Download SOLVING SCHL ¨OMILCH`S INTEGRAL EQUATION BY THE

MATHEMATICAL PHYSICS
SOLVING SCHLÖMILCH’S INTEGRAL EQUATION BY THE
REGULARIZATION-ADOMIAN METHOD
ABDUL-MAJID WAZWAZ
Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA
E-mail: [email protected]
Received September 3, 2014
We present a reliable study for the linear and the nonlinear Schlömilch’s integral equation and its generalized forms. We also introduce Schlömilch-type integral
equations and study it as well. The Schlömilch’s integral equation has been used for
many ionospheric problems, atmospheric and terrestrial physics. We use the regularization method combined with the Adomian decomposition method to handle all forms of
Schlömilch’s integral equations. The combined regularization-Adomian method can be
used in many applied physics, applied mathematics and engineering applications. We
justify the reliability of the regularization-Adomian method through illustrative examples.
Key words: Schlömilch’s integral equation; regularization; terrestrial physics.
1. INTRODUCTION
The ionosphere is a shell of electrons and electrically charged atoms that surrounds the earth in a region of the upper atmosphere. This region is ionized by solar
radiation and is important because it influences radio propagation to distant places on
the earth. The ionosphere shell owes its existence primarily to ultraviolet radiation
from the sun. The dynamics of the ionosphere shell is a very important area research
that has sustained and is still going strong for terrestrial physics. The ionosphere shell
has a unique structure in which electrons and electrically charged atoms coexist.
In this work we will study the Schlömilch’s integral equation that was used in
order to derive the electron density profile from the ionospheric for oblique incidence
for the quasi-transverse approximation [1–4]. The Schlömilch’s integral equation
reads
Z π
2 2
f (x) =
u(x sin t) dt,
(1)
π 0
where f (x) is a continuous differential coefficient for −π ≤ x ≤ π. It has been
proved in [1–4] that this equation has one solution given by
Z π
2
u(x) = f (0) + x
f 0 (x sin t) dt,
(2)
0
f0
where is the derivative of f with respect to the complicated argument ξ = x sin t.
The Schlömilch’s integral equation and its unique solution have been used to deRJP 60(Nos.
Rom.
Journ. Phys.,
1-2),
Vol.
56–71
60, Nos.
(2015)
1-2, P.(c)
56–71,
2015
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2015
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Solving Schlömilch’s integral equation by the regularization-Adomian method
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termine the electron density profile from the ionospheric for the case of the quasitransverse (QT) approximations [1–4]. The Schlömilch’s integral equation has been
used for many ionospheric problems, atmospheric and terrestrial physics.
There are four types of Schlömilch’s integral models that will be touched upon
in this paper. They are the standard model (1), a generalized Schlömilch’s integral
equation, a Schlömilch-type integral equation, where the cosine function replaces the
sine function of the standard form, and the nonlinear Schlömilch’s integral equation,
where the linear term u(x sin t) in the standard model is replaced by a nonlinear term
F (u(x sin t)) or by F (u(x cos t)).
The generalized Schlömilch’s integral equation takes the form
Z π
2 2
f (x) =
u(x sinn t) dt, n ≥ 1.
(3)
π 0
Moreover, using the sense of the Schlömilch’s integral equation we introduce
a Schlömilch-type integral equation given by
Z π
2 2
f (x) =
u(x cosn t) dt, n ≥ 1,
(4)
π 0
where the argument x sin t was replaced by x cos t.
It is worth noting that the Schlömilch’s integral equation may arise in a nonlinear form as
Z π
2 2
f (x) =
F (u(x sin t)) dt,
(5)
π 0
where f (x) is a continuous differential coefficient for −π ≤ x ≤ π, and F is a nonlinear form of u(x sin t) such as u2 (x sin t) and u3 (x sin t).
It is obvious that the Schlömilch’s integral equation is a special case of Fredholm
integral equation of the first kind. Recall that the linear Fredholm integral equations
of the first kind are of the form [5–16]
Z b
f (x) = λ
K(x, t)u(t) dt, x ∈ Ω,
(6)
a
and the nonlinear Fredholm integral equation of the first kind is given by
Z b
f (x) = λ
K(x, t)F (u(t)) dt, x ∈ Ω,
(7)
a
where Ω is a closed and bounded region. Fredholm integral equations of the first kind
(6) and (7) are characterized by the occurrence of the unknown function u(x) only
inside the integral sign. Some special difficulties arise from the existence of u(x)
only inside the integral sign. The Fredholm integral equations of the first kind appear
in many physical models such as radiography, spectroscopy, cosmic radiation, image
processing and in the theory of signal processing.
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The complete study of ionospheric problems, atmospheric and terrestrial physics
should cover both theoretical analysis, which exists reasonably in the literature, and
computational, which we believe needs more studies, and this is one of our aims of
this work. For this reason, we will combine the method of regularization [9–13] and
the Adomian decomposition method [14–16] to handle the linear and the nonlinear
Schlömilch’s integral equation and the other related equations (1) and (3)–(5). The
combined regularization-Adomian method proved to be reliable, effective [14–16],
and hence can be extremely useful in carrying out further works especially in illposed problems as Fredholm integral equations of the first kind. In what follows, we
give brief outlines of each method.
The Adomian decomposition method (ADM) and the method of regularization are given thoroughly in the literature. In what follows we will present a brief
summary for each method.
2. THE ADOMIAN DECOMPOSITION METHOD
The details of the ADM are now well known and widely applied in the literature; see, for example [14–16].
The ADM admits the use of the infinite decomposition series
u(x) =
∞
X
un (x),
(8)
n=0
for the solution u(x). The components ui (x), i ≥ 1 are determined by using a recursive relation, and the zero-th component u0 (x) is given all terms that are not included
inside the integral sign. Substituting (8) into both sides of the integral equation, and
assigning u0 (x) as indicated earlier, the classic ADM admits the use of the recursive
relation
u0 (x) = all terms not included inside the integral sign,
Z x
(9)
uj+1 (x) =
(uj (x sin t)) dx, j ≥ 0,
0
that will lead to the complete determination of the components un (x), for n ≥ 0, of
u(x). The series solution of u(x) follows immediately and converges to the closed
form solution if such a solution exists. For concrete problems, the obtained series
can be used for numerical purposes.
3. THE METHOD OF REGULARIZATION
In what follows we will present a brief summary of the method of regularization. Details can be found in [9–13].
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The method of regularization was established independently by Tikhonov [11,
12] and Phillips [13]. The method transforms a first kind Fredholm integral equation
to a second kind equation. The regularization method transforms the linear Fredholm
integral equation of the first kind
Z b
f (x) =
K(x, t)u(t) dt,
(10)
a
to the approximation Fredholm integral equation
Z b
K(x, t)uα (t) dt,
αuα (x) = f (x) −
(11)
a
where α is a small positive parameter called the regularization parameter. It is clear
that (11) is a Fredholm integral equation of the second kind that can be rewritten as
Z
1
1 b
uα (x) = f (x) −
K(x, t)uα (t) dt.
(12)
α
α a
Moreover, it was proved by Tikhonov [11, 12] and Phillips [13] that the solution uα
of equation (12) converges to the solution u(x) of (10) or (12) as α → 0. This means
that
u(x) = lim uα (x).
(13)
α→0
It was also found that the obtained series mostly includes infinite geometric series.
We will apply the aforementioned methods to the Schlömilch’s integral equation and its other related forms. Two examples will be given for each case to illustrate
the power of the proposed approach.
4. THE LINEAR SCHLÖMILCH’S INTEGRAL EQUATION
The method of regularization converts the linear Schlömilch’s integral equation
Z π
2 2
f (x) =
u(x sin t) dt, x ∈ D,
π 0
to the Schlömilch’s integral equation of the second kind in the form
Z π
2 2
αuα (x) = f (x) −
uα (x sin t) dt,
π 0
or equivalently
!
Z π
1
1 2 2
uα (x sin t) dt ,
uα (x) = f (x) −
α
α π 0
(14)
(15)
(16)
where α is a small positive parameter. It is obvious that the solution uα of equation
(16) converges to the solution u(x) of (14) as α → 0. Consequently, we can apply
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any method we studied before for solving the Schlömilch’s integral equations of the
second kind. However, we will use the Adomian decomposition method, where we
set the recurrence relation
1
uα0 (x) = f (x),
α
!
Z π
(17)
1 2 2
uαk+1 (x) = −
uαk (x sin t) dt , k ≥ 0.
α π 0
In what follows we will present two illustrative examples. Our focus will be on
transforming the first kind equation to a second kind equation by using the method of
regularization, and hence we can use any appropriate method such as the Adomian
decomposition method.
Example 1:
Solve the linear Schlömilch’s integral equation
Z π
2 2
2
1 + πx =
u(x sin t) dt, −π ≤ x ≤ π.
π 0
Using the method of regularization, Eq. (18) becomes
!
Z π
2
1
1
2
uα (x) = (1 + πx2 ) −
uα (x sin t) dt .
α
α π 0
(18)
(19)
We select the Adomian method for solving this equation. The Adomian method
admits the use of the recurrence relation
1
uα0 (x) = (1 + πx2 ),
α
!
Z π
(20)
1 2 2
uαk+1 (x) = −
uαk (x sin t) dt , k ≥ 0.
α π 0
This in turn gives the first few components
1
(1 + πx2 ),
α
2 + π2
uα1 (x) = −
,
2α2
4 + πx2
uα2 (x) =
,
4α3
8 + πx2
,
uα3 (x) = −
8α4
..
..
uα0 (x) =
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We have
uα (x) =
1
1
1
πx2
1
1
1
1
1− + 2 − 3 +··· +
1−
+ 2 − 3 + · · · , (22)
α
α α
α
α
2α 4α
8α
which gives
1
2π
+
x2 ,
(23)
1 + α 1 + 2α
obtained upon finding the summation of each infinite geometric series. The exact
solution is given by
u(x) = lim uα (x) = 1 + 2πx2 .
(24)
uα (x) =
α→0
Example 2:
Solve the linear Schlömilch’s integral equation
Z π
2 2
u(x sin(3t) dt, −π ≤ x ≤ π.
2x =
π 0
Using the method of regularization, Eq. (25) becomes
!
Z π
2
1 2 2
uα (x) = x −
uα (x sin(3t)) dt .
α
α π 0
(25)
(26)
We select the Adomian method for solving this equation. The Adomian method
admits the use of the recurrence relation
2
uα0 (x) = x,
α
!
Z π
(27)
1 2 2
uαk (x sin(3t)) dt , k ≥ 0.
uαk+1 (x) = −
α π 0
This in turn gives the first few components
2
x,
α
4
uα1 (x) = −
x,
3απ
8
uα2 (x) = 3 2 x,
9α π
16
uα3 (x) = −
x,
27α4 π 3
..
..
uα0 (x) =
(28)
We have
2
2
4
8
uα (x) = x 1 −
+
−
+··· ,
α
3απ 9α2 π 2 27α3 π 3
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which gives
6π
x,
(30)
2 + 3απ
obtained upon finding the summation of the infinite geometric series. The exact
solution is given by
u(x) = lim uα (x) = 3πx.
(31)
uα (x) =
α→0
5. THE GENERALIZED SCHLÖMILCH’S INTEGRAL EQUATION
We follow the discussion presented before to handle generalized Schlömilch’s
integral equation. The method of regularization converts this equation of the first
kind
Z π
2 2
u(x sinn t) dt, n ≥ 1, x ∈ D,
(32)
f (x) =
π 0
to the Schlömilch’s integral equation of the second kind in the form
Z π
2 2
αuα (x) = f (x) −
uα (x sinn t) dt,
(33)
π 0
or equivalently
!
Z π
1
1 2 2
uα (x) = f (x) −
(34)
uα (x sinn t) dt ,
α
α π 0
where α is a small positive parameter. As stated before, the solution uα of equation
(34) converges to the solution u(x) of (32) as α → 0. Proceeding as before, we use
the Adomian decomposition method that admits the use of the recurrence relation
1
uα0 (x) = f (x),
α
!
Z π
(35)
1 2 2
uαk+1 (x) = −
uαk (x sinn t) dt , k ≥ 0.
α π 0
In what follows we will present two illustrative examples.
Example 1:
Solve the Schlömilch’s integral equation
Z π
2 2
2
u(x sin2 t) dt, −π ≤ x ≤ π.
x + 3x =
π 0
Using the method of regularization, Eq. (36) becomes
!
Z π
1
1 2 2
2
2
uα (x) = (x + 3x ) −
uα (x sin t) dt .
α
α π 0
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Solving Schlömilch’s integral equation by the regularization-Adomian method
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We select the Adomian method for solving this equation. The Adomian method
admits the use of the recurrence relation
1
(x + 3x2 ),
α
!
Z π
1 2 2
uαk (x sin2 t) dt , k ≥ 0.
uαk+1 (x) = −
α π 0
uα0 (x) =
(38)
This in turn gives the first few components
1
(x + 3x2 ),
α
1
uα1 (x) = − 2 x(4 + 9x),
8α
1
uα2 (x) =
x(16 + 27x),
64α3
1
uα3 (x) = −
x(64 + 81x),
512α4
..
..
uα0 (x) =
(39)
We have
x
1
1
1
3x2
3
9
27
1−
+ 2 − 3 +··· +
1−
+
−
+
·
·
·
,
α
2α 4α
8α
α
8α 64α2 512α3
(40)
which gives
uα (x) =
2
24
x+
x2 .
1 + 2α
3 + 8α
(41)
u(x) = lim uα (x) = 2x + 8x2 .
(42)
uα (x) =
The exact solution is given by
α→0
Example 2:
Solve the Schlömilch’s integral equation
Z π
5 2 2 2
4x − x =
u(x sin3 (t) dt, −π ≤ x ≤ π.
16
π 0
(43)
Using the method of regularization, Eq. (43) becomes
1
5
1
uα (x) = (4x − x2 ) −
α
16
α
2
π
Z
π
2
!
3
uα (x sin (t)) dt .
0
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The Adomian method admits the use of the recurrence relation
1
5
(4x − x2 ),
α
16
!
Z π
1 2 2
3
uαk+1 (x) = −
uαk (x sin (t)) dt , k ≥ 0.
α π 0
uα0 (x) =
(45)
This in turn gives the first few components
1
5
(4x − x2 ),
α
16
16
25 2
x ,
uα1 (x) = − 2 x +
3α π
256α2
64
125 2
uα2 (x) = 3 2 x +
x ,
9α π
256α2
625 2
256
x+
x ,
uα3 (x) = −
4
3
27α π
4096α3
..
..
uα0 (x) =
(46)
Proceeding as before, and using the infinite geometric series gives
uα (x) =
5
12π
x−
x2 ,
4 + 3απ
5 + 16α
(47)
obtained upon finding the summation of the infinite geometric series. The exact
solution is given by
u(x) = lim uα (x) = 3πx − x2 .
α→0
(48)
6. THE LINEAR SCHLÖMILCH-TYPE INTEGRAL EQUATION
In this section we will examine a Schlömilch-type integral equation. We will
follow the analysis presented in the preceding sections, hence we skip details. The
method of regularization converts the Schlömilch-type integral equation
Z π
2 2
f (x) =
u(x cos t) dt, x ∈ D,
(49)
π 0
to the Schlömilch-type integral equation of the second kind in the form
!
Z π
1 2 2
1
uα (x) = f (x) −
uα (x cos t) dt .
α
α π 0
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It is obvious that the solution uα of equation (50) converges to the solution u(x) of
(49) as α → 0. The Adomian decomposition method gives the recurrence relation
1
f (x),
α
!
Z π
1 2 2
uαk+1 (x) = −
uαk (x cos t) dt , k ≥ 0.
α π 0
uα0 (x) =
In what follows we will examine two illustrative examples.
Example 1:
Solve the Schlömilch-type integral equation
Z π
2 2
1 + 2x =
u(x cos t) dt, −π ≤ x ≤ π.
π 0
Using the method of regularization, Eq. (52) becomes
!
Z π
1
1 2 2
uα (x) = (1 + 2x) −
uα (x cos t) dt .
α
α π 0
(51)
(52)
(53)
We select the Adomian method for solving this equation. The Adomian method
admits the use of the recurrence relation
1
uα0 (x) = (1 + 2x),
α
!
Z π
(54)
1 2 2
uαk+1 (x) = −
uαk (x cos t) dt , k ≥ 0.
α π 0
This in turn gives the first few components
1
(1 + 2x),
α
π + 4x
uα1 (x) = −
,
πα2
π 2 + 8x
uα2 (x) = 2 3 ,
π α
π 3 + 16x
uα3 (x) = − 3 4 ,
π α
..
..
uα0 (x) =
(55)
This in turn gives
1
1
1
1
2
2
4
8
uα (x) =
1− + 2 − 3 +··· + x 1−
+
−
+··· ,
α
α α
α
α
πα π 2 α2 π 3 α3
(56)
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which gives
1
2π
+
x,
(57)
1 + α 2 + πα
obtained upon finding the summation of each infinite geometric series. The exact
solution is given by
uα (x) =
u(x) = lim uα (x) = 1 + πx.
α→0
Example 2:
Solve the Schlömilch-type integral equation
Z π
1 2 2 2
x =
u(x cos(2t)) dt, −π ≤ x ≤ π.
2
π 0
(58)
(59)
Proceeding as before, we obtain the recurrence relation
1 2
x ,
2α
!
Z π
1 2 2
uαk+1 (x) = −
uαk (x cos(2t)) dt , k ≥ 0.
α π 0
uα0 (x) =
(60)
This in turn gives the first few components
1 2
x ,
2α
1
uα1 (x) = − 2 x2 ,
4α
1
uα2 (x) = 3 x2 ,
8α
1 2
x ,
uα3 (x) = −
16α4
..
..
(61)
1 2
1
1
1
x 1−
+
−
+··· ,
uα (x) =
2α
2α 4α2 8α3
(62)
uα0 (x) =
We have
which gives
1
x2 ,
(63)
1 + 2α
obtained upon finding the summation of the infinite geometric series. The exact
solution is given by
uα (x) =
u(x) = lim uα (x) = x2 .
α→0
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7. THE NONLINEAR SCHLÖMILCH’S INTEGRAL EQUATION
The nonlinear Schlömilch’s integral equation takes the form
Z π
2 2
f (x) =
F (u(x sin t)) dt,
π 0
(65)
where F (u(x sin t)) is a nonlinear function of u(x sin t), such as ui (x sin t), ui (x cos t),
i ≥ 2, and f (x) is a continuous differential coefficient for −π ≤ x ≤ π.
To handle this nonlinear equation, we will follow the same analysis presented
earlier for linear equations. To achieve this goal, we should first transform this nonlinear equation to a linear form.
To transform (65) to a linear form of the first kind, we first use the transformation
F (u(x sin t)) = v(x sin t), u(x sin t) = F −1 (v(x)),
(66)
u(x sin t) = F −1 (v(x sin t)),
(67)
such that
which will transform (65) to
2
f (x) =
π
Z
π
2
v(x sin t) dt.
(68)
0
The method of regularization transforms the linear Schlömilch’s integral equation of
the first kind (68) to to the Schlömilch’s integral equation of the second kind given
by
!
Z π
1
1 2 2
(69)
vα (x) = f (x) −
vα (x sin t) dt ,
α
α π 0
where α is a small positive parameter. Applying the Adomian decomposition method
gives the recurrence relation
1
f (x),
α
!
Z π
1 2 2
vαk+1 (x) = −
vαk (x sin t) dt , k ≥ 0.
α π 0
vα0 (x) =
The scheme that we presented will be illustrated by the following examples.
Example 1:
Solve the nonlinear Schlömilch’s integral equation
Z π
2 2 2
5x6 =
u (x sin t) dt, −π ≤ x ≤ π.
π 0
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Using the transformation v = u2 transforms the last equation to a linear equation
given by
Z π
2 2
5x6 =
v(x sin t) dt, −π ≤ x ≤ π.
(72)
π 0
Using the method of regularization, Eq. (72) becomes
!
Z π
1 2 2
6
vα (x) = 5x −
vα (x sin t) dt .
(73)
α π 0
The Adomian method allows us to use the recurrence relation
5
vα0 (x) = x6 ,
α
!
Z π
1 2 2
vαk+1 (x) = −
vαk (x sin t) dt , k ≥ 0.
α π 0
(74)
This in turn gives the first few components
5 6
x ,
α
25 6
x ,
vα1 (x) = −
16α2
125 6
vα2 (x) =
x ,
256α3
625 6
x ,
vα3 (x) = −
4096α4
..
..
(75)
5 6
5
25
125
vα (x) = x 1 −
+
−
+··· ,
α
16α 256α2 4096α3
(76)
vα0 (x) =
This in turn gives
which gives
80
x6 ,
(77)
5 + 16α
obtained upon finding the summation of each infinite geometric series. The exact
solution for v is given by
vα (x) =
v(x) = lim uα (x) = 16x6 .
α→0
(78)
Recall that v = u2 , hence the exact solution is given by
v(x) = x4 , u(x) = ±4x3 .
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(79)
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Solving Schlömilch’s integral equation by the regularization-Adomian method
Example 2:
Solve the nonlinear Schlömilch’s integral equation
Z π
35 8 2 2 4
x =
u (x sin t) dt, −π ≤ x ≤ π.
8
π 0
Using the transformation v = u4 converts the last equation to
Z π
35 8 2 2
v(x sin t) dt, −π ≤ x ≤ π.
x =
8
π 0
Using the method of regularization, Eq. (81) becomes
!
Z π
35 8 1 2 2
vα (x) =
vα (x sin t) dt .
x −
8α
α π 0
69
(80)
(81)
(82)
Using the Adomian method gives the first few components
35 8
x ,
8α
1225 8
x ,
vα1 (x) = −
1024α2
42875 8
vα2 (x) =
x ,
131072α3
1500625 8
vα3 (x) = −
x ,
16777216α4
..
..
vα0 (x) =
(83)
This gives the series solution
35
1225
35 8
x 1−
+
+··· .
vα (x) =
8α
128α 1024α
(84)
Proceeding as before, and recall that v = u4 , hence the exact solution is given by
v(x) = 16x8 , u(x) = ±2x2 .
Example 3:
Solve the nonlinear Schlömilch’s integral equation
Z π
4 3 2 2 3
x =
u (x cos t) dt, −π ≤ x ≤ π.
3π
π 0
Using the transformation v = u3 converts the last equation to
Z π
4 3 2 2
x =
v(x cos t) dt, −π ≤ x ≤ π.
3π
π 0
RJP 60(Nos. 1-2), 56–71 (2015) (c) 2015 - v.1.3a*2015.2.7
(85)
(86)
(87)
70
Abdul-Majid Wazwaz
Using the method of regularization, Eq. (87) becomes
!
Z π
4 3 1 2 2
vα (x) =
vα (x sin t) dt .
x −
3πα
α π 0
Using the Adomian method gives the first few components
4 3
vα0 (x) =
x ,
3πα
16
vα1 (x) = − 2 2 x3 ,
9π α
64
vα2 (x) =
x3 ,
27π 3 α3
256 3
vα3 (x) = −
x ,
81π 4 α4
..
..
15
(88)
(89)
Proceeding as before, and recall that v = u3 , hence the exact solution is given by
v(x) = x3 ,
u(x) = x.
(90)
8. DISCUSSION
In this work we employed a combination of the method of regularization and
the Adomian decomposition method for a reliable treatment of the linear and nonlinear Schlömilch’s integral equation. The proposed method showed reliability to
handle these two forms of the first kind. The combined regularization-Adomian method can be extremely useful in carrying out further works especially in ill-posed
problems as Fredholm integral equations of the first kind. A variety of examples, linear and nonlinear, were examined to illustrate the analysis that was presented. Exact
solutions were formally derived.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
H. Unz, J. Atmos. and Terrestrial Physics 28, 315 (1966).
P. J. Gething and R. G. Maliphant, J. Atmos. and Terrestrial Physics 29, 599 (1967).
S. De, B. Sarkar, M. Mal, M. De, B. Ghosh, and S. Adhikari, Jpn. J. Appl. Phys. 33, 4154 (1994).
L. Bougoffa, M. Al-Haqbani, and R. Rach, Kybernetes 41, 145 (2012).
D. Mirzadei and M. Dehghan, Applied Numerical Mathematics 60, 245 (2010).
H. Leblond, H. Triki, and D. Mihalache, Phys. Rev. A 86, 063825 (2012).
H. Leblond and D. Mihalache, Phys. Reports 523, 61 (2013).
R. F. Churchhouse, Handbook of Applicable Mathematics (Wiley, New York, 1981).
L. M. Delves and J. Walch, Numerical Solution of Integral Equations (Oxford University Press,
London, 1974).
RJP 60(Nos. 1-2), 56–71 (2015) (c) 2015 - v.1.3a*2015.2.7
16
Solving Schlömilch’s integral equation by the regularization-Adomian method
71
Y. Cherruault and V. Seng, Kybernetes 26, 109 (1997).
A. N. Tikhonov, Soviet Math. Dokl. 4, 1035 (1963).
A. N. Tikhonov, Soviet Math. Dokl. 4, 1624 (1963).
D. L. Phillips, J. Ass. Comput. Math. 9, 84 (1962).
A. M. Wazwaz, A First Course in Integral Equations (WSPC, New Jersey, 1997).
A. M. Wazwaz, The regularization-homotopy method for the linear and non-linear Fredholm integral equations of the first kind, ISPACS-Communications in Numerical Analysis 1, 1 (2011).
16. A. M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications (Springer
and HEP, Berlin and Beijing, 2011).
10.
11.
12.
13.
14.
15.
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