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S. S. Nourazar et al.
t
2
x−
−
1 1 e 4 4 −e
The Taylor series expansion for +
t
2
2 2 e 4 x − 4 + e−
2
t
x−
4
4
−
1 1e
−e
+
2
t
2 2
x−
−
e 4 4 +e
2
t
x+
4
4
2
t
x+
4
4
e
=
2
e
4
x
2
x
4
+e
−
2
x
−
4
t
2
x+
4
4
t
2
x+
4
4
1
2
e
2
is written as:
1
2
x
4
+e
2
x
4
−
2
2
2x
−
x
e 4 − e 4
1
t2
t−
3
2
8 2x
x
−
4
4
+e
e
(3.8)
2
2x
− 2x
e 4 − 4 +e 4
1
t 3 + .
−
4
2
2
48
x
−
x
e 4 + e 4
Combining Equation (3.8) with Equation (3.7), we get as follow:
2
x−
t
−
1 1 e 4 4 −e
υ ( x, t ) =
+
2 2 2 x− t
−
e 4 4 +e
2
t
x+
4
4
2
t
x+
4
4
1
1 1
t
=
+ tanh
x−
2 2
2
2 2
(3.9)
This is the exact solution of the problem, Equation (3.1). Table 1 shows the trend of rapid convergence of the
results of S1 ( x, t ) = v0 ( x, t ) to S6 ( x, t ) = ∑ i = 0 vi ( x, t ) using the HPM. The rapid convergence of the solution
5
toward the exact solution, the maximum relative error of less than 0.0000058% is achieved as shown in Table 1.
Case ІІ: In Equation (1.1) for α =
−1, n =
1, γ =
1, β =
1 , the Burgers-Huxley equation is written as:
∂u ∂ 2 u
∂u
=
+ u + u (1 − u )( u − 1)
∂t ∂x 2
∂x
(3.10)
Table 1. The percentage of relative errors of the results of S1 ( x, t ) = v0 ( x, t ) to S6 ( x, t ) = ∑ i=0 vi ( x, t ) of the HPM solu5
tion of Equation (3.1).
Percentage of relative error (%RE)
t = 0.1
t = 0.3
t = 0.4
x =1
x=2
x=3
S1 ( x, t )
0.01693168743
0.01002710463
0.005488150424
S3 ( x, t )
0.000002337346256
2.025644856e−7
9.527700575e−7
S5 ( x, t )
2.153484215e−10
3.464089104e−10
3.155246333e−10
S 6 ( x, t )
9.744801271e−12
3.920421861e−11
9.937961084e−11
S1 ( x, t )
0.05344388963
0.03164997413
0.01732302882
S3 ( x, t )
0.00006806597676
0.000003967422423
0.00002580410708
S5 ( x, t )
6.184344787e−8
9.467317545e−8
5.516785201e−8
S 6 ( x, t )
1.008793018e−8
1.166422821e−9
2.026398250e−9
S1 ( x, t )
0.07311570399
0.04329980772
0.02369935005
S3 ( x, t )
0.0001675410515
0.000007498069748
0.00006123279256
S5 ( x, t )
2.799248652e−7
4.009848425e−7
2.364525175e−7
S 6 ( x, t )
5.775483086e−8
6.758498122e−9
1.111128458e−8
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