$x^{*}$

Concentration $S^{*}$

when $\varphi =1$

when $\varphi =10$

when $\varphi =100$

Simulation

(HPM)Equation (13)

hyperbolic function method

Equation (22)

% of error deviation (HPM) Equation (13)

% of error deviation hyperbolic function method Equation (22)

Simulation

(HPM) Equation (13)

hyperbolic function method Equation (22)

% of error deviation (HPM) Equation (13)

% of error deviation hyperbolic function method Equation (22)

Simulation

Equation (13)

hyperbolic function method Equation (22)

% of Error deviation (HPM) Equation (13)

% of error deviation hyperbolic function method Equation (22)

0

1.0000

1.0000

1.0000

0.00

0.00

1.0000

1.0000

1.0000

0.00

0.00

1.0000

1.0000

1.0000

0.00

0.00

0.2

0.8998

0.9425

0.9236

4.53

2.64

0.6117

0.6188

0.6498

1.14

5.86

0.1700

0.1743

0.1810

2.46

6.07

0.4

0.8486

0.8924

0.8657

4.90

2.01

0.3535

0.3598

0.3701

1.75

4.48

0.0235

0.0243

0.0249

3.40

5.62

0.6

0.8082

0.8543

0.8252

5.39

2.06

0.2069

0.2107

0.2196

1.83

5.78

0.0032

0.0033

0.0034

3.12

5.88

0.8

0.7898

0.8306

0.8012

5.15

1.42

0.1338

0.1367

0.1389

2.16

3.67

0.0004

0.0004

0.0004

0.00

0.00

1

0.7768

0.8226

0.7932

5.56

2.06

0.1121

0.1147

0.1189

2.26

5.71

0.0001

0.0001

0.0001

0.00

0.00

Average error %

4.25

1.69

Average error %

1.52

4.25

Average error %

1.49

2.92