A SPECIAL TYPE OF APPROXIMATE SOLUTIONS FOR CERTAIN NONLINEAR POTENTIAL PDES IN HYPERGEOMETRIC FORM

(1)

A SPECIAL TYPE OF APPROXIMATE SOLUTIONS FOR CERTAIN NONLINEAR POTENTIAL PDES IN HYPERGEOMETRIC FORM

Chandrali Baishya

Assistant Professor, Department of Studies and Research in Mathematics, Tumkur University, Tumkur

Abstract: In the present paper a special type of approximate solution is obtained for Potential KdV equation and Potential KdV-Burger equation by employing a regular perturbation method. The perturbation series emerge in the form of geometric and hyper geometric series respectively. By introducing parameters suitably as coefficients in the Potential KdV equations, shockwave solution is obtained in the limiting case.

Key words: Potential KdV Equation, Potential KdV-Burger equation, regular perturbation series, Hyper geometric series, Shock wave.

INTRODUCTION

Burger equation and KdV equation are two classical and well-known among leading integrable nonlinear partial differential equations [1]. They posses exact solution in solitary wave form which exhibit shockwave in limiting case ([2], [4], [5]). In the present paper a special type of exact solution is obtained for Potential KdV equation and a approximate solution is obtained for Potential KdV-Burger equation by employing a regular perturbation method. By introducing parameters suitably as coefficient in the equations, shockwave solution is obtained for KdV equation in the limiting case. The perturbation series for potential KdV equation is geometric in nature and its formal sum provides exact solution.

The perturbation series for potential KdV-Burger equation is in the form of Standard Gauss hyper geometric series: ₂F₁(a,b;c;z) [3].

POTENTIAL KdV EQUATION

The KdV equation in the standard form is +6 + =0

xxx x

t uu u

u σ (1) If we substitute u=−V_x then the resulting equation is called potential KdV equation,

given by

−3 ² + =0

xxx x

t V V

V σ (2) First let us scale by perturbation parameter

Received Dec 24, 2012 * Published Feb 2, 2013 * www.ijset.net

(2)

V =εU (3) Substitution of (3) in (2) will lead to a perturbed potential KdV equation

U_t+σU_xxx =3εU_x² (4) Next, by seeking solution of (4) in the form of a perturbation series

U(x,t,ε)=U₀(x,t)+εU₁(x,t)+...+εⁿU_n(x,t)+...

one can break the equation (4) into following hierarchy of linear equations:

⁻

=

−

= −

+

= +

1

0

, , 1 ,

,

, 0 , 0

3 0

n

j

x j x j n xxx

n t n

xxx t

U U U

U

U U

σ σ

n = 1, 2, 3,…

Zero-order term:

Let us choose a particular traveling wave solution in the exponential form given by ⁽ ⁾

1 0

t x

e U

− −

= σ ^σ _,σ >₀ First-order term:

Let us choose

) 2( 1 1( , )

t x

e a t x U

−

= ^σ and

Substitute in U₁_,_t +σU₁_,_xxx =3U₀_,_x²

Then

2 ) 1( )

2(

1 2 8 3

−

− = +

− −

t x t

x

e e

a ^σ ^σ

σ σ σ σ

or 6 3

1 − =

a σ or

1 2 σ

= − a

Hence

) 2(

1( , ) 2

t x

e t

x U

−

= σ ^σ

Second-order term:

Let us choose ⁽ ⁾

3 2 2

t x

e a U

− −

= ^σ

and substitute in

U₂_,_t +σU₂_,_xxx =6U₀_,_xU₁_,_x

Then − = −

+

− −

) 2( ) 1( )

3(

2 3 27 ^x ^t 6 ^x ^t ^x ^t

e e

e

a ^σ ^σ ^σ

σ σ σ σ

(3)

or ² 4

= σ a

Hence ⁽ ⁾

3

2( , ) 4

t x

e t

x U

− −

= σ ^σ

By the method of mathematical induction, one can build in this way and obtain in general

) )( 1 (

2 ) 1 , (

t n x n

n x t e

U

+ −

−

=

σ

^σ

Hence

) 1(

1 2

t x

e e

− −

+

=

σ σ

ε

σ (5)

and

) 1(

1 2

t x

e V e

− −

+

=

σ σ

ε σ ε

It is interesting to note that if

) 1(

1 2 ) 2 (

2 ^x ^t

t x

e e a b a a V a b

v

−

− −

+

− +

− = +

=

σ σ

ε ε

σ

then v satisfies 6 2 0

= +

−

− x xxx

t v v

a

v b σ σ

and exhibits shockwave in the limiting case

<

−

>

= −

→ 0

) 0 , ( lim

0 b x t

t x t a

x

σ v

POTENTIAL KdV-BURGER EQUATION

Let us consider the potential KdV-Burger equation in the following form

u_t−δ u_xx+σu_xxx =3u_x² (6)

n

t x t

x

e e

t x U

∞

=

− −

−

=

0

) 1( )

1(

) 2 , ,

( ^σ ε ^σ

σ ε

(4)

The perturbed potential KdV-Burger equation is

U_t −δU_xx+σU_xxx =3εU_x² (7) where u =εU

By seeking a perturbation series solution

U(x,t,ε)=U₀(x,t)+εU₁(x,t)+...+εⁿU_n(x,t)+...

one can break (7) into the following hierarchy of linear equations:

⁻

=

−

= −

+

−

= +

−

1

0

, , 1 ,

, ,

, 0 ,

0 , 0

3 0

n

j

x j x j n xxx

n xx n t n

xxx xx

t

U U U

U U

U

σ δ

n = 1, 2, 3, ………..

Zero-order term:

Let us choose the traveling wave ₀ 1 _k⁽_x _t⁾

ke

U = ⁻ ⁻ ,

where σk²+ kδ −1=0. (8) First-order term:

As before, let us choose U₁ =a₁e⁻²^k⁽^x⁻^t⁾ and substitute in

U₁_,_t−δU₁_,_xx+σU₁_,_xxx =3U₀_,_x² Then using (8) we obtain

₁ ₂ ² ⁽ ⁾

) 2 1 ( 2

3 _k _x _t

k e

a k ⁻ ⁻

+

= −

σ

And hence ₁ ₂ ² ⁽ ⁾

) 2 1 ( 2 ) 3 ,

( e ^k ^x ^t

k t k

x

U ⁻ ⁻

+

= −

σ (9)

₁_, ₂ ² ⁽ ⁾

) 2 1 ( ) 3 ,

( ^k ^x ^t

x e

t k x

U ⁻ ⁻

+

= σ Second-order term Let us choose

U₂(x,t)=a₂e⁻³^k⁽^x⁻^t⁾ and substitute in

(5)

U₂_,_t −δU₂_,_xx+σU₂_,_xxx =6U₀_,_xU₁_,_x and using (8) we get

) 3 1 )(

2 1 (

3

2 2 2

k k

k a

σ

σ +

+

=

) 3 1 )(

2 1 (

4 . 3 2

1

2 2

2

k k

k + σ + σ

−

) ( 3 2 2

2

2 (1 2 )(1 3 )

4 . 3 2

) 1 ,

( e ^k ^x ^t

k k

k t

x

U ⁻ ⁻

+ +

−

=

σ

σ (10) Third-Order term:

Let us choose

U₃(x,t)=a₃e⁻⁴^k⁽^x⁻^t⁾ and substituting in

U₃_,_t −δU₃_,_xx+σU₃_,_xxx =3(2U₂_,_xU₀_,_x +U₁_,_x² we get

) 2 1 ( 30

) 63 27 ( )

4 1 )(

3 1 )(

2 1 (

5 . 4 . 3 2

) 1 ,

( 2

) 2 ( 3 2 2

2 3

3 k

e k k k

k k t

x

U ^k ^x ^t

σ σ σ

σ

σ +

× + +

+ +

−

= ⁻ ⁻ (11)

Fourth-Order term:

Choosing U₄(x,t)=a₄e⁻⁵^k⁽^x⁻^t⁾ and substituting in

U₄_,_t −δU₄_,_xx+σU₄_,_xxx =3(2U₃_,_xU₀_,_x +2U₂_,_xU₁_,_x) we get

) 2 1 ( 150

) 342 108 ( )

5 1 )(

4 1 )(

3 1 )(

2 1 (

6 . 5 . 4 . 3 2

) 1 ,

( 2

) 2 ( 5 2 2

2 2

4

4 k

e k k k

k k

k t

x

U ^k ^x ^t

σ σ σ

σ σ

σ +

× + +

+ +

+

−

= ⁻ ⁻ (12)

Similarly, the Fifth-order term gives

) 3 1 ( ) 2 1 (

) 5 1 )(

4 1 )(

2 1 ( 81 ) 5 1 )(

3 1 )(

126 54 ( 3 ) 3 1 )(

2 1 )(

171 54 ( 1575

2

) 6 1 )(

5 1 )(

4 1 )(

3 1 )(

2 1 (

7 . 6 . 5 . 4 . 3 2

) 1 , (

2 2

2

2 2

2

) ( 6 2 2

2 2

2 5

5

k k

k

e k k

k k

k k t

x

U ^k ^x ^t

σ σ

σ

σ σ

σ

+ +

+ + + +

+

× +

+ +

−

= ⁻ ⁻

(13) Numerical Calculation:

Choosing δ =1_andσ =₁ and using the relation (8)

(6)

we obtain the value

2 5

−1 +

k = (golden ratio).

Now in (11), (12), (13) if we put the value of k in

) 2 1 ( 30

) 63 27 (

2 2

k k σ

σ +

+ ,

) 2 1 ( 150

) 342 108 (

2 2

k k σ

σ +

+ ,

) 3 1 ( ) 2 1 (

) 5 1 )(

4 1 )(

2 1 ( 81 ) 5 1 )(

3 1 )(

126 54 ( 3 ) 3 1 )(

2 1 )(

171 54 ( 1575

2

2 2

2

2 2

2

k k

k

σ σ

σ

+ +

+ + + +

+

we obtain 0.9649, 0.9019, 0.6496 respectively which can be approximated to 1.

With this we can conclude that

U1, U₂, U₃, U₄, U₅ form a Hypergeometric solution of the equation (7) upto 5^th term.

By the method of mathematical induction we can build in this way that

) ( ) 1 ( 2 2

2)(1 3 )...(1 ( 1) ) 2

1 (

) 2 ...(

4 . 3 2

) 1 ,

( ⁿ ^k ^x ^t

n

n e

k n k

k k t n

x

U ⁻ ⁺ ⁻

+ + +

+

− +

=

σ σ

σ

Hence the perturbation series is

n t x k

n n n n

t x k

e n k

k k

t e x

u −

+

= ⁻ ⁻

∞

=

−

) ( 2 2

0 ) (

)! 2 ( 1 2

) 1 ( ) 3 ) (

, , (

σ ε

σ ε ε

⁼ ⁺ ⁻ ⁻ ⁻

−

) ( 2 1 2

2 ) (

2

; 1 2

; 1 ,

3 ^k ^x ^t

t x k

e k k

k F e

σ ε σ

ε

References

1. Ablowitz M.T. and Segur H., Solitons and the inverse scattering Transform, Siam, Philadelphia, 1981.

2. Achuthan P., Narasimhan R. and Rangarajan R., On KdV Solitons and Pade Approximants, Jour. Math. Phy. Sci., 25(1991), 287-304.

3. Rainville E.D., Special Functions, The Macmillan Company, New York, 1965.

4. Rosales R.R., Exact Solutions of Some Nonlinear Evolution Equations, Stud. Appl. Math.

59(1978), 117-151.

5. Zauderer E., Partial Differencial Equations of Applied Mathematics, John Wiley and Sons, New York 1985.