Gen. Math. Notes, Vol. 23, No. 1, July 2014, pp. 79-88
ISSN 2219-7184; Copyright © ICSRS Publication, 2014
www.i-csrs.org
Available free online at http://www.geman.in
Treatment for High-Order Linear Fredholm
Integro-Differential Equations of Degenerated
Kernel
Aloko MacDonald D1 and Abdulmalik Ibrahim Onuwe2
1
Department of Mathematics, University of Lagos, Lagos- Nigeria
E-mail: mac_aloko@yahoo.com
2
National Agency for Science and Engineering Infrastructure (NASENI), Abuja
Federal Ministry of Science and Technology, Nigeria
E-mail: Onuwe64@gmail.com
(Received: 2-4-14 / Accepted: 2-7-14)
Abstract
In this work, projection method with degenerated Kernel and its properties are
used to solve linear Fredholm Integro-Differential integral equations (LFIDE).
The kernel in the equation are degenerated and convert to matrix in which each
equations are integrated over the interval [0, 1] to produce the elements for
obtaining the Eigen value. It approaches linear Fredholm Integro-Differential
equations in a manner that gives the solution in an exact form and not in a series
form, the algorithm is simple and effective, and could also provide an accurate
approximate solution.
Keywords: Fredholm Integro-Differential Equations, Projection Method.
80
Aloko MacDonald D et al.
1.0
Introduction
Fredholm Integro-Differential equations (FIDE) arise in many scientific
applications. It was also shown that FIDE can be derived from boundary value
problems. Converting boundary value problems to Integro-Differential Fredholm
integral equations and converting Fredholm Integro-Differential equations (FIDE)
to equivalent boundary value problems are rarely used. Erik Ivar Fredholm (1866–
1927) is best remembered for his work on integral equations and spectral theory.
However, Adomian decomposition method (ADM) for solving integral equations
has been presented by Adomian [1-2] and then this has been extended by Wazwaz
to Volterra integral equation [13] and to boundary value problems for higherorder integro-differential equations. A Fredholm Integro-Differential equations
(FIDE) equation of the second kind may be written as
u x =f x +λ
k x, t u t dt
Subject to initial conditions u 0 =
(1.0)
,0 ≤
≤
−1
(1.1)
2.0
Formulation and Implementation Method
k ,
=
Definition 1: A kernel
, is called degenerate if it can be expressed as the
sum of a finite number of terms, each of which is the product of a function of x
only and a function of t only[16]: If we can write the kernel of (1) in the form:
!"
α
(2.0)
(or k = ∑%!" |α | ) We may assume that theα
are linearly independent. Any
continuous function k x, t can be uniformly approximated by polynomials in a
closed interval.
To illustrate the efficiency of the proposed numerical methods in studying the
model equation (1.0) subject to the initial conditions, and given the kernel in (2.0),
the general second kind Integro-Differential Fredholm equation,
u% x = f x + λ
k x, t u t dt, u 0 =
,0 ≤
≤
−1 ,
are the initial conditions
Equation (1.0) can be rewritten as
u% x = f x + λ
∑(
&!" α& x β& t u t dt
u% x = f x + λ ∑(
&!" α& x
β& t u t dt
(2.1)
(3.0)
Treatment for High-Order Linear Fredholm…
81
To discretized (2.1) let us denote
b& =
β& t u t dt
(4.0)
and since our focus is to solve for +, we need to find our +%,depending on the initial +
u x = f%,- x + λ ∑0
&!" b& α&
./
which is
x
(5.0)
from (3.0), we have
u% x = f x + λ ∑(
&!" b& α& x
(6.0)
L% x = f x + λ ∑0
&!" b& α& x
(6.1)
Operating with L2% on both sides of (6.1) and with the application of (1.1), it
follows
u x − ∑%!"
u x = ∑%!"
3 456
2" !
3 456
2" !
8
0 = L2% [f x + λ ∑0
&!" b& α& x ]
2"
8
0 + L2% [f x + λ ∑(
&!" b& α& x ]
2"
(6.2)
Let us multiply (6.2) by β; x , j = 1, 2, 3. . nintegrate from a to b over x
u x β; x = ∑%!"
B
u
C
D
B
C
3 456
2" !
8
x β; x A = ∑%!"
2"
0 β; x + L2% [f x + λ ∑(
&!" b& α& x ] β; x
B 3 456
8 2"
C 2" !
L2% f x β; x A + λ E
(
&!"
b&
B
C
0 β; x A +
L2% α& x β; x A F
(6.3)
where the differential operators L% = G3H , L%2" = G3H56.
GH
GH56
The inverse operator L2% is therefore considered an n−fold integral operator
defined by L2% . = . dx
Let
b; =
f; =
a&; =
B
u x β;
C
B 2%
L f x
C
L2% α&
x A
β; x A
x β; x dx
(7.0)
82
Aloko MacDonald D et al.
Putting (7.0) into (6.3), we have
b; = f; + λ E
b; − λ E
(I − λ E
(
&!"
(
&!"
(
&!"
b& a&;
b& a&; = f;
(8.0)
a&; b&; = f;
(I − λA B = F
There exist a unique solution for (8.0) if the determinant|I − λA| ≠ 0 and either
no solution or infinitely many solutions when |I − λA| = 0.
This is a system of n linear equations in the n unknowns f; . Suppose that there is a
unique solution f" , fO , fP … . f to this system.This system can be solved by finding
allb& , i = 1, 2, 3, … . n; and substituting into
u(x) = f%,- (x) + λ ∑0
&!" b& α&
./ (x)
b"
B = Sb.O T
b
a""
WaO"
A = VV .
V .
Ua "
to find u(x)
f"
F = Sf.O T
f
(8.1)
a"O .
aOO .
. .
. .
a O .
.
.
a"
aO
.
.
.
.
. a
Z
Y
Y
Y
X
.
L2% α
(9.0)
From (7.0)
L2% α"
./ (x)β" (x)dx
L2% αO
./ (x)β" (x)dx
.
W
V
V
A=V
V
V
U
L2% α"
./ (x)βO (x)dx
L2% α"
./ (x)β
3.0
Application and Numerical Results
.
.
L2% αO
./ (x)βO (x)dx
. .
. . .
. . .
. . .
L2% αO ./ (x)β (x)dx .
(x)dx
.
L2% α
./ (x)β" (x)dx Z
Y
./ (x)βO (x)dx Y
.
.
Y
Y
Y
2%
L α ./ (x)β (x)dxX
To illustrate the efficiency of the proposed Algorithm in studying the model
equation (1.0) subject to the initial conditions, we solve three examples of
Fredholm integro-differential equations by the method of solution above.
Example 3.1 Solve the following Fredholm integro-differential equation
u[[[ (x) = 2 + sin (x) −
]
(x
^
− t)u(t) dt, u(0) = 1, u[ (0) = 0, 8[[ (0) = −1
(10.0)
Treatment for High-Order Linear Fredholm…
83
The kernel is K x, t = x − t = ∑O!" α
α"
αO
=
= −1
=1
=t
"
O
=3
` = −1
From (6.3), 8
+%,-
+" =
+O =
`
=
]
^
O
3k
]
^ "
P
]h
−1+
+ cos
+%,-
+%,-
=
3a
O
A =
A =
bc
c
+ def g −
] ok
n
^ P
+ cos
]
^
ok
P
j − P! jO
i! "
3h
3k
p A = "O
(11.0)
]h
+ cos
A =
]q
"r
−2
s = S]q "O T
−2
"r
α"%,-
αO%,E
O
!"
t"" =
t"O =
tO" =
tOO =
=
3h
i!
= − P!
tu =
3k
]
α %,^
]
α
^ "%,-
]
α
^ O%,]
α
^ "%,-
]
α
^ O%,]h
"
"
O
A =
A =
A =
"ii
− P^
]q
A
] oh
A
^ i!
=
] oq
A
^ i!
= "ii
A =−
O
j
D " F = S]q "O T − S ]r!v
jO
−2
"r
u
− i!
]h
]q
] ok
A
^ P!
]
oh
−
A
^
P!
j
T D "F
jO
]q
r!
= − i!
]h
]v
= − P^!
]q
(12.0)
84
Aloko MacDonald D et al.
The determinant is non-zero and solving (12.0) to obtain
j" = 0, jO = −2
From (5.0)
u x = f%,- x + λ ∑&!" b& α&
8
8
=
3k
P
+ cos
−
= cos
3h
i!
./
∗0−
x
3k
P!
∗ −2
Example 3.2 Solve the following Fredholm integro-differential equation
z
a
u x x = 2x − π + sin x + cos x −
^
u 0 = u[ 0 = 1, 8[[ 0 = 8[[[ 0 = −1
x − 2t u t dt,
(13.0)
The kernel is K x, t = x − 2t = ∑O!" α
α"
αO
"
O
=
= −2
=1
=t
=4
` = −1
From (6.3), 8
−
}~
b|
+%,-
+" =
+O =
]b •
ۥ
z
a
3a
+ f•‚ b + def g −
= }~ −
b|
^
"
^
O
z
a
− 8 0 − 8[ 0 − O! 8[[ 0 − P! 8[[[ 0 =
ۥ
+%,-
+%,-
2 − i…^†
s = S]
]b •
]v
T
O‡]ˆ
−
O
POOr…^
3k
j − "O jO
"O^ "
3q
3h
(14.0)
+ f•‚ b + def g
A =
A =
z
a
^
n}~ −
ƒ|
z
a
^
]ƒ •
ۥ
−
}~
ƒ|
+ f•‚ ƒ + def „ p A = 2 − i…^†
]ƒ •
ۥ
]v
+ f•‚ ƒ + def „ A = O − POOr…^
]
O‡]ˆ
Treatment for High-Order Linear Fredholm…
α"%,-
αO%,"
O
E
O
=
=1
=t
!"
t"" =
t"O =
tO" =
tOO =
3q
r!
=−
z
a
"O
^
α"%,-
^
z
a
α %,-
z
a
z
a
j
D " F = S]
jO
−
O
A =
A =
O
]v
i…^†
O‡]ˆ
]v
T − S i…^†^
]ˆ
POOr…^
A
z q
o
a
"^‰rO^
A = i…^†^
r!
^
A =−
O
αO%,-
2−
A =
"
α"%,-
^
u
"
αO%,-
^
^
3h
z
a
tu =
85
]v
z h
o
a
^
z v
o
a
^ r!
z
a
^
A = − "‡O^
"O
]q
A = "^‰rO^
]ˆ
− "O A = − i…^†
−
oq
]v
]q
"‡O^ j"
TD F
]v
jO
− i…^†
(15.0)
The determinant is non-zero and solving (15.0) to obtain
j" = 2, jO =
Š
O
From (5.0)
u x = f%,- x + λ ∑&!" b& α&
8
8
= }~ −
b|
]b •
ۥ
./
x
+ f•‚ b + def g −
= f•‚ b + def g
∗ 2 − − "O ∗ O
r!
3q
3h
Š
Example 3.3: Solve the following Fredholm integro-differential equation
u [ x = 2x − x sin x + 2cos x −
u 0 = u[ 0 = 0
z
a
z
2
a
The kernel is K x, t = t − x = ∑O!" α
t − x u t dt,
(16.0)
86
Aloko MacDonald D et al.
α"
=1
αO
=
=t
"
O
=2
= −1
` = −1
From (6.3), 8
bc
c
+ gf•‚ b −
+%,-
+" =
+O =
z
a
z
2
a
=
z
a
z
2
a
]q
s = ‹ Oi^ Œ
−2
α"%,-
αO%,"
O
− 8 0 − 8[ 0 =
E
O
t"O =
c
+%,-
A =
+%,-
O
=
3k
+ gf•‚ b
"
3a
O!
A =
z
a
z
2
a
z
a
z
2
a
(17.0)
n + „f•‚ ƒ p A =
c
Oi^
ƒc
−
]q
ƒc
c
+ „f•‚ ƒ A = −2
= − P!
=t
= −1
!"
t"" =
bc
j − P! jO
O! "
3a
tu =
z
a
z
2
a
z
a
z
2
a
3k
z
a
z
2
a
α"%,-
αO%,-
α %,"
"
u
A
A =−
A =−
z
k
a o
z
2 O!
a
z
h
a o
z
2 P!
a
A =0
A = − i†^
]q
Treatment for High-Order Linear Fredholm…
z
a
z
2
a
tO" =
z
a
z
2
a
tOO =
α"%,-
O
αO%,-
j
D " F = ‹ Oi^Œ − S
jO
−2
]q
0
−
O
]k
Oi
z
a
a o
z
2 O!
a
A =−
A =
−
]q
z
k
a o
z
2 P!
a
j
T D "F
jO
A =−
A =0
87
]k
Oi
i†^
0
(18.0)
The determinant is non-zero and solving (15.0) to obtain
j" = 0, jO = −2
From (5.0)
u x = f%,- x + λ ∑&!" b& α&
8
8
4.0
=
bc
c
+ gf•‚ b −
= bf•‚ b
./
x
∗ 0 − − P! ∗ −2
O!
3a
3h
Conclusion
The present method reduces the computational difficulties of other traditional
methods and all the calculation can be made simple. The accuracy of the obtained
solution can be improved by taking more terms in the solution.
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