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: : 2980: 01.12.2005

PC:

: Numerov Algorithm

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::

m , :

2

2 2

34 3

d yy x

dx x =

( )xy :

i. ( )xy

: ( )7

151 =y ( )

7

101 =y .

ii.

Numerov Algorithm /. ;

//::

Fortran, Numerov Algorithm., compilation , :

Numerov Algorithm,

expdata.out ( )yx, , Gnuplot ( )xfy = .

::

Numerov Algorithm:

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Fortran

c234567program ergastiriaki_anafora_8implicit nonereal x(0:1000),y(0:1000),f,u,g,k,d,h,h2integer i,nprint*, ""

print*, "Doste ta x(initial), y(initial)."read*, x(0),y(0)print*, ""print*, "Doste to y'(",x(0),")."read*, dprint*, ""print*, "Doste ton arithmo ton zevgon timon (x,y) pou epithymeite.

&"read*, nprint*, ""print*, "Doste to vima, me to opoio thelete na prokypsoun oi diado

&xikes times tis metavlitis x."read*, hy(1)=y(0)+d*hprint*, ""

print*, "Me vasei ta dedomena pou eisagate, ta zevgi timon (x,y),&exoun os eksis:"print*, ""call Numerov_Algorithm(h,n,x,y)print*, ""end

subroutine Numerov_Algorithm(h,n,x,y)implicit nonereal x(0:1000),y(0:1000),f,u,g,k,h,h2integer i,nh2=h**2/12.0do i=0,n-1x(i) x(0)+i*h

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function f(u)implicit nonereal f,uf=-3./(4.*u**2)returnend

function g(k)implicit none

real g,kg=3.*k/4.returnend

i

2 22 3

2 2 2

3 3 34 3

4 4

d y d y y x x y xdx x dx

= =

, oy :

04

32

2

2 = ydx

ydx

ro xy = :

( ) ( )

( ) ( )2

21

2

4

3

411

04

30

4

310

4

310

4

3

2

2,1

222

2

2

2

=

=

==

==

r

rrrrxxxrrxxdx

xdx rrrr

r

, oy :

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7

1

4

3

4

210

4

3

4

360

4

3

4

36

4

3

4

3

64

3

4

3

4

3

4

3

3

33233

2

32

23

2

2

2

====

===

aa

aaaax

xaxaxxxaxdx

axd

xxydx

yd

x

, py :

( ) 37

1xxy

p

=

:

( ) ( ) ( )xyxyxy po +=

( )xy :

( ) 321

22

3

17

1 xxcxcxy ++=

, :

( )7

151 =y

( ) 2715

711 2121 =+=++= ccccy

( )7

101 =y

22

3

2

2

1

1

32

1

2

2

3

1

32

1

22

3

1

7

3

2

1

2

3

7

17

1

xxcxcdx

dx

dx

dx

cdx

dx

cdx

xxcxcd

dx

dy

+=++=

++

=

12

1

2

3

7

10

7

3

2

1

2

3

7

102121 ==+= cccc

d

dy

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ii

.

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0.95 1 1.05 1.1 1.15 1.2 1.25 1.3

y(x)

x

"expdata.out"x**(3./2.)+x**(-1./2.)+x**3/7.