.71860476 0.7202293 1.62454E-03
Example2. Consider the integral equation
.sin)()(
0
xdttxutu
x
 (17)
Laplace transform on both sides of eq. (17) gives
).(
1
1
)(
2
sF
s
uL 

 (18)
The condition 0)0( u is not satisfied
.1
1
1
lim)(lim)0(
2




s
sssFu
ss
As per provision of condition a possible function need to be subtracted from )(xu is 1. A function which takes the
value 1 at 0x is
x
e
. Therefore,
x
exuxU 
 )()( as 0)0( U . Since
1
1
)()(


s
sFsF . Thereby, the
solution )(xu can be obtained by computing the coefficients k the relation (12).
Table2.1. Coefficients in the Expansion of )(xu
k k
0 0.26369654
1 -0.0735987
2 -0.1482495
3 -0.0985091
4 -0.0769397
5 -0.0547004
6 -0.042268
7 -0.0310263
8 -0.0239681
9 -0.0176349
10 -0.0133825
11 -0.0095073
12 -0.0067188
13 -0.003956
14 -0.001397

ISSN 2378-704X
Table2.2. Computed Exact and Approximation Solution for Example2
x Exact solution Present method Absolute error
0 1 1 3.00000E-09
0.1 0.99750157 0.99722919 2.72375E-04
0.2 0.99002498 0.98950321 5.21765E-04
0.3 0.97762625 0.97849647 8.70222E-04
0.4 0.96039823 0.95925921 1.13902E-03
0.5 0.93846981 0.93944053 9.70723E-04
0.6 0.91200486 0.9125871 5.82237E-04
0.7 0.88120089 0.87896932 2.23157E-03
0.8 0.84628735 0.84616791 1.19441E-04
0.9 0.8075238 0.81065911 3.13531E-03
1 0.76519768 0.7670378 1.84012E-03
Example3. Consider the Volterra integral with a convolution kernel given by [13]
.sin)()cos()(
0
 
x
xdttutxxu (19)
As usual taking Laplace transform on both sides of eq.(19) yield
).(
1
1
)( 2
sF
ss
uL 

 (20)
The condition 0)0( u is satisfied as in Example 1, that is
.0
1
1
lim)(lim)0( 2



 ss
sssFu
ss
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.75
0.8
0.85
0.9
0.95
1
x
u(x)
Fig.2. Comparison of exact and present approximate solution
Exact solution
Present method

ISSN 2378-704X
Therefore, the solution )(xu is feasible. The coefficients k computed (12) are shown below.
Table3.1.Coefficients in the Expansion of )(xu
k k
0 0.42441318
1 0.03264717
2 -0.09372896
3 -0.07000573
4 -0.06126397
5 -0.04672609
6 -0.0391798
7 -0.03125062
8 -0.02649286
9 -0.02180741
10 -0.01875585
11 -0.01580044
12 -0.01381433
13 -0.01199908
14 -0.01103245
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
u(x)
Fig.3. Comaprison of Pade approximant [4/4] and present approximate solution
Pade approximant [4/4] method
Present method

ISSN 2378-704X
Table 3.2. Computed Pade approximant [4/4] and Approximation Solution for Example3
X Pade approximant [4/4] Present method Absolute error
0 0 4.12E-16 4.12297E-16
0.1 0.095004082 0.09158784 3.41624E-03
0.2 0.180063957 0.183098185 3.03423E-03
0.3 0.255316973 0.256151895 8.34922E-04
0.4 0.320980415 0.321179014 1.98599E-04
0.5 0.377341791 0.372097134 5.24466E-03
0.6 0.42474943 0.433955997 9.20657E-03
0.7 0.463603433 0.461889867 1.71357E-03
0.8 0.494347029 0.483582381 1.07646E-02
0.9 0.517458374 0.518579699 1.12133E-03
1 0.533442822 0.546451691 1.30089E-02
Example4. Consider the first order linear Volterra Integro-differential equation of the form [8]



x
x
tx
dttuexuxu
0
)()()( )(
 , 00 )( uxu  . (21)
If we choose 0 , 0 , 1 , 1 , 00 x , .10 u
Taking Laplace transform on both sides of eq. (21) and using derivative property and convolution theorem of Lap-
lace transform,
).(
1
1
)( 2
sF
ss
s
uL 


 (22)
The condition 0)0( u is not satisfied as
.1
1
1
lim)(lim)0( 2




 ss
s
sssFu
ss
In this case we choose a possible function that will be subtracted from )(xu to be 1. A function which takes the
value 1 at 0x is
x
e
. Therefore,
x
exuxU 
 )()( as .0)0( U Since ,
1
1
)()(


s
sFsF the solution
)(xu exists. The coefficients k can be calculated using the relation (12).
Table 4.1. Coefficients in the Expansion of )(xu
k k
0 0.212206589
1 -0.08161792
2 -0.12163703
3 -0.06349292
4 -0.04422927
5 -0.02637033
6 -0.01850358
7 -0.01134116
8 -0.00779918
9 -0.00443629
10 -0.00269506
11 -0.00095958

ISSN 2378-704X
12 -5.86E-05
Table4.2. Computed Exact and Approximation Solution for Example4
x Exact solution Present method Absolute error
0 1 1 0.00E+00
0.1 0.9951666 0.99489203 2.75E-04
0.2 0.9813308 0.98137139 4.06E-05
0.3 0.9594808 0.95941456 6.62E-05
0.4 0.930587 0.93094843 3.61E-04
0.5 0.8955945 0.89502027 5.74E-04
0.6 0.8554164 0.85534569 7.07E-05
0.7 0.8109282 0.81181337 8.85E-04
0.8 0.762963 0.76330526 3.42E-04
0.9 0.7123077 0.71129202 1.02E-03
1 0.6597002 0.65824129 1.46E-03
II. CONCLUSION
In this paper we have gone through a new insight into the use of Chebyshev polynomials. The series solutions in
terms of Chebyshev polynomials have been used as numerically inverting Laplace transform tool for finding solu-
tions of Volterra integral and integro-differential equations. The outcome of four test problems have been compared
with exact or Pade approximants and found to be numerically efficient.
REFERENCES
[1] Bellman, R.E., H.H. Kagiwada and R.E. Kalba, Numerical Inversion of Laplace Transforms and Some Inverse Problems in
Radiative Transfer, Journal of Atmospheric Sciences 23(1966), 555-559.
[2] Cohen, A.M., Numerical Methods for Laplace Transform Inversion, Springer, 2007.
[3] Dubner, H. and J. Abate, Numerical Inversion of Laplace Transforms by Relating them to the Finite Fourier Cosine Trans-
form, Jour. Assoc. Comput. Math. 15 (1968), 115-123.
[4] Durbin, F., Numerical Inversion of Laplace Transforms: An efficient improvement to Dubner and Abate's method, Comput.
Jour. 17 (1974), 371-376.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
x
u(x)
Fig.4. Comparison of exact solution and present approximate solution
Exact solution
Present method

ISSN 2378-704X
[5] Davies, Brian and Brian Martin, Numerical Inversion of Laplace Transform: A Survey and Comparison of Methods,
Journal of Computational Physics 33(1979), 1-32.
[6] Papoulis, A., A New Method of Inversion of the Laplace Transform, Quart. Appl. Math. 14(1956), 405-414.
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lution Type by using Operational Matrices of Piecewise Constant Orthogonal Functions, Journal of Computational and
Applied Mathematics 214 (2008), 495-508.
[8] Filiz, A. , Numerical Method for Linear Volterra Integro-differential Equation with Cash-karp Method, Asian Journal of
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[10] Raftari, B., Numerical Solution of Linear Volterra Integro-differential Equations: Homotopy Perturbation Method and
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pansion Method, World Journal of Modeling and Simulation 9 (2013), 289-301.
[12] Wazwaz, A.M., Linear and Nonlinear Integral Equations: Methods and Applications, Springer, 2011.
[13] Yang, Changqing and Jianhua Hou, Numerical Method for Solving Volterra Integral Equations with a Convolution Kernel,
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[14] Zarebnia, M. and Z. Nikpour, Solution of linear Volterra Integro-differential Equation via Sinc Functions, International
Journal of Applied Mathematics and Computation 2 (2010), 1-10.
[15] Piessens, Robert, Computing Integral Transforms and Solving Integral Equations using Chebyshev Polynomial Ap-
proximations, Journal of Computational and Applied Mathematics 121 (2000) 113-124.
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