SOLVING A DUAL INTEGRAL EQUATION INVOLVING FOURIER TRANSFORMS ENCOUNTERED IN A CRACK PROBLEMS FOR FRACTURE ELASTIC MATERIALS

SOLVING A DUAL INTEGRAL EQUATION INVOLVING
FOURIER TRANSFORMS ENCOUNTERED IN A CRACK
PROBLEMS FOR FRACTURE ELASTIC MATERIALS
NGUYEN VAN NGOC AND MA DINH TREN
Abstract. The aim of the present work is to consider a mixed bound-
ary value problem in a strip related to cracks of fracture materials. By
the Fourier transform, the problem are reduced to studying a dual inte-
gral equations on one edge of the strip. The uniqueness and existence
theorems of solution of dual integral equation are established in appro-
priate Sobolev spaces. A method for reducing the dual integral equation
to a hypersingular integral equation of second order is also proposed.
AMS Subject Classification: 45H05, 42A38, 46F05, 46F10, 47G30.
Key words: Biharmonic equation, mixed boundary value problems of
mathmatical physics, dual integral equations.
1. Introduction
During the last decade, numerous publics about crack problems of fracture
materials (see, [1], [2]). These problems often are reduced to hypersingular
integral equations [3], [4].
Formal technique for solving such problems have been developed exten-
sively, but their solvability problems have been not considered.
The aim of the present work is to consider a mixed boundary value prob-
lem of the linear elastic fracture mechanics in a strip by the dual integral
equations method [6], [8]. The problem may be intepreted as a plane crack
defect on the interval −a  x  a, y = 0. A siminary problem for the
half-plane worse considered in [1]-[2].
For investigating this problem, we use the pseudo-differemtial equations
approach [7] for investigation on solvability of a dual integral equations
involving Fourier transform, which are medial equations between the initial
mixed boundary value problems and hypersingular integral equations .
Our work is constructed as follows: In Section 2 we formulate the mixed
boundary value problem for the linear elastic fracture materials in a strip

and reduce it to a dual integral equation involving Fourier transforms. Sec-
tion 3 is intended for the solvability of dual integral equation with complex
inccreasing symbol in appropriate Sobolev spaces. Finally in the last sec-
tion, Section 4 we present a manners reducing the dual integral equation
involving Fourier transforms to a hypersingular integral equation of second
order.
13
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2. Formulations of the problem
2.1. Formulation of the mixed boundary value problem. We study
the solution Φ(x, y) of a boundary-value problem for the biharmonic equa-
tion
∂
2
Φ
∂x2
+
∂2Φ
∂y
2
+ β
∂Φ
∂x
= 0, β = const = 0 (2.1)
in the
Π = {(x, y) : −∞ < x < ∞, 0 < y < h}.
Let R be a real axis, Ω = (−a, a) denotes certain bounded interval and
Ω


= R \ Ω. Consider the following mixed boundary value problems.
To find solution Φ(x, y) of the equation (2.1) in the strip Π that satisfies
the boundary conditions
αΦ


y=h
+ γ
∂Φ
∂y



y=h
= g(x), x ∈ R, (2.2)





−
∂Φ
∂y



y=0
= f(x), x ∈ Ω,
Φ




y=0
= 0, x ∈ Ω

,
(2.3)
where α, γ are certain real constants such that α
2
+ γ
2
> 0.
This problem may be intepreted as a crack problem on the interval −a 
x  a, y = 0.
2.2. Reduction to a dual integral equation. We shall solve the for-
mulated problems by the method of Fourier transforms and reduce it to a
system of dual equations involving inverse Fourier transforms. For a suitable
function f(x), x ∈ R( for example, f (x) ∈ L
1
(R)), direct and inverse Fourier
transforms are defined by the formulas

f(ξ) = F [f](ξ) =

∞
−∞
f(x)e
ixξ
dx, (2.4)

˘
f(ξ) = F
−1
[f](ξ) =
1
2π

∞
−∞
f(x)e
−ixξ
dx. (2.5)
The Fourier transforms of tempered generalized functions may be found,
for example, in [5, 9].
Taking the Fourier transform with respect to the variable x for the equa-
tion (2.1), we obtain
∂
2
ˆ
Φ(ξ, y)
∂y
2
− (ξ
2
+ βiξ)Φ(ξ, y) = 0, (2.6)
where

Φ(ξ, y) = F
x


Φ(ξ, y) = A(ξ) cosh(λy) + B(ξ) sinh(λy), (2.7)
where A(ξ), B(ξ) are arbitrary functions of the variable ξ, and
[Φ(x, y)](ξ) is the Fourier transform with respect to x of
the function Φ(x, y). The general solution of the differential equation (2.6)
is taken in the form
NGUYEN VAN NGOC AND MA DINH TREN
14
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λ = λ(ξ) =
1
√
2


ξ
4
+ β
2
ξ
2
+ ξ
2
+
i
√
2
sgn(βξ)



ξ
4
+ β
2
ξ
2
− ξ
2
, (2.8)
where the sgn(.) is defined as
sgn(η) =





1, η > 0,
0, η = 0,
−1, η < 0.
The value
ˆ
Φ(0, y) is understood in the sense

Φ(0, y) = lim
ξ→0
ˆ
Φ(ξ, y).
We introduce notations
u(ξ) :=


Φ(ξ, 0) = A(ξ), u(x) := F
−1
[u](x), (2.9)
where u(ξ) is an unknown function.
Using boundary condition (2.2) and the relation (2.9), we express un-
known functions A(ξ) and B(ξ) in terms of u(ξ) and g(ξ). We have
A(ξ) = u(ξ), (2.10)
B(ξ) =
g(ξ)
α sinh(λh) + γλ cosh(λh)
−
α cosh(λh) + γλ sinh(λh)
α sinh(λh) + γλ cosh(λh)
u(ξ). (2.11)
From (2.7), (2.10) and (2.11), we have

Φ(ξ, y) = u(ξ) cosh(λy) +
[g(ξ) − u(ξ)(α cosh(λh) + γλ sinh(λh))] sinh(λy)
α sinh(λh) + γλ cosh(λh)
= u(ξ)
α sinh λ(h − y) + γλ cosh λ(h − y)
α sinh(λh) + γλ cosh(λh)
+ g(ξ)
sinh(λy)
α sinh(λh) + γλ cosh(λh)
, .
(2.12)
In order to determine the unknown function u(ξ), we use the condition
(2.3). Sastifying the mixed conditions (2.3), we obtain the dual integral

equation with respec to u(ξ) = F [u](ξ) :

F
−1

L(ξ)u(ξ)

(x) =

f(x), x ∈ Ω,
F
−1
[u(ξ)](x) = 0, x ∈ Ω

,
(2.13)
where
L(ξ) =
λ(ξ)[α + γλ(ξ) tanh(λh)]
α tanh(λh) + γλ(ξ)
, (2.14)

f(x) = f(x) + F
−1

G(ξ)g(ξ)

, (2.15)
G(ξ) =
λ(ξ)

α sinh(λh) + γλ(ξ) cosh(λh)
. (2.16)
SOLVING A DUAL INTEGRAL EQUATION
15
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Denote
µ = Reλ =
1
√
2


ξ
4
+ β
2
ξ
2
+ ξ
2
, (2.17)
ν = Imλ =
1
√
2
sgn(βξ)



ξ
4
+ β
2
ξ
2
− ξ
2
, (2.18)
D = α
2
(cosh 2µh − cos 2νh) + γ
2
(µ
2
+ ν
2
)(cosh 2µh + cos 2νh)
+ 2γα(µ sinh 2µh + ν sin 2νh). (2.19)
We have
L(ξ) =
α
2
(µ sinh 2µh + ν sin 2νh) + γ
2
(µ
2
+ ν
2
)(µ sinh 2µh − ν sin 2νh)

D
+
2αγ(µ
2
cosh 2µh + ν
2
cos 2νh)
D
+ i

α
2
(µ sinh 2µh + ν sin 2νh)
D
+
γ
2
(µ
2
+ ν
2
)(ν sinh 2µh + µ sin 2νh) + 2αγµν(cosh 2µh − cos 2νh)
D

.
(2.20)
3. Solvability of the dual integral equations
Our aim in this Section is to establish the solvability of the dual series
equation (??) in some appropriate Sobolev spaces.
3.1. Sobolev spaces. Let H

s
:= H
s
(R)(s ∈ R) be the Sobolev-Slobodeskii
space defined as a closure of the set C
∞
o
(R) of infinitely differentiable func-
tions with a compact support with respect to the norm [5,9]
||u||
s
:=


∞
−∞
(1 + |ξ|)
2s
|ˆu(ξ)|
2
dξ

1/2
< ∞, ˆu = F [u]. (3.1)
The space H
s
is Hilbert with the following scalar product
(u, v)
s
:=


∞
−∞
(1 + |ξ|)
2s
u(ξ)v(ξ)dξ. (3.2)
Let Ω be an interval in R. The subspace of H
s
(R) consisting of functions
u(x) with supp u ⊂ Ω is denoted by H
s
o
(Ω) [5] while the space of functions
v(x) = pu(x), where u ∈ H
s
(R) and p is the restriction operator to Ω is
denoted by H
s
(Ω). The norm in H
s
(Ω) is defined by
||v||
H
s
(Ω)
= inf
l
||lv||
s
,

where the infimum is taken over all possible extensions lv ∈ H
s
(R).
Theorem 3.1. Let Ω ⊂ R, u ∈ H
s
(Ω), f ∈ H
−s
(Ω) and let lf be an
extension of f from Ω to R belonging H
−s
(R). Then the integral
[f, u] := (lf, u)
o
:=

∞
−∞

lf (ξ)

u(ξ)dξ (3.3)
dos not depend on the choice of the extension lf. Therefore, this formula
defines a linear continuous functional on H
s
o
(Ω). Conversely, for every linear
NGUYEN VAN NGOC AND MA DINH TREN
16
.\WXF{QJWUuQKNKRDKeF3K?Q,
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7UuQJ9LKeF7KQJ/RQJ
continuous functional Φ(u) on H
s
o
(Ω) there exists an element f ∈ H
−s
(Ω)
such that Φ(u) = [f, u] and ||Φ|| = ||f ||
H
−s
(Ω)
.
Lemma 3.1. Let Ω be a bounded subset of intervals in R. Then the imbed-
ding H
s
(Ω) into H
s−ε
(Ω) is compact if and only if ε > 0.
Proof. See [10]. 
3.2. Pseudo-differential operators. Consider pseudodifferential opera-
tors of the form [5]
(Au)(x) := F
−1
[A(ξ)u(ξ)](x), (3.4)
where u = F [u], u ∈ H
s
(R). The function A(ξ) is called the symbol of the
pseudodifferential operator Au.
Definition 3.1. Let α ∈ R. We say that the function a(ξ) belongs to the
class σ

α
(R), if
|A(ξ)|  C
1
(1 + |ξ|)
α
, ∀ξ ∈ R, (3.5)
and belongs to the class σ
α
+
(R), if
C
2
(1 + |ξ|)
α
 A(ξ)  C
1
(1 + |ξ|)
α
, ∀ξ ∈ R, (3.6)
where C
1
and C
2
are some positive constants.
We shall need the following
Lemma 3.2. Assume that A(ξ) ∈ σ
α
(R), u(ξ) ∈ H
s

(R), A(ξ)u(ξ) ∈ S

(R).
Then (Au)(x) ∈ H
s−α
(R), where (Au)(x) is defined by the equation (3.4).
For the symbols of the dual integral equations (2.13) the following affir-
mations hold
Lemma 3.3. Let L(ξ) be determined by the formula (2.14). Then
i) L(ξ) = O

|ξ| +
i
2
sign(βξ)

, |ξ| → ∞, (3.7)
ii) L(ξ) ∈ σ
1
(R) ∩ C
∞
(R), (3.8)
iii) ReL(ξ) > 0, ∀ξ ∈ R, ξ = 0, αγ ≥ 0 (α
2
+ γ
2
> 0), (3.9)
iv) G(ξ) ∈ σ
−p
(p >> 1). (3.10)

By virtue of (3.8) and (3.10), using Lemma 3.2 we have.
Theorem 3.2. Suppose that
g(x) ∈ H
1
2
(R) and u(x) := Φ(x, 0) ∈ H
1
2
(R). (3.11)
Then we get
F
−1
[L(ξ)u(ξ)](x) ∈ H
−
1
2
(R), F
−1
[G(ξ)g(ξ)](x) ∈ H
t
(R), ∀t > 0. (3.12)
Due to Theorem 3.2, besides the assumptions in (3.11), we suppose that
f(x) ∈ H
−1/2
(Ω). (3.13)
Then in virtue of (2.15) and (3.11) we have

f(x) ∈ H
−1/2
(Ω).

SOLVING A DUAL INTEGRAL EQUATION
17
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3.3. Solvability of dual integral equations (2.13). Denote
(Lu)(x) = pF
−1
[L(ξ)u(ξ)](x), x ∈ Ω (3.14)
and rewite the (2.13) in the form

(Lu)(x) =

f(x), x ∈ Ω.,
u ∈ H
1/2
◦
(Ω).
(3.15)
Our purpose now is to establish the uniqueness and existen theorems of
solution for the pseudodifferential operator (3.15) in the space H
1/2
o
(Ω).
In virtue of Lemma 3.2 and 3.8, we have
L : H
1/2
o
(Ω) → H
−1/2

(Ω). (3.16)
Then we suppose the following assumption
f(x) ∈ H
−1/2
(Ω). (3.17)
Theorem 3.3. (Uniqueness). Let g(x) and f (x) satisfy the conditions
(3.11) and (3.17), respectively, and αγ ≥ 0(α
2
+ γ
2
> 0). Then equation
(3.15) has at most one solution with respect to u = F
−1
[ˆu] ∈ H
1/2
o
(Ω).
We introduce notations
L
1
(ξ) = Reλ(ξ) + 1, (3.18)
L
o
(ξ) = iImλ(ξ) −1 + L(ξ) −λ(ξ). (3.19)
It is not difficult show that
L(ξ) − λ(ξ) = λ(ξ)
[α − γλ][1 − tanh(λh)]
α tanh(λh) + γλ
∈ σ
−p

(∀p > 1),
L
1
(ξ) ∈ σ
1
+
(R), L
o
(ξ) ∈ σ
0
(R)
and
L(ξ) = L
1
(ξ) + L
o
(ξ). (3.20)
(3.21)
Lemma 3.4. Then the scalar and product in H
α/2
(R) can be defined by the
formulas
(u, v)
L
1
,1/2
=

∞
−∞

L
1
(ξ)F [u](ξ)F [v](ξ)dξ, , (3.22)
||u||
L
1
,1/2
=


∞
−∞
L
1
(ξ)|F [u](ξ)|
2
dξ

1/2
. (3.23)
Proof. The proof is based on the relations (3.1), (3.2) and (3.6). 
Theorem 3.4. (Existepnce). Let f(x) belong to H
−1/2
(Ω), g(x) ∈ H
1/2
(R),
and αγ ≥ 0 (α
2
+ γ
2

> 0). Then the dual equation (3.15) has a unique
solution u = F
−1
[ˆu] ∈ H
1/2
o
(Ω).
NGUYEN VAN NGOC AND MA DINH TREN
18
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4. Reduction to a hypersingular integral equation
The aim of this Section is to propose a method for reducing the dual
integral equation (2.13) to an hypersingular integral equation.
Definition 4.1. The Holder space C
m,α
(c, d), where (c, d) is an interval
on R, and m ≥ 0 an integer, consists of those functions on (c, d) having
continuous derivatives up to order m are Holder continuous with exponent
α, where 0 < α  1.
The Cauchy principle value (CPV) of integrals are defined following [??,
??]
Definition 4.2. If φ(x) ∈ C
0,α
(c, d), then
−

d
c

dt
t − x
= lim
ε→+0


x−ε
c
dt
t − x
+

d
x+ε
dt
t − x

= log
d − x
x − c
, c < x < d,
(4.1)
−

d
c
φ(t)
t − x
dt = lim
ε→+0



|t−x|≥ε
φ(t) − φ(x)
t − x
dt + φ(x)

|t−x|≥ε
dt
t − x

(4.2)
=

d
c
φ(t) − φ(x)
t − x
dt + φ(x) log
d − x
x − c
, c < x < d (4.3)
Finite part (HFP) integral is defined by [4]:
Definition 4.3. If φ(x) ∈ C
1,α
(c, d), then
=

d
c

φ(t)
(t − x)
2
dt (c < x < d) := (4.4)
lim
ε→+0


x−ε
c
φ(t)
(t − x)
2
dt +

d
x+ε
φ(t)
(t − x)
2
dt −
2φ(x)
ε

(4.5)
= lim
ε→+0

φ(c)
c − x

−
φ(d)
d − x
+

x−ε
c
φ

(t)
t − x
dt +

d
x+ε
φ

(t)
t − x
dt (4.6)
+
φ(x − ε) + φ(x + ε)
ε
−
2φ(x)
ε

(4.7)
=
φ(c)

c − x
−
φ(d)
d − x
+ φ

(x) log
d − x
x − c
+

d
c
φ

(t) − φ

(x)
t − x
dt. (4.8)
Theorem 4.1. [9]. Let S

and E

be the spaces of generalized functions
of slow growth and of generalized functions with compact support in R
n
,
respectively. Let f, ∈ S


and g ∈ E

. Then their convolution f ∗ g ∈ S

, and
its Fourier transform can be calculated from the formular
F [f ∗ g](ξ) = F [f](ξ)F [g](ξ) =

f(ξ)g(ξ). (4.9)
From the (4.9) we have
F
−1
[

fg](x) = (f ∗ g)(x). (4.10)
SOLVING A DUAL INTEGRAL EQUATION
19
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From the (4.9) we have
F
−1
[

fg](x) = (f ∗ g)(x). (4.11)
Represent the symbol L(ξ) of the dual integral equation (2.13) in the form
L(ξ) = L
∞
(ξ) + K(ξ), (4.12)

where
L
∞
(ξ) = |ξ|+
iβ
2
|ξ|
ξ
, (4.13)
K(ξ) = L(ξ) − L
∞
(ξ). (4.14)
We have
F
−1
[L
∞
(ξ)](x) =
1
2π

−
2
x
2
+
β
x

, . (4.15)

Using (4.11)-(4.15) we can prove the following Theorem.
Theorem 4.2. The dual integral equation (2.13) with respct to u(ξ) is equiv-
alent to the following hypersingular integral equation
−
1
π
=

a
−a
u(t)
(x − t)
2
dt +
β
2π
−

a
−a
u(t)
x − t
dt +
1
2π

a
−a
k(x −t)u(t)dt = f(x), (4.16)
where

x ∈ (−a, a), u(x) = F
−1
[u](x) = Φ(x, +0), k(x) = F
−1
[K(ξ)], (4.17)
References
[1] Y S. Chan, A. C., Fannjiang,G.H. Paulino, Integral Equations with Hypresingular
Kernel- Theory and Applications to Fracture Mechanics, it INt J. of Eng. Sci., 41,
(2003) 683-720.
[2] Y S. Chan, G.H. Paulino, A. C., Fannjiang, The crack problem for nonhomogeneous
materials under antiplane shear loading-A displacement based formulation, Int. J.
Solids Struct, 38 (2001) 2989-3005.
[3] L.K. Lifanov, I. N. Poltavskii and G. N. Vaniko, HYpersingular Integral Equations and
Their Applications, CRC, 2004.
[4] P. A. Martin, Exact solution of a simple hypersingular integral equation, Fournal of
Integral Equations and Applications bf 4(1992) 197-204.
[5] G. I. Eskin, Boundary Value Problems for Elliptic Pseudodifferential Equations, Nauka,
Moscow, 1973 (in Russian).
[6] Mandal B.N., Advances in dual integral equations, Chapman & Hall / CRC Press,
Boca Raton, 1999, 226 p.
[7] Nguyen Van Ngoc, On the solvability of dual integral equations involving Fourier
Transforms, Acta Math. Vietnamica, 13(2)(1988), 21-30.
[8] Ia. S. Ufliand, Method of Dual Equations in Problems of Mathematical Physics,
Leningrad, Nauka, 1977 (in Russian).
[9] V. S. Vladimirov , Generalized Functions in Mathematical Physics, Moscow, Mir, 1979
(in Russian).
[10] L. R. Volevich and B. P. Panekh , Some spaces of generalized functions and imbedding
theorems, Uspekhii Math. Nauk, 20 (1) (1965), 3-74(in Russian).
NGUYEN VAN NGOC AND MA DINH TREN
20

.\WXF{QJWUuQKNKRDKeF3K?Q,
7UuQJ9LKeF7KQJ/RQJ
7UuQJ9LKeF7KQJ/RQJ
Nguyen Van Ngoc
Hanoi Institute of Mathematics
18 Hoang Quoc Viet Road
P.O. Box 10307, BoHo, Hanoi, Vietnam
E-mail address:
SOLVING A DUAL INTEGRAL EQUATION
21
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