Page 151 - IJOCTA-15-2

A. Yoku¸s, H. Durur, M.H. Ekici / IJOCTA, Vol.15, No.2, pp.343-353 (2025)
is directly related to the wave speed. The x term where m is the integral constant. Using the bal-
represents the propagation of the wave vibration, ancing principle between the highest-order linear
the (-) sign indicates that the wave is propagating term and the strongest nonlinear term in equa-
to the right, and the (+) sign indicates that the tion (9), n = 4 is obtained. Considering equation
wave is propagating to the left. 47 We can trans- (6), the solution of equation (9) may be written
form this to the following ODE for U (ξ) : as follows.
′
′′
2

S U, U , U , U , ... = 0. (5) U (ξ) = α 0 + α 1 Q (ξ) + α 2 (Q (ξ)) 2
(10)
Step 2. For the sake of methodology, we as- 3 4
+ α 3 (Q (ξ)) + α 4 (Q (ξ)) ,
sume that the formal solution of equation (5) is
as follows
n the necessary derivatives of the solution presented
X q
U (ξ) = α 0 + α q Q(ξ) , (6) in equation (10) are calculated and substituted
in equation (9), which is a reflection of equation
q=1
(2). After some mathematical operations the fol-
here α q , q = {1, ..., n} are real or complex con-
lowing system can be constructed.
stants to be calculated by mathematical opera-
tions, such that at least one of these constants is
different from zero. Furthermore, Q (ξ) , the ker-
2
2 2
2
nel function of equation (6), is the solution of the Const : m + v α 0 − k κα 0 − k α = 0,
0
following basis equation. Q [ξ] : v α 1 − k κα 1 − k ln [a] α 1 − k ϵ ln [a] 4
4
6
2
2
2
′
2

Q (ξ) = Q (ξ) − Q (ξ) ln τ, (7) 2
α 1 − 2k α 0 α 1 = 0,
the analytical solution of the equation presented
2
2
4
4
2 2
6
by equation (7) is as follows. Q[ξ] : 3k ln [a] α 1 + 15k ϵ ln [a] α 1 − k α 1
2
2
4
6
2
1 + v α 2 − k κα 2 − 4k ln [a] α 2 − 16k ϵ
Q (ξ) = , (8)
4
2
1 ± τ ξ ln [a] α 2 − 2k α 0 α 2 = 0,
where τ > 0, τ ̸= 1 is a real number. 3 4 2 6 4
Q[ξ] : −2k ln [a] α 1 − 50k ϵ ln [a] α 1
Step 3. The computation of the positive in-
4
2
6
4
teger n in equation (6) is based on the homoge- + 10k ln [a] α 2 + 130k ϵ ln [a] α 2
neous balance between the nonlinear terms and 2 2 2
the highest-order derivatives in equation (5). 44 − 2k α 1 α 2 + v α 3 − k κα 3
4
2
6
4
Step 4. By transposing equation (6) − 9k ln [a] α 3 − 81k ϵ ln [a] α 3
into equation (5) with equation (7), we cal- − 2k α 0 α 3 = 0,
2
′
′′
culate all necessary U , U , ... derivatives of
2
4
4
4
6
U (ξ). Consequently, we attain a polynomial Q[ξ] : 60k ϵ ln [a] α 1 − 6k ln [a] α 2
j
of Q (ξ) , (j = 0, 1, 2, ...). In this polynomial − 330k ϵ ln [a] α 2 − k α
4
2 2
6
produced, if we add up all the terms with the 4 2 6 2 4
j
same powers Q (ξ) and set them equal to zero, + 21k ln [a] α 3 + 525k ϵ ln [a] α 3
2
2
2
we obtain a system of algebraic equations that − 2k α 1 α 3 + v α 4 − k κα 4
can be solved with a computer package program 4 2 6 4
to produce the unknown parameters k, v and − 16k ln [a] α 4 − 256k ϵ ln [a] α 4
2
α q , q = {1, ..., n}. Thus, we attain exact solu- − 2k α 0 α 4 = 0,
tions of the equation (2). 5 6 4 6 4
Q[ξ] : −24k ϵ ln [a] α 1 + 336k ϵ ln [a] α 2
3. Application of Kudryashov method − 12k ln [a] α 3 − 1164k ϵ ln [a] α 3
4
2
6
4
2
2
4
In this part, the Kudryashov method will be used − 2k α 2 α 3 + 36k ln [a] α 4
as a tool for the analytical solution of equation (2) 6 4 2
+ 1476k ϵ ln [a] α 4 − 2k α 1 α 4 = 0,
to better understand it and to lead to important
6 6 4 6 4
discussions in the field of fluid mechanics. equa- Q[ξ] : −120k ϵ ln [a] α 2 + 1080k ϵ ln [a] α 3
tion (4), presented as the wave transformation, 2 2 4 2
− k α − 20k ln [a] α 4
3
which plays a significant role in elucidating the
4
2
6
characteristics of nonlinear wave propagation, is − 3020k ϵ ln [a] α 4 − 2k α 2 α 4 = 0,
applied to equation (2) and integrated twice, the Q[ξ] : 6 4 6 4
7
following equation can be formed. −360k ϵ ln [a] α 3 + 2640k ϵ ln [a] α 4
2
− 2k α 3 α 4 = 0,
2
4
′′
2
2
2
2
4
2 2
6
8
m+v U −k κU −k U −k U −k U (4) = 0, (9) Q[ξ] : −840k ϵ ln [a] α 4 −k α = 0. (11)
4
346