Approximate Solution of Volterra Delay Integro-Differential Equations by using Hermite Polynomials Asma Shaikh*,Sarita Thakar** Department of Mathematics, Shivaji University, Kolhapur, Kolhapur (M.S.) 416 004 India. E-mail* :
[email protected] E-mail**:
[email protected] Abstract In this paper, a matrix method is presented to find an approximate solution of linear Volterra delay integro-differential equations with given initial conditions. The approximate solution is assumed to be a linear combination of Hermite polynomials. Hermite coefficients are obtained by collocation method. Approximate solution of two problems along with estimated error are obtained to illustrate the method .
Keywards : Volterra delay integro-differential equations, Collocation method, Approximate solutions, Hermite Polynomials, Maclaurin series.
1
Introduction :
Volterra Delay Integro-Differential Equations[VDIDEs] arise widely in scientific fields such as biology, ecology, medicine and physics. This class of equation plays an important role in modelling diverse problems of engineering and natural sceince[3]. In last few years, lots of work has been done to find the approximate solutions of integro-differential equations by polynomial method. In [10], Suayip Yuzbasi discussed Laguerre approach for solving pantograph type Volterra integro-differential equations. Niyazi Sahin and Suayip Yuzbasi[7] obtained the solution of general linear Fredholm integro-differential equations with delay by Bessel polynomial. Aysegul Akuz-Dascioglu[2] introduced Chebyshev polynomial solutions for system of higher order linear Fredholm-Volterra integro-differential equations. In [8], Salih Yalcinbas solved high order linear Fredholm integro-differential equations by using Legendre polynomial. Taylor collocation method for solution of system of high-order linear Fredholm-Volterra integro-differential equations has been discussed by Mustafa Gulsu and Mehmet Sezer[5]. Hermite Collocation method[4, 6, 9] is used to solve pantograph equations, high order linear differentialdifference equations and high order linear Fredholm integro-differential equations. The best advantages of Hermite Collocation method is that the Hermite coefficients of the solution are found very easily by using computer programs. In [1],Existence and Uniqueness of the solution of a general class of Volterra delay integro-differential equations are proved. In this paper, we obtained the approximate solution of linear Volterra Delay Integro-Differential Equations by using Hermite polynomials. The paper is organised as follows. In section 2, the fundamental relations associated with the problem are derived. The method of finding an approximate solution is described in section 3. In section 4, the error analysis technique based on residual function is developed. Some illustrative examples are presented in section 5. Finally conclusion is given in section 6. We consider Volterra Delay Integro-Differential Equations Z t 0 y (t) = α(t)y(t) + β(t)y(t − τ ) + K(t, s)y(s)ds + g(t), t0 ≤ t, s ≤ T < ∞, (1.1) t−τ
with initial condition, t ∈ [t0 − τ, t0 ],
y(t) = φ(t) 1
(1.2)
where y(t) is unknown function, the functions α(t), β(t), K(t, s) and g(t) are defined on the interval t0 ≤ t, s ≤ T < ∞, t0 ≥ 0, τ is a positive constant, we assume that the Kernel function K(t, s) can be expanded in Maclaurin series. The aim of this paper is to construct an approximate solution to the problem (1.1) and (1.2) in the form N +1 X y(t) = yN +1 (t) = an Hen (t), t0 − τ ≤ t ≤ T, (1.3) n=0
where an , n = 0, 1, 2, ...., N + 1 are unknown coefficients, N is a suitable positive integer and Hen (t), n = 0, 1, 2, ...., N + 1 are Hermite polynomials defined by, Hen (t) = (−1)n et
2
/2
dn −t2 /2 e . dtn
(1.4)
Observe that these functions are obtained by following recurrence relation. Hen (t) =
n X
bn,k tk ,
t0 − τ ≤ t ≤ T < ∞,
(1.5)
k=0
where, b0,0
2
=
1,
b1,0 = 0,
b1,1 = 1,
bn+1,k
= bn,k−1 − nbn−1,k
bn+1,k
= −nbn−1,k
k > 0, k = 0.
Fundamental Matrix Relations:
In this section, we represent equation(1.1) in the form of matrices. Firstly, we write the approximate solution (1.3) as, yN +1 (t) = He(t)A
(2.1)
where, He(t) = [He0 (t) He1 (t) He2 (t) .....
HeN +1 (t)]
and
A = [a0
a1
a2
aN +1 ]T .
.....
Hermite polynomials Hen (t) defined by equation (1.5) is He(t) = X(t)B T , where,
X(t) = [1
t
t2 .....
b0,0 b1,0 b2,0 .. .
B= bN +1,0
tN +1 ],
(2.2) 0
0 0
b1,1 b2,1 .. .
b2,2 .. .
..... ..... ..... .. .
bN +1,1
bN +1,2
.....
0 0 0 .. . bN +1,N +1
.
From equation (2.1), we have, 0 0 yN +1 (t) = He (t)A,
(2.3)
He0n (t) = nHen−1 (t).
where,
It is clearly seen that the relation between the matrix He(t) and its derivative He0 (t) is He0 (t) = He(t)M T ,
where,
0 1 M = 0 .. .
0 0 2 .. .
0 0 0 .. .
..... ..... .....
0 0 0 .. .
0
0
0
.....
N +1
2
0 0 0 . .. . 0
Thus equation (2.3) becomes, 0 T yN +1 (t) = He(t)M A.
(2.4)
From equation (2.1) and (2.2) we obtain, yN +1 (t)
=
X(t)B T A,
yN +1 (t − τ )
=
X(t − τ )B T A,
=
X(t)B ∗ (1, −τ )B T A,
where, B ∗ (1, −τ ) is given by [7], o 0 o (−τ ) 0 ∗ B (1, −τ ) = .. .
1 0
(−τ )1
2 0
(−τ )2
1 1
0
2 1
1
0
(−τ ) .. .
(−τ ) .. .
0
0
(2.5)
N +1 0
.....
(−τ )N +1
.....
N +1 1
.....
N +1 N +1
(−τ ) .. .
N
.
(−τ )0
By using relation (2.2), we write equation (2.5) as yN +1 (t − τ ) = He(t)(B T )−1 B ∗ (1, −τ )B T A.
2.1
(2.6)
Matrix relation for the integral part:
Since we assume that the kernel function K(t, s) can be expanded to the Maclaurin series, it can be approximated by truncated Taylor series. K(t, s)
=
N +1 N +1 X X
Kpq tp sq ,
(2.7)
p=0 q=0
where,
Kpq
=
1 ∂ p+q K(0, 0) , p!q! ∂tp ∂sq
p, q = 0, 1, 2, ...., N + 1.
The matrix form of (2.7) is written as K(t, s) = X(t) K X T (s),
K = [Kpq ].
(2.8)
If we approximate the kernel function K(t, s) by truncated Hermite polynomials, then K(t, s)
=
N +1 N +1 X X
∗ Kpq Hep (t) Heq (s)
p=0 q=0
represented by a matrix K(t, s)
=
He(t) K ∗ HeT (s),
∗ K ∗ = [Kpq ].
(2.9)
∗ Then coefficients Kpq in equation (2.9) are obtained from equation (2.8) and (2.9).
He(t) K ∗ HeT (s)
=
X(t) K X T (s).
By using equation (2.2), we write the above equation as X(t)B T K ∗ BX T (s)
= X(t) K X T (s).
Since B is non-singular, K ∗ = (B T )−1 K B −1 .
3
(2.10)
By substituting the matrix form (2.1) and (2.9) in the integral part of (1.1), we have, Z t Z t K(t, s) y(s) ds = He(t) K ∗ HeT (s) He(s) A ds, t−τ
t−τ
= He(t) K ∗ Q(t) A, t
Z where,
Q(t)
(2.11)
HeT (s) He(s) ds = [qij ],
= t−τ Z t
qij
Hei (s) Hej (s) ds
=
f or i, j = 0, 1, 2, ....., N + 1.
t−τ
2.2
Matrix relation for the initial condition:
From equation (2.1), the corresponding matrix form for initial condition (1.2) is constructed as He(t) A = φ(t).
3
(2.12)
Method of Solution:
To obtain the approximate solution given by (1.3) to equation (1.1), we compute the unknown coefficients by using the following collocation method. On substituting (2.1), (2.4), (2.6) and (2.11) in equation (1.1), we obtain He(t)M T A = α(t) He(t) A + β(t) He(t)(B T )−1 B ∗ (1, −τ ) B T A + He(t) K ∗ Q(t) A + g(t).
(3.1)
To obtain collocation points, choose N such that if we divide the interval [t0 − τ, T ] into N equal parts then points t0 and T will belong to the set of mesh points. This is achieved by solving the integer equation i =
N τ T −t0 +τ ,
for i, N ∈ N. The mesh size h = τi .
The mesh points in the interval [t0 − τ, t0 ] are denoted by t∗i = t0 − τ + ih,
i = 0, 1, 2, ...., k.
Simillarly in the interval [t0 , T ] the mesh points are denoted by tj = t0 + jh,
j = 0, 1, ...., N − k.
At collocation points tj , j = 0, 1, ...., N − k, equation (3.1) becomes, He(tj )M T A = α(tj ) He(tj ) A + β(tj ) He(tj )(B T )−1 B ∗ (1, −τ ) B T A + He(tj ) K ∗ Q(tj ) A + g(tj ). The above equations can be written in the matrix form as [He M T − α He − β He (B T )−1 B ∗ (1, −τ ) B T − C] A = G,
(3.2)
where, α(t0 ) 0 0 ... 0 α(t ) 0 ... 1 α= . . . .. .. . . . .. 0 0 0 ...
He0 (t0 ) He0 (t1 ) .. .
0 0 .. .
β(t0 ) 0 0 β(t 1) β= . . .. .. 0 0
,
α(tN −k )
He1 (t0 ) He1 (t1 ) .. .
He2 (t0 ) He2 (t1 ) .. .
... ... .. .
He = He0 (tN −k ) He1 (tN −k ) He2 (tN −k ) . . .
a0 a1 a2 .. .
A= , aN +1
g(t0 ) g(t1 ) g(t2 ) .. .
G= , g(tN −k ) 4
HeN +1 (t0 ) HeN +1 (t1 ) .. .
0 ... 0 ... .. . . . . 0 ...
0 0 .. .
, β(tN −k ) He(t0 ) He(t1 ) .. .
= , HeN +1 (tN −k ) He(tN −k )
He(t0 )K ∗ Q(t0 ) He(t1 )K ∗ Q(t1 ) He(t2 )K ∗ Q(t2 ) .. .
C= . ∗ He(tN −k )K Q(tN −k )
Let, W = He M T − α He − β He (B T )−1 B ∗ (1, −τ ) B T − C, then equation (3.2) can be expressed as follows. WA = G
(3.3)
Equation (3.3) are (N − k + 1) linear algebraic equations with unknown coefficients a0 , a1 , a2 , ......, aN +1 . By using collocation points t∗i , i = 0, 1, 2, ...., k, the initial condition given by (2.12) is written as He∗ A = φ.
(3.4)
where, He0 (t∗0 ) He0 (t∗1 ) He∗ = .. .
He1 (t∗0 ) He1 (t∗1 ) .. .
He2 (t∗0 ) He2 (t∗1 ) .. .
HeN +1 (t∗0 ) HeN +1 (t∗1 ) , .. .
... ... .. .
He0 (t∗k ) He1 (t∗k ) He2 (t∗k ) . . .
φ(t∗0 ) φ(t∗1 ) φ = . . ..
HeN +1 (t∗k )
φ(t∗k )
The assembled system (3.3) and (3.4) are N + 2 linear equations in N + 2 unknowns a0 , a1 , a2 , ......, aN +1 , we have, W G A = . He∗ φ ˜ A W
i.e
˜ = G,
where, ˜ = W
W He∗
G ˜ G= . φ
and
˜ ˜ is non-singular and A = (W ˜ )−1 G. Since Hermite polynomials are orthogonal, W Substituting the determined coefficients A = [a0 the approximate solution of equation (1.1)-(1.2).
4
a1
a2
.....
aN +1 ]T
in equation (1.3), we get
Error analysis based on residual function:
In this section an error estimator based on residual function for the approximate solution of a linear Volterra delay integro-differential equations is obtained. For our purpose, we define residual function for the method as Z t 0 RN +1 (t) = yN +1 (t) − α(t) yN +1 (t) − β(t) yN +1 (t − τ ) − K(t, s) yN +1 (s)ds − g(t), (4.1) t−τ
where, yN +1 is the approximate solution of the equation (1.1)-(1.2). Note that yN +1 satisfies the equation 0 yN +1 (t) = α(t)yN +1 (t) + β(t)yN +1 (t − τ ) +
Z
t
K(t, s)yN +1 (s)ds + g(t) + RN +1 (t), t0 ≤ t, s ≤ T < ∞,(4.2) t−τ
Let, the error function eN +1 (t) be defined as, eN +1 (t) = y(t) − yN +1 (t),
t0 − τ ≤ t ≤ T.
(4.3)
where y(t) is the exact solution of equation (1.1)-(1.2). Now subtracting (4.2) from equation (1.1), we have, e0N +1 (t) = α(t)eN +1 (t) + β(t)eN +1 (t − τ ) +
Z
t
K(t, s)eN +1 (s)ds − RN +1 (t), t0 ≤ t, s ≤ T < ∞, (4.4) t−τ
5
From equation (1.2) we have, eN +1 (t)
φ(t) − yN +1 (t),
=
t ∈ [t0 − τ, t0 ].
(4.5)
Finally, we apply the method defined in section 3 for equation (4.4)-(4.5), where RN +1 is obtained from equation (4.1), and we get the approximate solution in the form eN +1 (t) =
N +1 X
a∗n Hen (t).
n=0
From equation (4.3), we observe that the corrected solution of equation (1.1)-(1.2) is y(t) = yN +1 (t) + eN +1 (t).
5
Illustrative Examples:
Example 5.1. For equation, π y (t) = cos(t) y(t) + sin(t) y(t − ) + 2 0
Z
t
t− π 2
1 cos(t + s) y(s)ds + cos(t) + (2cos(3t) + πsin(t)), 4
t ∈ [0, π],
with initial condition, π t ∈ [− , 0], 2
y(t) = sin(t), y(t) = sin(t) is exact solution.
On compairing above equation with equation (1.1)-(1.2),we have, τ=
π 2,
α(t) = cos(t), β(t) = sin(t), g(t) = cos(t) + 14 (2cos(3t) + πsin(t)), k(t, s) = cos(t + s)
and φ(t) = sin(t). The approximate solutions of the above problem given by equation (1.3) are F or N = 9,
y10 (t) = −2.2659 × 10−13 + 1.t − 1.43758 × 10−6 t2 − 0.166667 t3 + 5.06429 × 10−6 t4 + 0.00833219 t5 −4.76076 × 10−6 t6 − 0.000196346 t7 + 1.20479 × 10−6 t8 + 2.06205 × 10−6 t9 − 8.55198 × 10−9 t10 .
F or N = 12,
y13 (t) = 9.4022 × 10−13 + 1.t − 3.1814 × 10−9 t2 − 0.166667 t3 + 1.82033 × 10−8 t4 + 0.00833333 t5 −3.17774 × 10−8 t6 − 0.000198391 t7 + 1.58077 × 10−8 t8 + 2.73806 × 10−6 t9 + 9.0011 × 10−10 t10 −2.13167 × 10−8 t11 − 1.30232 × 10−9 t12 + 2.73974 × 10−10 t13 .
F or N = 15,
y16 (t) = −4.90818 × 10−10 + 1.t − 1.26182 × 10−9 t2 − 0.166667 t3 + 3.12427 × 10−9 t4 + 0.00833333 t5 −1.09643 × 10−9 t6 − 0.000198413 t7 + 1.28258 × 10−10 t8 + 2.75572 × 10−6 t9 − 3.45049 × 10−12 t10
−2.50458 × 10−8 t11 − 2.62991 × 10−12 t12 + 1.59997 × 10−10 t13 + 5.73624 × 10−13 t14 − 9.01501 × 10−13 t15 +2.31586 × 10−14 t16 . The error in computation is as follows. F or N = 9, e10 (t) = 1.84202 × 10−14 + 1.60502 × 10− 7 t + 2.10336 × 10−6 t2 − 1.50381 × 10−10 t3 −7.20427 × 10−6 t4 + 2.83824 × 10−6 t5 + 6.22018 × 10−6 t6 − 3.87225 × 10−6 t7 − 1.03777 × 10−6 t8 +1.10608 × 10−6 t9 − 1.73659 × 10−7 t10 . F or N = 12, e13 (t) = −9.39507 × 10−13 − 2.37003 × 10−11 t + 3.57675 × 10−9 t2 + 2.79019 × 10−9 t3 −2.13537 × 10−8 t4 + 4.76615 × 10−9 t5 + 3.76297 × 10−8 t6 − 2.5128 × 10−8 t7 − 1.83772 × 10−8 t8 +2.10753 × 10−8 t9 − 1.5142 × 10−9 t10 − 4.3719 × 10−9 t11 + 1.66437 × 10−9 t12 − 1.76105 × 10−10 t13 . 6
F or N = 15, e16 (t) = 4.90818 × 10−10 − 2.16926 × 10−9 t + 1.26172 × 10−9 t2 + 1.37612 × 10−9 t3 −3.12378 × 10−9 t4 − 2.12739 × 10−10 t5 + 1.09603 × 10−9 t6 + 1.52706 × 10−12 t7 − 1.2985 × 10−10 t8 +9.45093 × 10−12 t9 + 6.46213 × 10−12 t10 − 5.32127 × 10−12 t11 + 9.96908 × 10−13 t12 + 9.78984 × 10−13 t13 −3.72855 × 10−13 t14 − 1.51127 × 10−15 t15 + 8.52172 × 10−15 t16 . The approximate solution of example 5.1 and exact solution and error in computation are listed in table 5.1, Fig.1. and table 5.2, Fig.2.
Table 5.1: Exact and approximate solution of example 5.1 . Exact Solution Approximate Solution N =9 N = 12 N = 15 −13 −13 0 0 -2.2659 × 10 9.4022 × 10 -4.90818 × 10−10 π/5 0.587785 0.587785 0.587785 0.587785 2π/5 0.951057 0.951056 0.951057 0.951057 3π/5 0.951057 0.951052 0.951057 0.951057 4π/5 0.587785 0.587633 0.587785 0.587785 π 0 -0.00133623 -0.0000125963 8.6274 × 10−8 t
1.0
0.5
1
-1
2
3
-0.5
1.pdf
-1.0
Fig.1. Table 5.2: Error in the approximate solution of example 5.1 . t Error Error Error N =9 N = 12 N = 15 0 1.84202 × 10−14 -9.39507 × 10−13 4.90818 × 10−10 π/5 3.09333 × 10−7 4.11129 × 10−10 -4.75884 × 10−10 −7 2π/5 2.68963 × 10 6.46992 × 10−10 -2.35485 × 10−9 −6 −9 3π/5 4.52241 × 10 -2.19492× 10 -1.73195 × 10−9 −7 4π/5 0.000152858 2.14766 × 10 7.75987 × 10−9 π 0.00133416 0.0000125912 -8.62686 × 10−9
7
2. × 10-6
1.5 × 10-6
1. × 10-6
5. × 10-7
1
-1
2
3
-5. × 10-7
2.pdf Fig.2.Error in solution of example 5.1 Example 5.2. For equation, Z y 0 (t) = et y(t) + t y(t − 1) +
t
t−1
1 et+s y(s)ds + et − e2t − t e(t−1) − (e3t − e(3t−2) ), 2
t ∈ [0, 3],
with initial condition, y(t) = et ,
t ∈ [−1, 0],
y(t) = et is exact solution. On comparing above equation with equation (1.1)-(1.2), we have, τ = 1, α(t) = et , β(t) = t, g(t) = et − e2t − t e(t−1) − 12 (e3t − e(3t−2) ) k(t, s) = et+s and φ(t) = et . The approximate solutions of the above problem given by equation (1.3) are F or N = 4,
F or N = 8,
y5 (t) = 1. + 1.00338 t + 0.531623 t2 + 0.124104 t3 + 0.00879832 t4 + 0.0450581 t5 .
y9 (t) = 1. + 1.t + 0.500009 t2 + 0.166681 t3 + 0.041636 t4 + 0.00829964 t5 + 0.00144749 t6 +0.000207197 t7 − 8.9232 × 10−6 t8 + 0.000015345 t9 .
F or N = 12,
y12 (t) = 1. + 1.t + 0.5 t2 + 0.166667 t3 + 0.0416666 t4 + 0.00833339 t5 + 0.00138892 t6 +0.000198268 t7 + 0.0000248933 t8 + 2.79705 × 10−6 t9 + 1.95571 × 10−7 t10 +6.88974 × 10−8 t11 − 9.7248 × 10−9 t12 + 1.64898 × 10−9 t13 .
The error in the computation is as follows. F or N = 4,
e5 (t) = 1.26704 × 10−14 − 0.00298495 t − 0.0288731 t2 + 0.0370253 t3 + 0.0313884 t4 − 0.0315251 t5 .
F or N = 8,
e9 (t) = 5.61472 × 10−11 − 1.23538 × 10−8 t − 0.0000103122 t2 − 0.0000119396 t3 + 0.0000363398 t4 +0.0000154916 t5 − 0.0000505867 t6 + 2.51653 × 10−6 t7 + 0.0000217713 t8 − 8.84386 × 10−6 t9 .
F or N = 12,
e12 (t) = 8.75133 × 10−8 + 1.73373 × 10−7 t − 1.90596 × 10−7 t2 − 8.87541 × 10−8 t3 + 6.08901 × 10−8 t4 −5.26808 × 10−8 t5 − 2.96756 × 10−8 t6 + 1.4505 × 10−7 t7 − 8.72605 × 10−8 t8 − 4.46058 × 10−8 t9 +7.89559 × 10−8 t10 − 4.15644 × 10−8 t11 + 1.07122 × 10−8 t12 − 1.27046 × 10−9 t13 .
The approximate solution of example 5.2 and exact solution and error in computation are listed in table 5.3, Fig.3. and table 5.4, Fig.4.
8
Table 5.3: Exact and approximate solution of example 5.2 . t Exact Solution Approximate Solution N =4 N = 8 N = 12 0 1 1 1 1 1/2 1.64872 1.65207 1.64872 1.64872 1 2.71828 2.71296 2.71829 2.71828 3/2 4.48169 4.50677 4.48177 4.48169 2 7.38906 7.70871 7.38998 7.38906 5/2 12.1825 13.5141 12.1911 12.1825 3 20.0855 23.8073 20.1417 20.0857 25
20
15
10
5
3.pdf
1
-1
2
3
Fig.3. Table 5.3: Error in the approximate solution of example 5.2 . t Error Error Error N =4 N =8 N = 12 0 1.26704 × 10−14 -5.61472 × 10−11 8.75133 × 10−8 1/2 -0.00310597 -2.02424 × 10−6 -1.17917 × 10−7 1 0.00503055 -5.57536 × 10−6 2.00867 × 10−8 3/2 -0.0249714 -0.000077138 -1.83111 × 10−7 2 -0.331848 -0.000929667 -1.06288 × 10−6 5/2 -1.46191 -0.0086493 -0.00001565 3 -4.38727 -0.0563137 -0.000199638 0.005
1
-1
2
-0.005 -0.010 -0.015 -0.020 -0.025
4.pdf
-0.030
Fig.4.Error in solution of example 5.2 9
3
6
Conclusion:
In this paper, we obtained the approximate solution of a linear Volterra delay integro-differential equations by using Hermite polynomials. An error analysis technique based on residual function is also developed. The improved solution is obtained by adding error term. The method is illustrated for two examples. Since the absolute difference between two consecutive solutions becomes sufficiently small for N sufficiently large, the approximate solution {yN +1 } converges to exact solution.
References [1] Asma Shaikh, Sarita Thakar, Analytical and Numerical Stability of Volterra Delay IntegroDifferential Equations, Journal of Indian Academy of Mathematics, 37(2015) [2] Aysegul Akyuz-Dascioglu, Mehmet Sezer, Chebyshev polynomial solution of system of higher-order linear Fredholm-Volterra integro-differential equations, Journal of Franklin Institute, 342(2005), 688701. [3] Hermann Brunner, Theory and numerical solution of Volterra functional integral equations, July 2010. [4] Mustafa Gulsu, Hatice Yalman, Yalcin Ozturk, and Mehmet Sezer, A New Hermite Collocation Method for Solving Differential Difference Equations, Applications and Applied Mathematics: An International Journal, 6(2011), 116-129. [5] Mustafa Gulsu, Mehmet Sezer, Taylor collocation method for solution of systems of high-order linear Fredholm -Volterra integro-differential equations, International Journal of Computer Mathematics, 83(2006), 429-448. [6] Nilay Akgonullu, Niyazi Sahin, Mehmet Sezer, A Hermite Collocation Method for the Approximate Solutions of High-Order Linear Fredholm Integro-Differential Equations, Wiley Online Library (wileyonlinelibrary.com)(2010),DOI 10.1002/num.20604. [7] Niyazi Sahin, Suayip Yuzbasi, Mehmet Sezer, A Bessel polynomial approach for solving general linear Fredholm integro-differential-difference equations, International Journal of Computer Mathematics, 88(2011), 3093-3111. [8] Salih Yalcinbas, Mehmet Sezer, Huseyin Hilmi Sorkun, Legendre polynomial solutions of high-order linear Fredholm integro-differential equations, Applied Mathematics and Computation, 210(2009), 334-349. [9] Salih Yalcinbas, Muge Aynigul, Mehmet Sezer, A collocation method using Hermite polynomials for approximate solution of pantograph equations, Journal of the Franklin Institute, 348(2011),1128-1139. [10] Suayip Yuzbasi, Laguerre approach for solving pantograph-type Volterra integro-differential equations, Applied Mathematics and Computation, 232(2014), 1183-1199.
10
