Gen. Math. Notes, Vol. 28, No. 1, May 2015, pp. 59-71
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2015
www.i-csrs.org
Available free online at http://www.geman.in
Dependency of the Solution of a Class of Quartic
Partial Differential Quasilinear Equation With
Periodic Boundary Condition on ε
Hüseyin Halilov1 , Bahadır Özgür Güler2 and Kadir Kutlu3
1
Recep Tayyip Erdogan University
Faculty of Arts and Science, Department of Mathematics
E-mail: huseyin.halilov@erdogan.edu.tr
2
Karadeniz Technical University
Faculty of Science, Department of Mathematics
E-mail: boguler@ktu.edu.tr
3
Recep Tayyip Erdogan University
Faculty of Arts and Science, Department of Mathematics
E-mail: kadir.kutlu@erdogan.edu.tr
(Received: 14-1-15 / Accepted: 27-4-15)
Abstract
In this paper, we study the multidimensional mixed problem with periodic
4u
2
boundary condition for quasilinear Euler-Bernoulli equation ∂∂t2u − εb2 ∂t∂2 ∂x
2 +
∂2u
2 ∂4u
2 ∂4v
a ∂x4 = f (t, x, u). We also consider the mixed problem ∂t2 +a ∂x4 = f (t, x, v).
Finally we prove that the solution function u(t, x, ε) of the quasilinear EulerBernoulli equation is convergent to the solution function v(t, x) of the quasilinear quartic equation, as ε → 0.
Keywords: Mixed problem, Periodic boundary condition, Quasi-linear
quartic partial differential equation.
1
Introduction
In recent years, mathematical modeling of sound wave distribution problems
and also the vibration, buckling and dynamic behavior of various building
60
Hüseyin Halilov et al.
elements widely used in nano-technology are formulated with following EulerBernoulli equations
4
4
∂ 2u
2 ∂ u
2∂ u
−
εb
+
a
= f (t, x, u)
(1)
∂t2
∂t2 ∂x2
∂x4
Due to the new and exceptionally its electronic and mechanical properties,
carbon nanotubes are considered to be one of the most useful material in future. Nowadays, nanotubes are used as atomic force microscopy, nanofillers for
composite materials, nanoscale electronic devices and even frictionless nanoactuators, nanomotors, nanobearings and nanosprings [4,9,13,16,17]. These elements are tackled by different boundary conditions depending on different
loading conditions. Therefore, investigation of existence and uniqueness of the
solution of Euler-Bernoulli equations with different boundary conditions used
in the mathematical modeling of the structural components of nano-materials
continues to be a focus of interest amongst mathematicians.The reader is refereed to [2,6,7,8] for some relevant previous work on linear and quasi-linear
equations. The textbooks [3,11,12,14,15] also contain important results.
In mathematics, the classical statement of Euler-Bernoulli equation
4
∂ 2v
2∂ u
+a
=0
∂t2
∂x4
(2)
which is taken ε = 0 from (1) is used for beam vibration equation. As well as
the homogeneous equation, quasilinear and non-linear equations can be handled in this case. Various problems for equations of this type were investigated
and many results have been obtained in different ways.
We first discuss the following mixed problem for the Euler-Bernoulli equation with the nonlinear source term f = f (t, x, u):
4
4
∂ 2u
2 ∂ u
2∂ u
− εb 2 2 + a
= f (t, x, u), (t, x) ∈ D{0 < t < T, 0 < x < π} (3)
∂t2
∂t ∂x
∂x4
u(0, x, ε) = ϕ(x), ut (0, x, ε) = ψ(x), (0 ≤ x ≤ π)
(4)
u(t, 0, ε) = u(t, π, ε), ux (t, 0, ε) = ux (t, π, ε),
ux2 (t, 0, ε) = ux2 (t, π, ε), ux3 (t, 0, ε) = ux3 (t, π, ε), (0 ≤ t ≤ T )
(5)
and also we consider the following mixed problem for a quasilinear quartic
equation with the nonlinear source term f = f (t, x, v):
4
∂ 2v
2∂ v
+
a
= f (t, x, v), (t, x) ∈ D{0 < t < T, 0 < x < π}
∂t2
∂x4
(6)
v(0, x) = ϕ(x), vt (0, x) = ψ(x), (0 ≤ x ≤ π)
(7)
Dependency of the Solution of a Class of Quartic...
v(t, 0) = v(t, π), vx (t, 0) = vx (t, π),
vx2 (t, 0) = vx2 (t, π), vx3 (t, 0) = vx3 (t, π), (0 ≤ t ≤ T )
61
(8)
where ϕ(x), ψ(x) and f (t, x, u) are given functions defined on [0, π] and D̄{0 ≤
t ≤ T, 0 ≤ x ≤ π} × (−∞, ∞), respectively, and the functions u(t, x, ε) and
v(t, x) are solutions of the problems(3)-(5) and (6)-(8).
In [7], the authors have studied the solution of the problem (3)-(5) for
ε > 0. The same authors have considered the solution of the problem (6)(8) for ε = 0 in [8]. In the present paper, we discuss the solution of the
problem (3)-(5) and (6)-(8) and show that the solution function of u(t, x, ε) of
the quasilinear Euler-Bernoulli equation is convergent to the solution function
v(t, x) of the quasilinear quartic equation, as ε → 0 (i.e., limε→0 u(t, x, ε) =
v(t, x), (t ∈ [0, t])).
2
Preliminaries
Definition 2.1 [1, 10] The function w(t, x) ∈ C(D̄) is called test function
if it has continuous partial derivatives of order contained in equation (1) and
satisfies both following conditions
w(T, x) = wt (T, x) = wx2 (T, x) = wtx2 (T, x) = 0
and the boundary condition;
w(t, 0) = w(t, π), wx (t, 0) = wx (t, π), wx2 (t, 0) = wx2 (t, π), wx3 (t, 0) = wx3 (t, π)
Definition 2.2 The function u(t, x, ε) ∈ C(D̄ × [0, ε0 ]) satisfying the integral identity
Z T Z πn 2
o
4
4
∂ w
2 ∂ w
2∂ w
− εb 2 2 + a
u
− f (t, x, u)w dx dt+
∂t2
∂t ∂x
∂x4
0
0
Z π
Z π
4
4
2 ∂ w
2 ∂ w
ϕ(x) vt (0, x) − εb 2 2 dx −
ψ(x) v(0, x) − εb 2 2 dx = 0
∂t ∂x
∂t ∂x
0
0
(9)
for an arbitrary test function w(t, x) is called weak generalized solution of
problem (3)-(5).
Definition 2.3 The function v(t, x) ∈ C(D̄) satisfying the integral identity
Z T Z πn 2
o
4
∂ w
2∂ w
v
+
a
−
f
(t,
x,
u)w
dx dt+
∂t2
∂x4
0
0
(10)
Z π
Z π
ϕ(x)vt (0, x) dx −
ψ(x)v(0, x) dx = 0
0
0
for an arbitrary test function w(t, x) is called weak generalized solution of
problem (6)-(8).
62
3
Hüseyin Halilov et al.
Solution of Problem
The set
1
ū(t, ε) =
u0 (t, ε), uc1 (t, ε), us1 (t, ε), . . . , uck (t, ε), usk (t, ε), . . .
2
of functions satisfying the condition
∞ X
1
max |u0 (t, ε)| +
max |uck (t, ε)| + max |usk (t)| < ∞.
t∈[0,T ]
t∈[0,T ]
2 t∈[0,T ]
k=1
for all ε ∈ [0, ε0 ] is denoted by BT . Let
ku(t, ε)kBT
∞ X
1
= max |u0 (t, ε)| +
max |uck (t, ε)| + max |usk (t, ε)|
t∈[0,T ]
t∈[0,T ]
2 t∈[0,T ]
k=1
be the norm in BT . It can be shown that BT is Banach space.
Looking for the weak solutions of (3)-(5) and (6)-(8) of which existence and
uniqueness are proven under certain conditions as following respectively;
∞
X
1
u(t, x, ε) = u0 (t, ε) +
[uck (t, ε) cos 2kx + usk (t, ε) sin 2kx]
2
k=1
(11)
∞
X
1
v(t, x) = u0 (t) +
[uck (t) cos 2kx + usk (t) sin 2kx] ,
2
k=1
(12)
and
we get the following infinite system of integral equations for the unknown
functions u0 (t, ε), uck (t, ε), usk (t, ε) and v0 (t), vck (t), vsk (t), (k = 1, ∞)
Dependency of the Solution of a Class of Quartic...
63
Z Z
2 t π
1
v0 (t) = ϕ0 + ψ0 t +
(t − τ )f τ, ξ, u0 (τ, ε)+
π 0 0
2
∞
X
[ucn (τ, ε) cos 2nξ + usn (τ, ε) sin 2nξ] dξ dτ,
n=1
uck (t, ε) = ϕck cos αk t +
2
παk
Z tZ
0
0
π
ψck
sin αk t+
αk
∞
X
1
[ucn (τ, ε) cos 2nξ + usn (τ, ε) sin 2nξ] ×
f τ, ξ, u0 (τ, ε) +
2
n=1
cos 2kξ sin αk (t − τ ) dξ dτ,
ψsk
usk (t, ε) = ϕsk cos αk t +
sin αk t+
αk
Z tZ π
∞
X
2
1
[ucn (τ, ε) cos 2nξ + usn (τ, ε) sin 2nξ] ×
f τ, ξ, u0 (τ, ε) +
παk 0 0
2
n=1
sin 2kξ sin αk (t − τ ) dξ dτ,
a(2k)2
αk =
, k = 1, ∞. (13)
1 + ε(2k)2
and
Z Z
2 t π
1
v0 (t) = ϕ0 + ψ0 t +
(t − τ )f τ, ξ, v0 (τ )+
π 0 0
2
∞
X
[vcn (τ ) cos 2nξ + vsn (τ ) sin 2nξ] dξ dτ,
n=1
vck (t) = ϕck cos α
fk t +
2
πf
αk
Z tZ
0
0
π
ψck
sin α
fk t+
α
fk
∞
X
1
f τ, ξ, v0 (τ ) +
[vcn (τ ) cos 2nξ + vsn (τ ) sin 2nξ] ×
2
n=1
cos 2kξ sin α
fk (t − τ ) dξ dτ,
ψsk
vsk (t) = ϕsk cos α
fk t +
sin α
fk t+
α
fk
Z tZ π
∞
X
2
1
f τ, ξ, v0 (τ ) +
[vcn (τ ) cos 2nξ + vsn (τ ) sin 2nξ] ×
πf
αk 0 0
2
n=1
sin 2kξ sin α
fk (t − τ ) dξ dτ,
α
fk = a(2k)2 , k = 1, ∞. (14)
64
Hüseyin Halilov et al.
For simplicity, let
∞
X
1
Au(τ, ξ, ε) = u0 (t, ε) +
[ucn (τ, ε) cos 2kx + usn (τ, ε) sin 2nx]
2
k=1
and
∞
X
1
Av(τ, ξ) = u0 (t) +
[ucn (τ ) cos 2kx + usn (τ ) sin 2nx] .
2
k=1
To examine the difference of the systems (13) and (14), we write as following;
2
u0 (t, ε)−v0 (t) =
π
Z tZ
0
π
(t−τ ) f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) dξ dτ,
0
fk t
sin αk t sin α
−
αk
α
ek
sin αk t sin α
fk t
−
αk
α
ek
uck (t, ε) − vck (t) = ϕck (cos αk t − cos α
fk t) + ψck
Z tZ π
2
f {τ, ξ, Au(τ, ξ, ε)} cos 2kξ sin αk (t − τ ) dξ dτ −
παk 0 0
Z tZ π
2
f {τ, ξ, Av(τ, ξ)} cos 2kξ sin α
fk (t − τ ) dξ dτ
πf
αk 0 0
usk (t, ε) − vsk (t) = ϕsk (cos αk t − cos α
fk t) + ψsk
+
+
Z tZ π
2
f {τ, ξ, Au(τ, ξ, ε)} sin 2kξ sin αk (t − τ ) dξ dτ −
παk 0 0
Z tZ π
2
f {τ, ξ, Av(τ, ξ)} sin 2kξ sin α
fk (t − τ ) dξ dτ
πf
αk 0 0
then we have
2
u0 (t, ε)−v0 (t) =
π
Z tZ
0
π
(t−τ ) f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) dξ dτ,
0
fk t
sin αk t sin α
−
αk
α
ek
uck (t, ε) − vck (t) = ϕck (cos αk t − cos α
fk t) + ψck
+
Z Z
1
1 1 t π
−
f {τ, ξ, Au(τ, ξ, ε)} cos 2kξ sin αk (t − τ ) dξ dτ +
αk α
fk π 0 0
Z tZ π
2
f {τ, ξ, Av(τ, ξ)} cos 2kξ[sin αk (t − τ ) − sinf
αk (t − τ )] dξ dτ +
πf
αk 0 0
65
Dependency of the Solution of a Class of Quartic...
2 f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) cos 2kξ sin αk (t − τ ) dξ dτ,
πf
αk
usk (t, ε) − vsk (t) = ϕck (cos αk t − cos α
fk t) + ψck
sin αk t sin α
fk t
−
αk
α
ek
+
Z Z
1
1 1 t π
f {τ, ξ, Au(τ, ξ, ε)} sin 2kξ sin αk (t − τ ) dξ dτ +
−
αk α
fk π 0 0
Z tZ π
2
f {τ, ξ, Av(τ, ξ)} sin 2kξ[sin αk (t − τ ) − sinf
αk (t − τ )] dξ dτ +
πf
αk 0 0
Z tZ π
2
f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) sin 2kξ sin α
fk (t−τ ) dξ dτ.
πf
αk 0 0
Take the absolute values of differences and after making the necessary grouping, we form a sum as following;
1
|u(t, ε) − v(t)| = |u0 (t, ε) − v 0 (t)|+
2
∞
X
[|uck (t, ε) − vck (t)| + |usk (t, ε) − vsk (t)|] ≤
T
π
k=1
π
Z tZ
0
|f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) | dξ dτ +
0
∞
∞
X
X
(|ψck | + |ψsk |) sinααk k t −
(|ϕck | + |ϕsk |)| cos αk t − cos α
fk t| +
k=1
k=1
∞
X
sin α
fk t α
ek +
Z Z
t 2 π 1
1 − αfk
f τ, ξ, Au(τ, ξ, ε) cos 2kξdξ| +
|
αk
π
0
0
k=1
Z
2 π |
f τ, ξ, Au(τ, ξ, ε) sin 2kξdξ| dτ +
π 0
Z t Z π
∞
X
2
1
f τ, ξ, Au(τ, ξ, ε) cos 2kξdξ| +
|
α
fk 0
π 0
k=1
Z π
2
|
f τ, ξ, Au(τ, ξ, ε) sin 2kξdξ| | sin αk (t − τ ) − sinf
αk (t − τ )|dτ +
π 0
Z t Z π
∞
X
1
2
|
f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) cos 2kξdξ| +
α
fk 0
π 0
k=1
Z π
2
|
f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) sin 2kξdξ| dτ. (15)
π 0
66
Hüseyin Halilov et al.
1 kt 1
−
Each of the statements | cos αk t−cos α
fk t|, sinααk k t − sinαeαf
, | sin αk (t−
,
α
α
f
k
k
k
τ ) − sinf
αk (t − τ )| of the right side of in the equality (15) are bounded for k, τ
and t (0 ≤ τ ≤ t ≤ T ), as ε → 0. Let us denote these statements by
δ1 (ε), δ2 (ε), δ3 (ε), δ4 (ε), respectively, and then write last inequality as following;
T
|u(t, ε) − v(t)| ≤
π
Z tZ
0
δ1 (ε)
π
|f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) |+
0
∞
X
k=1
(|ϕck | + |ϕsk |) + δ2 (ε)
∞
X
(|ψck | + |ψsk |)+
k=1
Z
∞ Z t
X
2 π f τ, ξ, Au(τ, ξ, ε) cos 2kξdξ| +
δ3 (ε)
|
π 0
k=1 0
Z
2 π |
f τ, ξ, Au(τ, ξ, ε) sin 2kξdξ| dτ +
π 0
Z t Z π
∞
X
1
2
δ4 (ε)
f τ, ξ, Au(τ, ξ, ε) cos 2kξdξ| +
|
α
fk 0
π 0
k=1
Z π
2
|
f τ, ξ, Au(τ, ξ, ε) sin 2kξdξ| dτ +
π 0
Z t Z π
∞
X
2
1
f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) cos 2kξdξ| +
|
α
fk 0
π 0
k=1
Z π
2
|
f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) sin 2kξdξ| dτ.
π 0
(16)
Under the assumptions of the theorem given below, we can take δ(ε) as the
sum of the 2nd, 3rd, 4th, 5th sums in the right side of the inequality (14), then
we have
|u(t, ε) − v(t)| ≤ δ(ε)+
Z
Z
T t 2 π |
f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) dξ|dτ +
2 0 π 0
∞
X 1 Z t 2 Z π |
f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) cos 2kξdξ| +
α
fk 0
π 0
k=1
Z
2 π f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) sin 2kξdξ| dτ.
π 0
Dependency of the Solution of a Class of Quartic...
67
Applying Hölder inequality to third sum in the right side of the inequality
above, we obtain
|u(t, ε) − v(t)| ≤ δ(ε)+
!1/2
Z
Z
∞
X
T t2 π 1
|f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) |dξdτ +
2 0 π 0
α
f2
k=1 k
( ∞ Z Z
t
X
2 π |
f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) cos 2kξdξ +
π 0
0
k=1
2 )1/2
Z
2 π f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) sin 2kξdξ| dτ
. (17)
π 0
Applying Cauchy inequality to the integrals in the right side of (17), then
applying following inequality
(a1 + a2 + . . . + an )2 ≤ n(a21 + a22 + . . . + a2n )
(18)
to third sum for n = 2, we obtain
|u(t, ε) − v(t)| ≤ δ(ε)+
(Z Z
2 ) 21
t
2 π T 3/2
f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) dξ dτ
+
2
π 0
0
!1/2
(∞ Z Z
∞
t
X
√ X
2 π 1
2
T
f
τ,
ξ,
Au(τ,
ξ,
ε)
−
π 0
α
f2
0
k=1
k=1 k
2
f τ, ξ, Av(τ, ξ) cos 2kξdξ dτ +
2 )1/2
Z π
Z t
∞
X
2
T
f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) sin 2kξdξ dτ
π 0
0
k=1
(19)
and then,
|u(t, ε) − v(t)| ≤ δ(ε)+
(Z Z
2 ) 21
t
2 π T 3/2
f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) dξ dτ
+
2
π 0
0
!1/2 (Z " ∞ Z
∞
t X
X
√
1
2 π 2T
T
f τ, ξ, Au(τ, ξ, ε) −
2
π 0
α
f
0
k=1 k
k=1
2
f τ, ξ, Av(τ, ξ) cos 2kξdξ +
68
Hüseyin Halilov et al.
2
π
Z
π
f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) sin 2kξdξ
)1/2
2 #
dτ
. (20)
0
Taking square and using the inequality (18) to the inequality above for n = 3,
we have
|u(t, ε) − v(t)|2 ≤ 3δ(ε)2 +
2
Z Z
3T 3 t 2 π f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) dξ dτ +
4 0 π 0
( Z
2
Z tX
∞
∞
X
2 π 1
6T
f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) cos 2kξdξ +
2
π
α
f
0
0
k
k=1
k=1
2 )
Z π
2
f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) sin 2kξdξ
. (21)
π 0
Supposing
max
∞
X
3T 3
1
, 6T
2
α
f2
k=1 k
!
= µ0 ,
let us combine the integrals of the right side of (21) as follows,
|u(t, ε) − v(t)|2 ≤ 3δ(ε)2 +
2
Z t ( Z π
2
µ0
f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) dξ dτ +
π 0
0
" Z
2
∞
X
2 π f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) cos 2kξdξ +
π 0
k=1
2 #)
Z
2 π f τ, ξ, Au(τ, ξ, ε) − f τ, ξ, Av(τ, ξ) sin 2kξdξ
. (22)
π 0
Applying Bessel inequality to the sum in (22), we obtain
2
|u(t, ε)−v(t)| ≤ 3δ +µ0
π
2
2
Z tZ
0
π
2
f τ, x, Au(τ, x, ε) − f τ, x, Av(τ, x)
dx.
0
Using Lipshitzs inequality, we have
2
|u(t, ε) − v(t)| ≤ 3δ(ε) + µ0
π
2
2
Z tZ
0
0
π
b2 (t, x) [Au(τ, x, ε) − Av(τ, x)]2 dxdτ
69
Dependency of the Solution of a Class of Quartic...
or
2
|u(t, ε) − v(t)| ≤ 3δ(ε) + µ0
π
2
2
Z tZ
0
π
b2 (t, x)|u(τ, ε) − v(τ )|2 dxdτ.
(23)
0
Applying Gronwall inequality to (23), we obtain
Z Z
2 t π 2
2
2
b (t, x)dxdτ,
|u(t, ε) − v(t)| ≤ 3δ(ε) exp µ0
π 0 0
or
√
µ0
|u(t, ε) − v(t)| ≤ 3δ(ε) exp exp
k b(t, x) k2L2 dxdτ.
π
In the last inequality, taking into account δ(ε) → 0 for ε → 0, we see that
lim u(t, ε) = v(t).
ε→0
Hence, the following theorem is proved.
Main Theorem: Suppose the following conditions are satisfied;
a) f (t, x, u) is continuous respect to all arguments on D × (−∞, ∞), and
satisfies the following condition |f (t, x, u) − f (t, x, v)| ≤ b(t, x)|u − v|
where b(t, x) ∈ L2 (D), b(t, x) ≥ 0,
b) [f (t, x, u)]x ∈ L2 (D),
c) f (t, x, 0) ∈ L2 (D),
d) The functions ϕ(x), ψ(x) with ϕ(x) ∈ C 1 [0, π], ψ(x) ∈ C[0, π] satisfy the
following conditions;
ϕ(0) = ϕ(π), ϕ0 (0) = ϕ0 (π), ψ(0) = ψ(π).
In this case, limε→0 u(t, ε) = v(t), i.e. limε→0 u(t, x, ε) = limε→0 v(t, x) is true
References
[1] H.I. Chandrov, On mixed problem for a class of quasilinear hyperbolic
equation, PhD Thesis, Tbilisi, (1970).
[2] İ. Çiftçi and H. Halilov, Dependency of the solution of quasilinear pseudoparabolic equation with periodic boundary condition on , Int. J. Math.
Anal.(Ruse), 2(17-20) (2008), 881-888.
[3] E.A. Conzalez-Valesco, Fourier Analysis and Boundary Value Problems,
Academic Press, New York, (1995).
70
Hüseyin Halilov et al.
[4] I. Elishakoff and S. Candan, Apperently first closed-form solution for vibrating inhomogeneous beam, Int. Jour. of Solids and Structures, 38(19)
(2001), 3411-3441.
[5] H.M. Halilov, Solution of the mixed non-linear problem for a class of
Quasi-linear equation 4. Order, Jour. of Mathematical Physics and Functional Analysis, Alma Ata, (1966), 27-32 (In Russian).
[6] H. Halilov, On the mixed problem for a class of quasilinear pseudoparabolic equations, Applicable Analysis, 75(1-2) (2010), 61-71.
[7] H. Halilov, K. Kutlu and B.Ö. Güler, Solution of a mixed problem with
periodic boundary condition for a Quasi-linear Euler-Bernoulli equation,
Hacettepe Journal of Mathematics and Statistics, 39(3) (2010), 417-428.
[8] H. Halilov, K. Kutlu and B.Ö. Güler, Examination of mixed problem
with periodic boundary condition for a class of quartic partial differantial Quasi-linear equation, International Electronic Journal of Pure and
Applied Mathematics, 1(1) (2010), 47-59.
[9] X.Q. He, S. Kitipornchai and K.M. Liew, Buckling analysis of multi-walled
carbon nanotubes: A continuum model accounting for van der Waals
interaction, Journal of the Mechanics and Physics of Solids, 53(2005),
303-326.
[10] V.A. Il’in, Solvability of mixed problem for hyperbolic and parabolic equations, Uspekhi Math. Nauk., 92(1960), 97-154 (in Russian).
[11] D.A. Ladyzhenskaya, Boundary Value Problem of Mathematical Physics,
Springer, New York, (1985).
[12] R. Lattes and J.L. Lions, Methode de Quasi Reversibility et Applications,
Dunod, Paris, (1967).
[13] T. Natsuki, Q.Q. Ni and M. Endo, Wave propagation in single-and doublewalled carbon nano tubes filled with fluids, Journal of Applied Physics,
101(034319) (2007), 1-5.
[14] S.L. Sobolev, Applications of functional analysis in mathematical physics,
American Mathematical Soc., Providence, R.I., (1963).
[15] K.H. Shabadikov, Issledovanie rashennie smashannikh zadach dlya kvazilineaynikh differentsialnikh urevneniy malim parametrom pri starshey
smessonoi preizvodnoi, PhD Thesis, Fargana, (1984).
Dependency of the Solution of a Class of Quartic...
71
[16] Y. Yana, X.Q. Heb, L.X. Zhanga and C.M. Wang, Dynamic behavior of
triple-walled carbon nano-tubes conveying fluid, Journal of Sound and
Vibration, 319(2010), 1003-1018.
[17] J. Yoon, C.Q. Ru and A. Mioduchowski, Sound wave propagation in multiwall carbon nanotubes, Journal of Applied Physics, 93(2003), 4801-4806.