Page 152 - IJOCTA-15-2

Effect of diffusion parameters on traveling wave solutions of singular perturbed boussinesq equation
α 0 , α 1 , α 2 , α 3 , α 4 and m, v, k, `o, κ constants
are attained from equation (11) the system uti-  6 3a 4tv + 3a 4kx − 122a 2(tv+kx) + 12a 3tv+kx + 12a tv+3kx k ln [a] 2 
2
 − 4 
lizing a software program. u 2 (x, t) = 1  (a tv + a kx ) 
 .

 q 
−169m + 324k 6 ln [a] 4 
13 
−
k
(15)
Case 1: If Case 3: If
2
2
2
2
α 1 = 6k ln [a] , α 2 = −6k ln [a] ,
α 1 = 0,
4
2
2 2
α 3 = 0, α 4 = 0, m = −k ln [a] α 0 − k α ,
0
q
840 2 2 2 2
α 2 = k ln [a] , v = k κ + k ln [a] + 2α 0 , ` o = 0,
13 (16)
for equation (1), we derive traveling wave soliton
1680 2 2
α 3 = − k ln [a] , by substituting values equation (16) into equation
13
(10).
840 2 √ 2
2
2
α 4 = k ln [a] , k x+t κ+k ln [a] +2α 0 2 2
13 u 3 (x, t) = 6a k ln [a] + α 0 .
√ 2 2 2
a kx + a kt κ+k ln [a] +2α 0
1 4 2 2 2
m = −36k ln [a] α 0 − 13k α 0 , (17)
13
Case 4: If
q √
2
2
k 13κ + 36k ln [a] + 26α 0 260v − 260k κ − 291k ln [a] − 3i 31k ln [a] 2
2
4
2
2
4
v = − √ , α 0 = ,
13 520k 2
21 2 2 √ 2 2
α 1 = 17k ln [a] + i 31k ln [a] ,
1 13
´
U = − , (18)
2
13k ln [a] 2
21 √
(12) α 2 = − i −79ik ln [a] + 7 31k ln [a] 2 ,
2
2
2
13
84 √
2
2
2
α 3 = 31k ln [a] + 3i 31k ln [a] 2 ,
for equation (1), we derive traveling wave soliton 13
42 √
2
by substituting values equation (12) into equation α 4 = − i −31ik ln [a] + 3 31k ln [a] 2 ,
2
2
13 √
(10). −33800v + 67600k v κ − 33800k κ + 42201k ln [a] + 873i 31k ln [a] 4
4
8
4
4 2
2 2
8
m = 2 ,
√ 135200k
31 + 3i 31
` o = ,
2
u 1 (x, t) = 260k ln [a] 2
4
q q (19)
36 2 36 2
2kx+2kt κ+ 13 k 2 ln [a] +2α 0 2 2 kx+kt κ+ 13 k 2 ln [a] +2α 0
840a k ln [a] + 13 1 + a α 0
. for equation (1), we derive traveling wave soliton
4
q
36 2
kx+kt κ+ 13 k 2 ln [a] +2α 0
13 1 + a by substituting values equation (19) into equation
(13) (10).
Case 2: If 2
1 v
u 4 (x, t) = − κ
q
 4  2 k 2
6
1 −169m + 324k ln [a]
2
2
α 0 =  −18k ln [a] −  , "
13 k 1 √
+ 3 (−97 − i 31)a 4tv
840 1680 tv kx 4
2
2
2
2
α 1 = 0, α 2 = k ln [a] , α 3 = − k ln [a] , 520(a + a )
13 13 √
q 4kx
 4  + (−97 − i 31)a
6
840 1 13v 2 2 −169m + 324k ln [a]
2
2
α 4 = k ln [a] , κ =  +  ,
13 13 k 2 k
√
1 2(tv+kx)
` o = − , + (−8422 − 1126i 31)a
13k ln [a] 2 √
2
(14) + 4(1093 + 69i 31)a 3tv+kx
#
for equation (1), we derive traveling wave soliton √ 2
2
+ 4(1093 + 69i 31)a tv+3kx k ln [a] .
by substituting values equation (14) into equation
(10).
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