Numerical Solution of The Korteweg de Vries Equation - IJENS

International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 11 No: 02

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Numerical Solution of The Korteweg de Vries Equation Jamrud Aminuddin, Lecturer, Physics UNSOED, and Sehah, Lecturer, Physics UNSOED Abstract — The dynamics of solitary waves is modeled by the Korteweg de Vries (KdV) equation. In this work, we seek the solution of the KdV equation no perturbation term. We start by discreetizing the KdV equation using the finite difference method. The discreet form of the KdV equation is put into a matrix form. The solution the of matrix is determined using the Gauss-Jordan method. The plot of the set of the numerical solution shows a wave envelope, which represents the dynamics of the solitary wave. As the wave envelope η(ζ, ) moves away, its amplitude (ζ) is found to be distorted after a certain period of time ( ). This distortion indicates that such a wave packet is not stable. Index Term — Solitary waves, Korteweg de Vries, Finite difference, Gauss-Jordan.

I. INTRODUCTION The dynamical model of solitary waves can be represented by the Korteweg de Vries (KdV) equation [1], [2], [3]. Its characteristic is determined by modifying the perturbaration term of the KdV equation [4]. The numerical method which can be used to determined the solution of the KdV equation is the finite difference method. This method can minimize the truncation error in the Taylor series. Its evaluation is done iteratively to make computer programming convenient[5]. The compulation is accomplished using MATLAB 7.0 due to its simplicity in data computing. It also enable 3-dimensional data visualization conveniently done[6],[7].

The solution of the equation (1) are obtained by assuming an ansatz [9], [10]. This ansatz is representation of wave packet (η(ζ)) as a function of range (χ) in secant hyperbolic from which treats time( ) as constant (APPENDIX A).

1      0   2 

    3 sech 2  (2)

Where and ω is the speed, which is propotional to amplitude. Equation (2) is an exact solution of the KdV equation representation a non dispersive travelling wave packet [3], [8], [9]. Using some random numbers as input values of speed (ω), initial wave packet( , and wave length, computation in MATLAB 7.0 results in some graphical plots (Fig.1 to Fig.6). It can be seen from the figures that the height of the packet increases with speed. To determine whether it is in fact a soliton requires further evaluation of the KdV equation that includes time as a parameter. It has been achieved by numerical calculation.

II. KORTEWEG DE VRIES EQUATION The KdV equation is a nonlinear, dispersive, non dissipative equation which has soliton solutions. The mathematical form of the equation is third order nonlinier partial deifferential equation [1], [4], [8].

t  x  xxx (1) Fig. 1. wave packet with parameter: J. Aminuddin is with Program Study of Physics, Faculty of Science and Engineering, University of Jenderal Soedirman, Jl. dr. Suparno 61 Karangwangkal, Purwokerto 53123, INDONESIA. E-mail: [email protected] / [email protected] Sehah is with Program Study of Physics, Faculty of Science and Engineering, University of Jenderal Soedirman, Jl. dr. Suparno 61 Karangwangkal, Purwokerto 53123, INDONESIA. E-mail: [email protected]/ [email protected]

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  1,  0  200,

and

  400.

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Fig. 2. wave packet with parameter:

Fig. 3. Wave packet with parameter:

  1,  0  400,

  2,  0  200,

and

  800. Fig. 5. Wave packet with parameter:

and

  400.

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  3,  0  200,

and

  1,  0  400,


  400.

and

  800.

t n 1

m, n 1

n

m  1, n

m, n

m 1, n

m  2, n

Dt n 1

m, n 1

m 1


  2,  0  400,

and

  800.

x

m 1

m Dx

Fig. 7. Implicit scheme of the finite difference methods


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III. NUMERICAL APPROXIMATION The numerical solution of the KdV equation is determined by using the implicit scheme Of the finite difference method, the grid of which is shown in fig. 7[5], [8], [11]. The first step in disceetizing the KdV is by constructing the equation into a simple difference form.

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 3      0, 3 x x t

(3)

  m  2,n  3 m 1,n  3 m ,n   m 1,n    m 1,n   m ,n    m ,n   m ,n 1         0. 3 D x Dx Dt      

(4)

The equation (4) can be written in the following form (5)

m 2,n  3m1,n  3m,n  m 1,n Dx 2   m 1,n m,n  

To make equation (5) appears simple in the next step α and β are defined respectively as , and . Hence, m  2,n  3m 1,n  3m,n  m 1,n   m 1,n 2

(6)

    m 1,n   m,n   m,n 1  0   2

Equation (6) is solved iteratively in ascending order from index to .

 2    3     1    0       0  

  2  2,n   3    3,n   4,n    2,n 1 

  2,n    3   2 

  2 3,n   3     4,n  5, n   3, n 1 

 3,n    3   2 

  2  4,n   3    5,n  6,n    4,n 1 

  4,n    3   2 

  2 5,n   3    6,n  7,n   5,n 1 

(7)

    M 1,n    3   2  M ,n   3   2  M 1,n   M  2,n   M ,n 1    

These equations constitutes a system of equations, called iteration formulas, of the KdV equation formulated from its non linear differential equation from [5],[6],[7],[10]. These iteration formulas in equation (7) were expressed in matrix for

 2  3

1

0

  3   2 

 2  3

1

1

  3   2 

 2  3

0

1

       0    1     2  3   0

  3   2 

 1,n      1,n 1  0, n              2,n    2,n 1             3,n 1    3,n                              M ,n    M ,n 1   

   0,n    3   2 1, n   3   2  2, n  3, n  1, n 1   

 1,n    3   2 

(5)

Dx3 m,n m,n1   0. Dt

(8)

IV. THE NUMERICAL SOLUTION IN GRAPHICAL REPRESENTATIONS The numerical solution of the KdV equation is found by determining the values of in equation (8) as some wave packets. In the matrix, there are two elements which pair up with one another, i.e. and , where the former depends on the later as the numerical solution of the KdV equation. In this calculation, the -index as a number of column of both elements increases with time . Thus, these wave packets are calculated with time parameter t varying. To determine their values the Gauss-Jordan methods are used by assuming equation (20) as the ansatz function for the initial condition, except for , which is chosen to be 0, (APPENDIX B).


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The constant parameters used in this calculation are and . By inserting some random numbers in the variables of speed , and time

, initial wave packet

, wavelength

in MATLAB 7.0, some graphical

representations are produced (Fig.8 – Fig.13). The surface graphics in the figures depict that the significant changes of wave packet occurs due to time parameter. The series of graphics shows that the amplitude of envelope wave distorts after a long period of time. These solutions have the same charecteristics as those in references [2], [5], and [10].

Fig. 11. Wave packet with parameter:   2,   400,   800, and 0

t  3.

V. CONCLUSION According to this modeling, the wave envelope η( ζ, t) is found to propagates with its amplitude (ζ) distorted. This distortion gives evidence that, after a certain period of time (t), the wave packet is unstable. The distortion is notably small for wider range of time. t  2.

Fig. 8. Wave packet with parameter:   1,  0  200,   400, and


t  2.


t  3.


t  4.


t  2.


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 ' ' " '  0

APPENDIX A: ANSATZ FUNCTION

(11)

Equation (11) is integrated to yield The first step to determine the ansatz function is by writing parameter ζ of the KdV equation.

 t   x   xxx

1 2

(12)

   2   "  A

(9) Where A is an undetermined constant. By expressing

In the form

 x, t    x  t     

(10)

1 d ' 2 d d 1  '   '"  " 2 d d d '

(13)

Where ω is speed parameter. Based on the ansatz in equation (10), equation (9) can be written in the form Equation (13) can be represented in the form 1 d ' 2 1     2  A 2 d 2

From equation (8), a matrix can be arranged with assumptions  0,n  0 and M  m . The index n runs from n  1 to n  N for (14)

Parameter η in equation (14) is integrated to result in

1 2 1 1  '   2   3  A  B 2 2 6

(15)

Where B is a new undetermined constant. By using some boundary conditions, i.e:  , ' ,"  0 to   , A  0 in equation (12), B  0 in equation (13) and  

1  3 

(16)

 '2   2    

Also, by including  

d



 3

d  3  

(17)

Then by using the ansatz   3 sec h 2

(18)

And the algebraic principle  0 

2



 d 

2





(19)

the left matrix and from n  0 to n  n for the right one. Then, equation (8) can be written in the form:  2 2   3      3  1    0           0  

1,1 1, 2 1,3   2,1  2, 2  2,3   3,1  3, 2  3,3       m,1  m , 2  m,3     1,1     2,1      3,1        m ,1 

 1, 2    2, 2    3, 2  

  m,2 

1

0



            

0

1

      0 A 1  2   3     3   2    0

 1,n    2,n    3, n   B       m ,n     1,3  1,n        2,3   2,n     C    3, 3   3, n     





(21)



   m ,3   m ,n    

The Gauss-Jordan algorithm to find the solution matrix can be done by determining the diagonal elements of matrix A as the pivot, and then divide all of the rows parallel to the pivot. This 1   (20)     3 sec h 2  3  0    2 procedure makes the diagonal has elements with the value of 1.   Subsequently, all of the columns within the pivot are used as an APPENDIX B: GAUSS-JORDAN METHODS eliminator by multiplying all elements parallel to the eliminator The Gauss-Jordan method is an algorithm to solve a system by all elements parallel with to pivot that has the value 1. The of linear equations in its matrix representation such that the results are used to subtract all elements parallel to the pivot. solution in found when the matrix becomes an identity matrix. Their past result are put on the eliminator row. This procedure 114502-9292 IJBAS-IJENS © April 2011 IJENS IJENS It is obtained that


will make the element in the pivot row has the value of 0. When this procedure is continued to the next row within the pivot column, it will make all columns within the pivot have 0 values. These calculation are proceeded for all elements in the A matrix such that all columns have 0 values except for the pivot, which has the value of 0. As a results, the A matrix will become an identity matrix. Correspondingly, B’s elements will change. Based on the rule of matrix multiplication, it will be found that A = B. Thus, matrix B is the solution of the system of linear equations.

1 0  0   0

0

0

1 1 0

0

0   1,1 1,2 1,3    2,2  2,3   2,1  0   3,1 3,2 3,3   0  1   m ,1  m ,2  m ,3

* * *  1,1 1,2 1,3  * * *  2,1  2,2  2,3 * * *  3,1 3,2 3,3    *  *  m* ,3 m ,2  m ,1

1,n  2, n  3, n 

   m,n  (22)

1,* n   2,* n  3,* n 

   m* ,n 

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[8]. Jose VG and JS. Eugene, 2002, Clasical Dynamics: a contemporary approach, Cambridge University Press, New York, USA. [9]. Kataoka T and M. Tsutahara, 2004, Tranverse Instability of surface Solitary Wave, J. fluid. Mech. 512, 211-221, Cambridge University, USA. [10]. Li Y and F. Raichlen, 2002, Non-Breaking and Breaking Solitary Wave RunUp, J. fluid. Mech., 456, 295-318, Cambridge University, USA. [11]. Aminuddin J. and FP. Zen, 2006, Application of Finite Difference Methods for Solving Nonlinear Schrdinger Equation, Workshop on Theoretical Physics and 70th Birthday of Prof.Muslim, UGM, Yogyakarta, Indonesia. Jamrud Aminuddin was born in Kasambang, West Sulawesi, INDONESIA on June 11, 1977. He graduated from Department of Physics, Faculty of Mathematics and Natural Science, Hasanuddin University (UNHAS) Makassar, INDONESIA. He has studied in advance to get his master degree at Department of Physics faculty of Mathematics and Natural Science, Institute Technology of Bandung (ITB), INDONESIA. Now, his position as a lecturer in Physics Study Program, Faculty of Science and Engineering, Jenderal Soedirman University (UNSOED), Purwokerto, INDONESIA. Sehah was born in Rembang, Jawa Tengah, INDONESIA on August 6, 1971. He graduated from Department of Faculty Mathematics and Natural Science, Diponegoro University (UNDIP) Semarang, INDONESIA. He has studied in advance to get his a master degree at Department of Physics faculty of Mathematics and Natural Science, Gadjah Mada University (UGM), Yogyakarta, INDONESIA. Now, his position as a lecturer in Physics Study Program, Faculty of Science and Engineering, Jenderal Soedirman University (UNSOED), Purwokerto, INDONESIA

ACKNOWLEDGEMENT We thank the Directorate General of Higher Education, Ministry of National Education Republic of Indonesia for its support in our research by its grant, namely HIBAH PEKERTI. REFERENCES [1]. Lamb JGL, 1980, Elements on soliton theory, John Wiley & Sons, Inc, New York, USA. [2]. Malfiet W, 1992, Solitary wave Solutions of Nonlinear Wave Equation, Am. J. Phys. 60, 650. USA. [3]. Johnson RS, 1994, Solitary wave, solution and shelf evolution over variable depth, J. fluid. Mech. 276, 125-138, Cambridge University, USA. [4]. Soewono E, Andonowati, SR. Pudjaprasetya, HJ. Wospakrik. dan FP. Zen., 2000, Soliton and Other Solution Related to the Korteweg de Vries Equation, Proc. ITB Vol. 32, No. 2, Bandung, Indonesia. [5]. PirozzoliS, 2007, Performance Analysis and Optimization of Finite Difference Scheme for Wave Propagation Problem, J. Comp. Phys, Vol. 222, 809-831. [6]. Martha LA and PB. James, 1992, Mathematica by Example, Academic Press. Inc, San Diego, USA. [7]. Lindfield G. and J. Penny, 1995, Numerical Methods Using MATLAB, Ellis Horwood Limited, New York, USA.

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