International Journal of Applied Mathematics & Statistical Sciences (IJAMSS) ISSN(P): 2319-3972; ISSN(E): 2319-3980 Vol. 3, Issue 1, Jan 2014, 35-56 © IASET

SEMI-ANALYTICAL APPROACH TO SOLVE NON-LINEAR DIFFERENTIAL EQUATIONS AND THEIR GRAPHICAL REPRESENTATIONS M. TAHMINA AKTER1, A. S. M. MOINUDDIN2 & M. A. MANSUR CHOWDHURY3 1

Department of Mathematics, Chittagong University of Engineering & Technology, Chittagong, Bangladesh 2

1,2,3

Department of Mathematics, University of Chittagong, Chittagong, Bangladesh

Research Centre for Mathematical & Physical Sciences, University of Chittagong, Chittagong, Bangladesh

ABSTRACT In this paper we have applied a new approximation technique to solve non-linear partial differential equation. The approximation technique is called Homotopy perturbation method (HPM). The main difference between traditional perturbation method and this one is that it can be applied for even higher values of the parameter, where as traditional perturbation method can be applied only for lower values of parameter. It means that when the value of the parameter is less than one only then the traditional perturbation can be applied. In this paper we have considered a highly non-linear partial differential equation and found the approximate solution using HPM for two types of initial conditions. Then we have drawn two-dimensional and three dimensional graphs from the solution of the equations which demonstrates the physical situation of the solution for different values of the parameter. This gives us the clear picture of the range of the variables for which the normal solution exists and for what values of the variable the chaotic situation arises.

KEYWORDS: Approximation Solution, Chaotic Solution, Homotopy, Homotopy Perturbation, Perturbation 1. INTRODUCTION Chaos or chaotic system received great attention among mathematicians and physicists. Because it stemps out from natural phenomena. Mathematically one can get this studying linear or non-linear dynamical system or non-linear partial differential equations. Unfortunately to get a close form of the solution of any of them is almost beyond our present knowledge. However, one can use some kind of analytical or numerical methods to get an approximate solution. Recently, J. H. He [1, 2] found an ingenious method to solve non-linear differential equation, which is called “Homotopy Perturbation Method” (HPM). He applied this method to solve different types of non-linear equations, such as Van der Pol- duffing equations [3] for different types of oscillator [4, 5, 6] problems and so on. Here we have applied HPM technique to find the analytical solution of a non-linear differential equation with different types of initial conditions. Taking different values of the parameters and for a wide range of time we have found the graphical representation of the analytical solution for all domain of response. Following He’s paper [2] we have presented the basic prescription of the method in section 2. In section 3 we have applied this technique to solve non-linear partial differential equation with two different types of initial conditions and found an approximate solution of the problem. In section 4 we have given the graphical representation of the solution for two different initial conditions. In last section we have discussed the nature of the solution for different values of the parameters and range of variables Homotopy Perturbation Prescription J. H. He [1] gave a prescription to solve nonlinear differential equation which can be expressed in the following way.

36

M. Tahmina Akter, A. S. M. Moinuddin & M. A. Mansur Chowdhury

Let us consider a nonlinear differential equation of the form

A(u ) f (r ) 0 , r

(1)

Subject to the boundary conditions:

B(u , u / n) 0 , r where

(2)

A is a general differential operator, B

the boundary of the domain

a boundary operator,

f (r )

a known analytical function and

is

.

In general one can divide the operator

A into two parts: linear and non- linear. That means

A L N where

L

is linear and

N

is non-linear.

Hence, equation (1) can now be rewritten as

L(u ) N (u ) f (r ) 0 , r

(3)

By the homotopy technique, one can construct a homotopy in the following way

v(r , p) : [0,1] R

which satisfies

H (v, p) (1 p)[ L(v) L(u 0 )] p[ A(v) f (r )] 0 , p [0,1] , r or

H (v, p ) L(v) L(u 0 ) pL(u 0 ) p[ N (v) f (r )] 0

Where

p [0,1]

is an embedding parameter,

u0

(4) (5)

is an initial approximation of equation (1) which satisfies the

boundary conditions. Obviously, from equations (4) and (5) we will have:

H (v,0) L(v) L(u 0 ) 0

(6)

H (v,1) A(v) f (r ) 0

(7)

The changing process of called deformation, and

from zero to unity is just that of

L(v) L(u 0 )

the embedding parameter a power series in

p

p

and

A(v) f (r )

v(r , p) from u 0 ( r ) to u (r ) . In topology, this is

are called homotopy. According to the HPM, we can first use

as a "small parameter", and assume that the solution of equations (4) and (5) can be written as

p

v v0 pv1 p 2 v2 .... ... ... Setting

(8)

p 1 results in the approximate solution of equation (1):

u lim v v0 v1 v2 .... ... ... p 1

(9)

37

Semi-Analytical Approach to Solve Non-Linear Differential Equations and their Graphical Representations

The combination of the perturbation method and the homotopy method is called the homotopy perturbation method (HPM), which has eliminated the limitations of the traditional perturbation methods. On the other hand, this technique can have full advantage of the traditional perturbation techniques. The series (9) is convergent for most cases.

2. SOLUTION OF NON-LINEAR PARTIAL DIFFERENTIAL EQUATION In this section we will show how He’s homotopy method can be applied non-linear partial differential equation with different types of initial conditions. In this view we have considered a highly non-linear partial differential equation with two types of initial conditions, and used He’s homotopy method to solve these problems. The two different types of initial conditions considered here is just to see the sensitivity of the behaviour of the solutions for different types of initial conditions. Because in any dynamical system it is very important to check whether this types of equations is sensitive to initial conditions. 3.1. Solution of Non-Linear Equation with Initial Conditions Let us consider a highly non-linear partial differential equation

U tt U xx U U 3 0

(10)

and try to find the solution with the following initial conditions:

U ( x ,0) x 3

and

U t ( x,0) 0

(11)

In order to solve equation (10) using HPM, a homotopy- perturbation method can be constructed as follows

2v 2v 2v H (v, p ) (1 p ) 2 p ( 2 2 v v 3 ) 0 t t x Substituting

u lim v lim [v0 p v1 p 2 v2 .... ... ... p 1

equation based on powers of

p 1

into equation (12) and rearranging the resultant

p -terms, we can find the following equations:

2 v0 p : 2 0 t 0

p1 :

(12)

2 v0 2 v1 3 v0 v0 0 2 2 t x

2 v2 2 v1 2 p : 2 2 v1 v0 v1 0 t x 2

(13)

(14)

(15)

Solving equation (13) we get

v 0 A t

(16)

v0 At B

(17)

38

M. Tahmina Akter, A. S. M. Moinuddin & M. A. Mansur Chowdhury

Before using the initial condition in (16) and (17) let us clarify one important point. If we back to earlier substitution that is

u( x, t ) lim v v0 v1 v2 .... ... ... p 1

U ( x ,0) x 3

and taking the initial conditions : which implies,

and

U t ( x,0) 0

v0 x 3 and v1 v2 .... ... ... 0

Using the initial conditions (11), at

t 0

we get

A0

and

B x3

Hence, finally we get the solution

v0 x 3

(18)

From this equation we find 2 v0 2 v0 6x 3x , t 2 x

After substituting the value of

we get the equation for

v0 ,

v 0 x

and

2 v0 t 2

in (14)

v1

2 v1 (6 x x 3 x 9 ) 2 t Integrating

v1 (6 x x 3 x 9 )t C t Again integrating we arrived at

1 v1 (6 x x 3 x 9 )t 2 C t D 2 Using the initial conditions in the above equations we get

at

t 0,

v1 0 C , v1 ( x,0) 0 D t

Putting the values of

C

and

D

we get

1 v1 (6 x x 3 x 9 )t 2 2

(19)

Semi-Analytical Approach to Solve Non-Linear Differential Equations and their Graphical Representations

which is the solution for

39

v1

Now to solve equation (15) for

v2

let us find

v1 1 (6 3 x 2 9 x 8 )t 2 x 2 2 v1 1 (6 x 72 x 7 )t 2 2 x 2 Using these in equation (15) we get

2 v2 2 v1 2 ( v1 3 v0 v1 ) 2 2 t x

1 (12 x 2 x 3 90 x 7 4 x 9 3 2 x15 )t 2 2 Integrating,

v2 1 (12 x 2 x 3 90 x 7 4 x 9 3 2 x15 )t 3 C t 6 Again integrating,

v2

1 (12 x 2 x 3 90 x 7 4 x 9 3 2 x15 )t 4 C t D 24

Again the initial conditions (11) implies that

C 0 and D 0 Therefore

v2

1 (12 x 2 x 3 90 x 7 4 x 9 3 2 x15 )t 4 24

According to HPM we can write the solution of (10) up to second order of

u ( x, t ) lim v lim [v0 p v1 p 2 v2 ] p 1

Setting

p 1

p 1 the above equation becomes,

u ( x, t ) v0 v1 v2 Substituting the values of

u ( x, t )

v 0 , v1

and

v2

1 [24 x 3 (72 x 12 x 3 12 x 9 )t 2 24

(20)

p

as :

40

M. Tahmina Akter, A. S. M. Moinuddin & M. A. Mansur Chowdhury

(12 x 2 x3 90 x7 4 x9 3 2 x15 )t 4 ]

(21)

This is the approximate solution of the non-linear partial differential equation (10) with initial conditions (11). 3.2. Solution of Same Non-Linear Differential Equation with another Initial Conditions Again considering the same differential equation

U tt α U xx β U γ U 3 0

(22)

with different type of initial conditions:

u ( x,0) 0

and

U t ( x,0) e x

(23)

Following the same procedure for this problem we get,

U ( x, t )

1 x e [120 t (20 2 20 ) t 3 ( 2 4 2 2 2 )t 5 ] 120

1 3 x 1 2 5 x 9 e [42 t 5 (19 2 11 )t 7 ] e t 840 480

(24)

This is the approximate analytical solution of the partial differential equation (22) with another type of initial conditions. With the help of HPM we are able to solve highly nonlinear higher order partial differential equation with different types of initial conditions. In the following sections we will find the graphical representation of the solutions of non-linear differential equations (10) and (22) found in (21) and (24) for different values of parameters

α, β, γ

and for a wide range of x and

t values. From which we can draw some conclusion of the results.

4. GRAPHICAL REPRESENTATION OF THE SOLUTIONS OF (10) AND (22) In this section we have drawn three-dimensional and two-dimensional graphs from the solution (21) of non-linear differential equation (10) with the given initial conditions taking larger and smaller values of the parameter and for a wide range of the values of the variables x and t . After a close look to the graphs we have divided the shape of the graphs in four kinds. The graphs of similar kind for different parametric values but for a particular range of x and t are depicted below.

41

Semi-Analytical Approach to Solve Non-Linear Differential Equations and their Graphical Representations

First Kind 15 12.5 10

15 0.2

10 5

0.15

0 0

7.5 5

0.1 0.5 1

2.5

0.05

1.5 2 2.5

0

0.5

Figure 1

1

1.5

2

2.5

Figure 1'

Figure 1: The Surface Shows the Solution U (x, t) when the Values of the Parameters α 1 , β

1 and

γ 1 and Values of the Variables x (0,2.5) and t (0,.2) . Figure 1' Shows the Corresponding Two-Dimensional Graph 14 12 10 15 10 5 0 0

0.2 0.15 0.1 0.5

1 2 2.5

6 4

0.05

1.5

8

2

0

0.5

Figure 2

1

1.5

2

2.5

Figure 2'

Figure 2: The Surface Shows the Solution U (x, t) when the Values of the Parameters α .5 , β .6 and

γ .7 and Values of the Variables x (0,2.5) and t (0,.2) . Figure 2' Shows the Corresponding Two-Dimensional Graph 70 60 50 20 15 10 5 0 0

0.2 0.15 0.1 0.5 1

0.05

1.5 2 2.5

0

40 30 20 10 0.5

Figure 3

1

1.5

2

2.5

Figure 3'

Figure 3: The Surface Shows the Solution U (x, t) when the Values of the Parameters α 1 , β

2 and

γ 3 and Values of the Variables x (0,2.5) and t (0,.2) . Figure 3' Shows the Corresponding Two-Dimensional Graph

42

M. Tahmina Akter, A. S. M. Moinuddin & M. A. Mansur Chowdhury

50 40 15

0.2

10 5

0.15

0 0

30 20

0.1 0.5 1

10

0.05 1.5 2 2.5

0

0.5

1

1.5

Figure 4 Figure 4' Figure 4: The Surface Shows the Solution U (x, t) when the Values of the Parameters

2

2.5

α 5 , β 1 and

γ 2 and Values of the Variables x (0,2.5) and t (0,.2) . Figure 4' Shows the Corresponding Two-Dimensional Graph 17.5 15 12.5 15 0.2

10 5

0.15

0 0

10 7.5 5

0.1 0.5 1

2.5

0.05

1.5 2 2.5

0

0.5

1

1.5

Figure 5 Figure 5' Figure 5: The Surface Shows the Solution U (x, t) when the Values of the Parameters

2

2.5

α 1 , β 5 and

γ 1 and Values of the Variables x (0,2.5) and t (0,.2) . Figure 5' Shows the Corresponding Two-Dimensional Graph Second Kind

1.5 ´ 10 1.25 ´ 10 1 ´ 10

53

8 ´ 10 53 6 ´ 10 53 4 ´ 10 53 2 ´ 10 0 0

1000

7.5 ´ 10

750 500 250 250

500

5 ´ 10 2.5 ´ 10

37 37 37 36 36 36

750 1000

0

200

400

600

800

1000

Figure 6 Figure 6' Figure 6: The Surface Shows the Solution U (x, t) when the Values of the Parameters α 1 , β 1 and

γ 1 and Values of the Variables x (0,1000) and t (0,1000) Figure 6' Shows the Corresponding Two-Dimensional Graph

43

7 ´ 10 6 ´ 10 5 ´ 10

53

4 ´ 10 53 3 ´ 10 53 2 ´ 10 53 1 ´ 10 0 0

1000 750

3 ´ 10

500 250

2 ´ 10

250

500 750

4 ´ 10

1 ´ 10

36 36 36 36 36 36 36

0

1000

200

400

600

800

1000

Figure 7 Figure 7' Figure 7: The Surface Shows the Solution U (x, t) when the Values of the Parameters α .5 , β .6 and γ .7 and Values of the Variables x (0,1000)and t (0,1000). Figure 7' Shows the Corresponding Two-Dimensional Graph 1.4 ´ 10 1.2 ´ 10 1 ´ 10 6 ´ 10

54

1000

54

4 ´ 10 54 2 ´ 10 0 0

750

6 ´ 10 4 ´ 10

500 250

2 ´ 10

250

500

8 ´ 10

38 38 38 37 37 37 37

750 1000

0

200

400

600

Figure 8 Figure 8' Figure 8: The Surface Shows the Solution U (x, t) when the Values of the Parameters

800

1000

α 1 , β 2 and

γ 3 and Values of the Variables x (0,1000)and t (0,1000). Figure 8' Shows the Corresponding Two-Dimensional Graph 1.5 ´ 10 1.25 ´ 10 1 ´ 10

53

8 ´ 10 53 6 ´ 10 53 4 ´ 10 53 2 ´ 10 0 0

1000

7.5 ´ 10

750 500 250 250

500

5 ´ 10 2.5 ´ 10

37 37 37 36 36 36

750 1000

0

200

400

Figure 9 Figure 9' Figure 9: The Surface Shows the Solution U (x, t) when the Values of the Parameters

600

800

1000

α 1 , β 5 and γ 1 and Values of the Variables x (0,1000)and t (0,1000). Figure 9' Shows the Corresponding Two-Dimensional Graph

44

M. Tahmina Akter, A. S. M. Moinuddin & M. A. Mansur Chowdhury

Third Kind 2 ´ 10 1.5 ´ 10

41

41

55

1.5 ´ 10 55 1 ´ 10 54 5 ´ 10 0 700

1000

1 ´ 10

41

750 500 800

5 ´ 10

40

250 900 1000

0

750

800

850

900

Figure 10 Figure 10' Figure 10: The Surface Shows the Solution U (x, t) when the Values of the Parameters

950

1000

α 1 , β 1 and

γ 1 and Values of the Variables x ��� (700,1000) and t (0,1000). Figure 10' Shows the Corresponding Two-Dimensional Graph 1 ´ 10 8 ´ 10 54

8 ´ 10 54 6 ´ 10 54 4 ´ 10 54 2 ´ 10 0 700

1000 750

6 ´ 10 4 ´ 10

41 40 40 40

500 800

2 ´ 10

250

40

900 1000

0

750

800

850

Figure 11 Figure 11' Figure 11: The Surface Shows the Solution U (x, t) when the Values of the Parameters

900

950

1000

α .5 , β .6 and

γ .7 and Values of the Variables x (700,1000)and t (0,1000). Figure 11' Shows the Corresponding Two-Dimensional Graph

1.75 ´10 1.5 ´10 1.25 ´10 1.5 ´ 10

56

1 ´ 10

56

5 ´ 10

1000

55

750

0 700

500 800

250

1 ´10 7.5 ´10 5 ´10 2.5 ´10

42 42 42 42 41 41 41

900 1000

0

750

800

850

Figure 12 Figure 12' Figure 12: The Surface Shows the Solution U (x, t) when the Values of the Parameters

900

950

α 1 , β 2 and

γ 3 and Values of the Variables x (700,1000) and t (0,1000). Figure 12' Shows the Corresponding Two-Dimensional Graph

1000

45

Fourth Kind

0.84736 0.847355 12

0.84735

4 ´ 10 12 3 ´ 10 12 2 ´ 10 12 1 ´ 10 0 1

1000

0.847345

750 500 1

1

250

1.00001

1

1.00001 1.00001 1.00001

0.847335

1.00001 1.00001

0

Figure 13 Figure 13' Figure 13: The Surface Shows the Solution U (x, t) when the Values of the Parameters α 1 , β 1 and γ 1 and Values of the variables x (1,1.00001)and t (0,1000). Figure 13' Shows the Corresponding Two-Dimensional Graph

1

1

1.000011.000011.00001

0.916595 1.5 ´ 10

12

1 ´ 10

12

5 ´ 10

1000

11

750

0 1

0.91659 0.916585

500 1 1.00001 1.00001

0.91658

250

0.916575

1.00001 0

Figure 14 Figure 14' Figure 14: The Surface Shows the Solution U (x, t) when the Values of the Parameters α .5 , β .6 and γ .7 and Values of the Variables x (1,1.00001)and t (0,1000). Figure 14' shows the Corresponding Two-Dimensional Graph

0.80329 0.803285 1 ´ 10

13

5 ´ 10

1000

12

0.80328

750

0 1

500 1 1.00001

0.803275

250

1

1.00001 1.00001

1

1.00001 1.00001 1.00001

0

Figure 15 Figure 15' Figure 15: The Surface Shows the Solution U (x, t) when the Values of the Parameters α 1 , β 2 and γ 3 and Values of the Variables x (1,1.00001)and t (0,1000). Figure 15' Shows the Corresponding Two-Dimensional Graph

46

M. Tahmina Akter, A. S. M. Moinuddin & M. A. Mansur Chowdhury

0.79722 0.797215 13

2 ´ 10 13 1.5 ´ 10 13 1 ´ 10 12 5 ´ 10 0 1

1000 750 500 1 1.00001

0.79721 0.797205

250 1.00001 1.00001

0

1

1

1.00001 1.00001 1.00001

Figure 16 Figure 16' Figure 16: The Surface Shows the Solution U (x, t) when the Values of the Parameters α 1 , β 1 and γ 5 and Values of the Variables x (1,1.00001)and t (0,1000). Figure 16' Shows the Corresponding Two-Dimensional Graph Analysis of the Solution of Equation (22) The solution of equation (22) is given in equation (24). From this solution we have drawn many three-dimensional and two dimensional graphs for different values of the parameters and for a wide range of the variables. The significant graphs are shown below. If we look very carefully these graphs, we can see that the shape of the graphs changes for different range of variables. A slight change of the graphs has also been seen for positive and negative values of the parameters. First Kind

2 ´10 1.5 ´10

193

2 ´ 10 193 1.5 ´ 10 193 1 ´ 10 192 5 ´ 10 0 0

100 80

1 ´10

176

176

176

60 40

20 40

5 ´10

175

20

60 80 100

0

20

40

60

Figure 17 Figure 17' Figure 17: The Surface Shows the Solution U (x, t) when the Values of the Parameters

80

α 1, β 1 ,

γ 1 and λ 1 and Values of the Variables x (0,100) and t (0,100). Figure 17' Shows the Corresponding Two-Dimensional Graph

100

47

5 ´ 10 4 ´ 10 2 ´ 10

178 178

193 193

1.5 ´ 10 193 1 ´ 10 192 5 ´ 10 0 0

100

3 ´ 10

178

80

2 ´ 10

60 40

20 40

1 ´ 10

20

60 80 100

178 178

0

20

40

60

80

Figure 18 Figure 18' Figure 18: The Surface Shows the Solution U (x, t) when the Values of the Parameters

100

α 1 , β 1 ,

γ 1 and λ 1 and Values of the Variables x (0,100) and t (0,100). Figure 18' Shows the Corresponding Two-Dimensional Graph

1.4 ´ 10 1.2 ´ 10 1 ´ 10 4 ´ 10

156

2 ´ 10

100

156

80

0 0

8 ´ 10 6 ´ 10

60

4 ´ 10

40

20 40

2 ´ 10

20

60 80 100

142 142 142 141 141 141 141

0

20

40

60

Figure 19 Figure 19' Figure 19: The Surface Shows the Solution U (x, t) when the Values of the Parameters

80

100

α .5 , β .6 ,

γ .7 and λ .8 and Values of the Variables x (0,100) and t (0,100). Figure 5.24' Shows the Corresponding Two-Dimensional Graph Second Kind 8 ´ 10

6 ´ 10

249

249

288

1 ´ 10 287 7.5 ´ 10 287 5 ´ 10 287 2.5 ´ 10 0 0

1000

4 ´ 10

249

750 500 250 250

500

2 ´ 10

249

750 1000

0

20

40

60

80

Figure 20 Figure 20' Figure 20: The Surface Shows the Solution U (x, t) when the Values of the Parameters

100

120

140

α 1, β 1 ,

γ 1 and λ 1 and Values of the Variables x (0,1000)and t (0,1000). Figure 20' Shows the Corresponding Two-Dimensional Graph

48

M. Tahmina Akter, A. S. M. Moinuddin & M. A. Mansur Chowdhury

8 ´ 10

6 ´ 10

249

249

288

1 ´ 10 287 7.5 ´ 10 287 5 ´ 10 287 2.5 ´ 10 0 0

1000

4 ´ 10

249

750 500

2 ´ 10

250

249

250

500 750 1000

0

20

β

40

60

80

100

120

140

Figure 21 Figure 21' Figure 21: The Surface Shows the Solution U (x, t) when the Values of the Parameters α 1 , 1 , γ 1 and λ 1 and Values of the Variables x (0,1000)and t (0,1000). Figure 21' Shows the Corresponding Two-Dimensional Graph

Third Kind

1.2 ´ 10 1 ´ 10 1.510254 110254 510253 0 0

1000 750 500 50

100

200

6 ´ 10 4 ´ 10

250

150

8 ´ 10

2 ´ 10

250 250 249 249 249 249

2500

25

50

75

100

Figure 22 Figure 22' Figure 22: The Surface Shows the Solution U (x, t) when the Values of the Parameters

125

150

175

α .5 , β .6 ,

γ .7 and λ .8 and Values of the Variables x (0,250) and t (0,1000). Figure 22' Shows the Corresponding Two-Dimensional Graph Fourth Kind

1.2 ´ 10 1 ´ 10 8 ´ 10

244

7.5 ´ 10 244 5 ´ 10 244 2.5 ´ 10 0 0

1000 750 500 50

250

6 ´ 10 4 ´ 10 2 ´ 10

267 267 266 266 266 266

100 0

10

20

30

40

Figure 23 Figure 23' Figure 23: The Surface Shows the Solution U (x, t) when the Values of the Parameters

50

60

70

α .5 , β .6 ,

γ .7 and λ 2 and Values of the Variables x (0,136) and t (0,1000). Figure 23' Shows the Corresponding Two-Dimensional Graph

49

4 ´ 10

3 ´ 10 1.5 ´ 10

260

260

248

1 ´ 10

248

5 ´ 10

1000

247

750

0 0

2 ´ 10

500 200

1 ´ 10

250 400

260

260

0

5

10

15

20

Figure 24 Figure 24' Figure 24: The Surface Shows the Solution U (x, t) when the Values of the Parameters

25

α 1,β 1 ,

γ 1 and λ 5 and Values of the Variables x (0,500) and t (0,1000). Figure 24' Shows the Corresponding Two-Dimensional Graph Fifth Kind

3.5 ´ 10 3 ´ 10 6 ´ 10

2.5 ´ 10

-61

4 ´ 10

1000

-61

2 ´ 10

-61

750

0 0

1.5 ´ 10

500

1 ´ 10

250 250

500

2 ´ 10

5 ´ 10

-73 -73 -73 -73 -73 -73 -74

750 1000

0

200

400

600

Figure 25 Figure 25' Figure 25: The Surface Shows the Solution U (x, t) when the Values of the Parameters

800

1000

α 1,β 1 ,

γ 1 and λ 1 and Values of the Variables x (0,1000)and t (0,1000). Figure 25' Shows the Corresponding Two-Dimensional Graph Sixth Kind 7 ´ 10 6 ´ 10 5 ´ 10 2 ´ 10

307

1 ´ 10

1000

307

750

0 130

500 135

250 140 145

4 ´ 10 3 ´ 10 2 ´ 10 1 ´ 10

303 303 303 303 303 303 303

0

132

134

136

138

140

Figure 26 Figure 26' Figure 26: The Surface Shows the Solution U (x, t) when the Values of the Parameters α 1 ,

142

β 1 ,

γ 1 and λ 1 and Values of the Variables x (130,145)and t (0,1000). Figure 26' Shows the Corresponding Two-Dimensional Graph

50

M. Tahmina Akter, A. S. M. Moinuddin & M. A. Mansur Chowdhury

Seventh Kind

1 ´ 10 8 ´ 10 307

4 ´ 10 307 3 ´ 10 307 2 ´ 10 307 1 ´ 10 0 130

1000

6 ´ 10

750

4 ´ 10

289 288 288 288

500 131 132

2 ´ 10

250

133 134 135

288

0

131

132

133

Figure 27 Figure 27' Figure 27: The Surface Shows the Solution U (x, t) when the Values of the Parameters

134

135

α 1,β 1 ,

γ 1 and λ 1 and Values of the Variables x (130,135)and t (0,1000). Figure 27' Shows the Corresponding Two-Dimensional Graph

7 ´ 10 6 ´ 10 5 ´ 10 2 ´ 10

307

1 ´ 10

1000

307

750

0 130

500 135

3 ´ 10 2 ´ 10

250

1 ´ 10

140 145

4 ´ 10

303 303 303 303 303 303 303

0

132

134

136

138

Figure 28 Figure 28' Figure 28: The Surface Shows the Solution U (x, t) when the Values of the Parameters

140

142

α 1 , β 1 ,

γ 1 and λ 1 and Values of the Variables x (130,145)and t (0,1000). Figure 28' Shows the Corresponding Two-Dimensional Graph

1 ´ 10 8 ´ 10 307

4 ´ 10 307 3 ´ 10 307 2 ´ 10 307 1 ´ 10 0 130

1000

6 ´ 10

750

4 ´ 10

289 288 288 288

500 131 132

250

133 134 135

2 ´ 10

288

0

131

132

133

134

135

Figure 29 Figure 29' Figure 29: The Surface Shows the Solution U (x, t) when the Values of the Parameters α 1 , β 1 , γ 1 and λ 1 and Values of the Variables x (130,135)and t (0,1000). Figure 29' Shows the Corresponding Two-Dimensional Graph

51

Eighth Kind

1 ´ 10 8 ´ 10 2 ´ 10

-45

1000

1 ´ 10

-45

6 ´ 10

-57 -58 -58

750 0 130

4 ´ 10

500 132 134

2 ´ 10

250

136 138 140

-58 -58

0

132

134

136

Figure 30 Figure 30' Figure 30: The Surface Shows the Solution U (x, t) when the Values of the Parameters

138

140

α 1, β 1 ,

γ 1 and λ 1 and Values of the Variables x (130,140)and t (0,1000). Figure 30' Shows the Corresponding Two-Dimensional Graph Ninth Kind

1.5 ´ 10 1.25 ´ 10 1 ´ 10

303

3 ´ 10 303 2 ´ 10 303 1 ´ 10 0 -135

1000

7.5 ´ 10

750

5 ´ 10

500 -130

2.5 ´ 10

250

285 285 285 284 284 284

-125 -120

0

-132

-130

-128

-126

Figure 31 Figure 31' Figure 31: The Surface Shows the Solution U (x, t) when the Values of the Parameters

-124

-122

-120

α 1,β 1 ,

γ 1 and λ 1 and Values of the Variables x (-135,-120) and t (0,1000). Figure 31' Shows the Corresponding Two-Dimensional Graph Tenth Kind 3 ´ 10 2.5 ´ 10

1.5 ´ 10

2 ´ 10

234

1 ´ 10

234

5 ´ 10

1000

233

1.5 ´ 10

290 290 290 290

750

0 500

200 400

250

600 800 1000

1 ´ 10 5 ´ 10

290 289

0

100

110

120

Figure 32 Figure 32' Figure 32: The Surface Shows the Solution U (x, t) when the Values of the Parameters

130

140

α 1, β 1 ,

γ 1 and λ 1 and Values of the Variables x (97,1000)and t (0,1000). Figure 32' Shows the Corresponding Two-Dimensional Graph

52

M. Tahmina Akter, A. S. M. Moinuddin & M. A. Mansur Chowdhury

2.5 ´ 10

2 ´ 10 254

1.5 ´ 10

2 ´ 10 254 1.5 ´ 10 254 1 ´ 10 253 5 ´ 10 0

296

296

296

1000 750

1 ´ 10

296

500

200 400

250

600

5 ´ 10

295

800 1000

0

150

160

170

180

Figure 33 Figure 33' Figure 33: The Surface Shows the Solution U (x, t) when the Values of the Parameters α .5 , β .6 , γ .7 and λ .8 and Values of the Variables x (133,1000)and t (0,1000). Figure 33'Shows the Corresponding Two-Dimensional Graph

2 ´10

1.5 ´10 3 ´ 10

275

275

140

2 ´ 10

1000

140

1 ´ 10

140

750

1 ´10

275

0 500 200 400

5 ´10

250

600 800 1000

274

0

40

50

60

70

Figure 34 Figure 34' Figure 34: The Surface Shows the Solution U (x, t) when the Values of the Parameters α .5 , β .6 , γ .7 and λ 2 and Values of the Variables x (27,1000)and t (0,1000). Figure 34' Shows the Corresponding Two-Dimensional Graph Eleventh Kind

1 ´ 10 8 ´ 10 1 ´ 10

238

5 ´ 10

1000 237

6 ´ 10

750 0 500

200 400

250

600 800 1000

4 ´ 10 2 ´ 10

292 291 291 291 291

0

100

110

120

Figure 35 Figure 35' Figure 35: The Surface Shows the Solution U (x, t) when the Values of the Parameters

130

140

α 1,β 1 ,

γ 1 and λ 1 and Values of the Variables x (99,1000)and t (0,1000). Figure 35' Shows the Corresponding Two-Dimensional Graph

53

1 ´ 10 8 ´ 10 1 ´ 10

238

1000

5 ´ 10

6 ´ 10

292 291 291

237

750

4 ´ 10

0

291

500

200 400

2 ´ 10

250

600 800 1000

291

0

100

110

120

Figure 36 Figure 36' Figure 36: The Surface Shows the Solution U (x, t) when the Values of the Parameters

130

140

α 1 , β 1 ,

γ 1 and λ 1 and Values of the Variables x (99,1000)and t (0,1000). Figure 36' Shows the Corresponding Two-Dimensional Graph 8 ´ 10

6 ´ 10

2 ´ 10

144

1 ´ 10

1000

144

4 ´ 10

269

269

269

750 0 500 200

2 ´ 10

400

269

250

600 800 1000

0

40

α

50

60

70

Figure 37 Figure 37' Figure 37: The Surface Shows the Solution U (x, t) when the Values of the Parameters .5 , β .6 , γ .7 and λ 2 and Values of the Variables x (28,1000)and t (0,1000). Figure 37' Shows the Corresponding Two-Dimensional Graph

RESULTS AND DISCUSSIONS To find the nature of the graphs and to give the physical meaning of the solution of non-linear differential equation (10) we have drawn 16 numbers of three dimensional and two dimensional graphs for a wide or close range of values of the variables x and t and for different values of the parameters

α, β, γ . These graphs have shown that the

approximate solution (21) is valid for all values of x and t and for lower and higher parametric values of

α, β

and

γ.

These graphs also indicate that the shape of the graph depends on the values of the variables but not on parametric values. Here we can see four different shapes of the graph which are mostly depends on the range of the values of x and t but not on any parameter. We can classify these four kinds in the following way. Say for all parametric values but for close range of x and t that is if x is from 0 to 2.5 and t is from 0 to 0.2 then let us represent the graph by first kind, which are depicted in figures 1 to 5. Similarly, for a wide range values of x and t that means say x is from 0 to 1000 and t is also from 0 to 1000 the graphs are same which we call second kind which are depicted in figures 6 to 9. Another shape of the graph one can find for close range of x but wide range of t, which we call as third kind. That is the x values say is from 700 to 1000 but t is from 0 to 1000, which are depicted in figs. 10 to 12. But when the range of x is from 1 to 1.00001 and

54

M. Tahmina Akter, A. S. M. Moinuddin & M. A. Mansur Chowdhury

the range of t is from 0 to 1 or 0 to 1000 we have got another shape of the graphs and we call these types of the shape as fourth kind, which are depicted in figures of 13 to 16. The two-dimensional graphs that is for a fixed values of t but for a wide range of x the corresponding figures of first kind are depicted in figures from 1' to 5' and that of second kind are depicted in figures from 6' to 9' and the third kind are depicted in figures from 10' to 12' and the fourth kind are depicted in figures from 13' to 16'. All these graphs shows that the approximate solution that found by HPM method are most accurate and there is no chaotic solution in this type of highly non-linear differential equation with this type of initial conditions. Also it is clear from the graphs that the solution is valid for lower and higher values of the parameter, which is the main advantage of the HPM method. The approximate solution of (22) that is for the same non-linear equation with the other type of initial conditions, the solution found by HPM method is given in equation (24). To find the nature of the solution we have drawn many three dimensional graphs for a wide or short range of values of the variables x and t and for different values of the parameters and corresponding two- dimensional graphs for fixed values of t. From these graphs we have chosen only 21 numbers of both three-dimensional and two-dimensional graphs to show the differences of shape of the graphs. In the first case that is for the differential equation (10) we got only four types of different graphs, which are almost same but a minor difference in bend point, which is negligible. But in the second case that is for other type of initial conditions we got various types of graphs. This is mainly because of the different type of initial conditions. The main point here is that the solution of equation (22) exists for all values of the parameters except negative values of the parameter

λ

for a particular range of values of x and t which are depicted in figures from 17 to 24. After certain range of x and t values the solution does not exists. And these can be seen for the different range of the values of the x variables and the different values of the parameters. This can be easily understood if we look very carefully to the figures: 17 to 24, we can see that the solution exists only if x lies between 0 to 136 approximately but for all t and after that there is no solution. This one can see from the graphs in figure 35 to figure 37. For negative exponential initial conditions the solution exists in any range of values of x and t, which can be seen in figure 25. The solution (24) gives us some irregular shape of the graphs and also chaotic solution arises with these initial conditions, whereas in the solution (21) there was no such type of graphs found. This is only because of the initial conditions. The chaotic solution found for a very close range of x values but almost for all t and even for any parametric value. This one can find in the graphs of figures 26 to 29. Again if we compare figure 25, 30, 31 we can see that for

- that is for negative exponential the graph depends on x -values but not on t .

It is clear from these calculations that the solution is sensitive to the initial conditions. For the same equation due to the first type of initial conditions we got regular solution for all x and t. But in the second case we got both regular and irregular or chaotic solution. This proves the sensitivity of the initial conditions. From these solutions it is clear that HPM is a good semi-analytic approximation method to solve non-linear differential equation or dynamical system.

REFERENCES 1.

He J. H, Some Asymptotic Methods for Strongly Nonlinear Equations, International Journal of Modern Physics B 20 (2006) 1141–1199.

2.

He J.H, Non-Perturbative Methods for Strongly Nonlinear Problems, Dissertation.de-Verlag im Internet GmbH, Berlin, (2006).

3.

Sajadi H., Ganj D.D. & Shenas Y.V., Application of Numerical semi-Analytical approach on Van der Pol-Duffing Oscillators, Journal of Advanced Research to Mathematical Engineering Vol-1, (2010/iss3), 136-141.

4.

55

Njah A. N, Vincent U.E, Chaos Synchronization between Single and Double Wells Duffing-Van der Pol oscillators using active control, Chaos, Solitons & Fractals (2008); 37:1356-61.

5.

Ji J., Zhang N., Additive Resonances of a Controlled Van der Pol-Duffing oscillator, J Sound Vibr (2008); 315:22-23.

6.

Kimiaeifar, A., Saidi, A.R., Bagheri G.H., Rahimpour M., Domairry D.G., Analytical Solution for Van der Pol-Duffing Oscillators, Chaos, Solitons and Fractals 42 (2009) 2660-2666.

7.

Ganji D.D, Sadighi, A., Application of Heâ€™s Homotopy- Perturbation Method to Nonlinear Coupled Systems of Relation-Diffusion Equations, Int. J. Nonlinear Sci. Numer, Simul, 7, (2006) 411-418.