Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://ani.cs.msu.su/files/maple-echkina.pdf Äàòà èçìåíåíèÿ: Fri Aug 29 22:59:45 2008 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 19:47:22 2012 Êîäèðîâêà:

. .

Maple

1. ple.
Maple ­ [1]. , , . . , , . , : · ; · ; · ; · ; · ; · ; · ; · ; · . . Maple . Maple . , , Latex, . le · ; · ; · ;
. . http://ani.cs.msu.su 1


. .

Maple

· ; · ; · ; · ; · ; · ; · ; · ; · ; · ; · ; · Maple ­ «» , Waterloo Maple, Inc ().

2. .
. , , . . ­ . : · ­ , ; · ­ , ; · ­ , .
. . http://ani.cs.msu.su 2


. .

Maple

u ( x, y , z )
u u u ,, x y z r u grad (u ) = i + x r u r u j +k . y z


u l rr rr rr cos(l , ex ) , cos(l , e y ) , cos(l , ez

[

grad (u ) ] =

=

rr rr rr u u u cos(l , ex ) + cos(l , e y ) + cos(l , ez ) x y z

) - , ,
r

l . A
r A A A divA = x + y + z , x y z r r r r A( Ax , Ay , Az ) = iAx + jAy + kAz .

A
r i r rot A = x Ax r j y Ay r k . z Az

r

div grad = .
= 2 2 2 + 2+ 2. x 2 y z

2u 2u 2u u, u = 2 + 2 + 2 . x y z

( )
r r r = i +j +k . x y z

u, u = i
r

r u r u r u +j +k = grad (u ). x y z

A
r A r r A r A r A = i x + j y + k z = divA. x y z

A
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r


. .

Maple

r r A A r A A r A A r â A = i z - y + j x - z + k y - x = rot A. z x y z y x

xyz. . S ­ - , D. u v M D ,
v u [uv - vu ]d = u - v D S n n d .

. S ­ - , - . , S ­ . a - , S,

r


S

r rr ( rot a ) n ds = ( a , t )dl
C

. a - M D . , D S. , ,
D

r



r div a d = an d
S

d - S, d - D, an = a n , n - S. , , . .

rrr

. .

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. .

Maple

. : z, , z, , Oz. r, , z x, y , z
x = r cos( ), y = r sin( ), z = z.

u =

1 u r r r r

2 2 1 u u +2 + 2. 2 z r

. : O
, ,

z , , z xz. , , :
x = sin( ) cos( ), y = sin( ) sin( ), z = cos( ).

x, y , z


,

, , .
u 2u 1 2 u 1 1 u = 2 + sin( ) . + 2 sin( ) 2 2 sin( )

3.
. 1.
2u 2u 2u + + = 0. x 2 y 2 z 2









,

,

-



, ,



. , , , .
. . http://ani.cs.msu.su 5


. .

Maple

2.
2 u 2u 2u + + = - f ( x , y , z ). x 2 y 2 z 2

­ , . , , ­ . 3.
u - 1 2u = - f ( x , y , z , t ). v 2 t 2

.

.

4.
u - 1 u = - f ( x , y , z , t ). v 2 t

, , , . 5.
u + k 2u = - f ( x, y , z, t ), k = co n s t.

- ; k = w v - . 6.
+ 2m ( E - V ) = 0. h2

. ( x ) - ; m ­ ; E ­ , V(x) ­ , h - . 7.
u - 1 u u - b - c u = - f ( x , y , z , t ). 2 v t t

.
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. .

Maple



i , j =1



n

aij ( x )

2u + ( x, u, gradu ) = 0 xi x j

x0 = BT x,

B - , y = B
n

i , j =1



aij ( x0 ) yi y

j

.

2u 2u 2u + c ( x , y ) 2 = ( x , y , u , u x , u y ), a ( x , y ) 2 + 2 b( x , y ) x xy y a + b + c 0
, b 2 - ac > 0, , b 2 - ac = 0, , b2 - ac < 0.

(1)

a ( x, y )( dy ) 2 - 2b( x, y )dxdy + c( x, y )( dx ) 2 = 0 ,
(1) , :

ady - (b + b2 - ac )dx = 0, ady - (b - b2 - ac )dx = 0.
. : b2 - ac > 0.

(2)

( x , y ) = c1 , ( x , y ) = c2 (2) . (1). = ( x, y ), = ( x, y )
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(1)


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Maple

2u = 1 ( , , u, u , u ).
: b2 - ac = 0. ( x, y ) = c (2) (1). = ( x, y ), = ( x, y ) ,

( x, y ) -- ,
, (1)

2u = 1 ( , , u, u , u ). 2
: b2 - ac < 0. ( x, y ) + i ( x, y ) = c ,

( x, y ), ( x, y ) -

. = ( x, y ), = ( x, y ) (1)

2u 2u + = 1 ( , , u, u , u ). 2 2
:
2u 2u 2u u u - 2 + 2 + 3 = 0; a) 3 2 + 2 x xy y x y

b) c)

2u 2u 2u u u +2 + 2 + 3 - 5 + 4u = 0; 2 x xy y x y 2u 2u 2u u u +4 +3 2 +5 + + 4u = 0; 2 x xy y x y
2

2u 2 x 2u u -e - 4 y2 + 4u = 0; d) 4 y 2 2 x y y

e) f)

2u 2u 2u 2u 2u +2 -2 + 2 2 + 6 2 = 0; x 2 yx xz y z 2u 2u u u u - + + - = 0; yx xz x y z

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Maple

g)

2u 2u 2u 2u 2u 2u 2u +2 +2 2 +2 +2 + 2 2 + 3 2 = 0; x 2 yx y yt yz z t

2u 2u 2u 2u 2u +2 + + + = 0; h) x 2 yx xt xz tz

i) j)

2u 2u - y 2 = 0; x 2 y y 2u 2u - x 2 = 0; x 2 y
2

2 2u 2u = 0; k) x 2 + y x y 2

l) (1 + x 2 ) m) 4 y
2

2u 2u u + (1 + y 2 ) 2 + y = 0; 2 x y y

2u 2 x 2u -e = 0; x 2 y 2

4. u(x,t) C 2 (t > 0) C 1 (t 0) , t>0

2u = a 2 u + f ( x, t ) 2 t

= u0 ( x ), ut t =0 = u1 ( x ) f , u0 1 - . u
t =0

n=1

1 1 u( x, t ) = [u0 ( x + at ) + u0 ( x - at )] + 2 2a
n=2

x + at

x - at



1 u1 ( )d + 2a f ( , )d d



t x + a ( t - )

f ( , )d d ;

0 x - a ( t - )

u( x, t ) = 1 2 a

2 a

1

t

0 - x < a ( t - )



a (t - ) - - x
2 2 2

2

+ u0 ( )d ;

- x < at



u1 ( )d a t --x
22

+

1 2 a t

- x < at



a t --x
22

2

n=3
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Maple

u ( x, t ) = 1 4 2 a 2 t

1 4 a

2

- x < at



-x 1 f ,t - d + -x a
- x = at

- x = at



1 1 u1 ( )dS + 4 a 2 t t



u0 ( )dS

.

u(x,t) C 2 (t > 0) C (t 0) , t>0

u = a 2 u + f ( x, t ) t
u
t =0

= u0 ( x ).

f , u0 - .
2

u ( x, t ) =

1 (2a t )
n

R



u0 ( )e
x -
2

-

x -
2

4a t

d +
.

n


0R

t

f ( ,

)
n

n

2a ( t - )

e

-

4 a 2t

d d

. : u, :
u = 0


u = f,

f ­ . , f u . ( r, ) .
1 u 1 2u = 0, 0 r < a, - < < . + r r r r 2 2

. .

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. .

Maple

:
u
r =0

= O (1), u

r =a

= f ( ).

, , ­ :
u
=-

=u

=

,

u

=
=-

u

.
=

.
u = R ( r ) ( ) .


'' ( ) + ( ) = 0, r ( rR ' ( r ))' - R( r ) = 0.



=-

=

=

, '

= -

= '

=

.

.
( ) = A cos( ) + B sin( ), 0.


2 B sin( ) = 0, 2 A sin( ) = 0.

sin( ) 0 , ==0 . ,
sin( ) = 0 = n = n 2 , n .
n ( ) = Bn sin( n ) + An cos( n ).

r ( rR ' ( r ))' - R( r ) = 0 , = n = n 2 , n . R ( r ) = r s .

s 2 - n 2 = 0, s = ± n. , Rn ( r ) = Cn r n + Dn r - n , n , R0 ( r ) = C0 + D0 ln( r ).
, , u ,

un = ( N n sin(n ) + M n cos(n )), n , u0 = M 0 , n = 0.
, u
r =a



= f ( ) .
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. .

Maple

u

r =a

= M0 +

(
n =1



N n sin( n ) + M n cos( n ) ) a n = f ( ) .

f ( ) ,

a Mn =
n

1




1

-



f ( ) cos( n )d , f ( ) sin( n )d , f ( )d .

an Nn = M0 =





-



1 2



-



, . -. , , . , , , -. ­ .

X '' ( x ) + X ( x ) = 0, 0 < x < 2 ; X (0) = X (2 ), X ' (0) = X ' ( 2 ).
sin( x )
, ,

( x, ) = cos( x ), ( x, ) =



.

-

X ( x ) = A cos( x ) + B

sin( x )



.

,

A[1 - cos( 2 )] - B

sin( 2 )



= 0,

A sin( 2 ) + B[1 - cos( 2 )] = 0.
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. .

Maple



( ) =

1 - cos( 2 ) sin( 2 )

= 0 ( ) = 2(1 - cos( 2 )) = 0 1 - cos( 2 )

-

sin( 2 )

cos( 2 ) = 1, n = n 2 , n = 0,1, 2...


X n ( x ) = An cos( nx ) + Bn
.

sin(nx ) . n

1. (n=1)

2u 2u = + 6, u a. t 2 x 2

t =0

= x 2 , ut

t =0

= 4 x; = 0; = 0; = ( x ).

2u 2u = + sin x, u b. t 2 x 2

t =0

= sin x, ut

t =0

2 2u 2u =a + sin wx, u c. t 2 x 2 2 2u 2u =a + f ( x , t ), u d. t 2 x 2

t =0

= 0, ut

t =0

t =0

= ( x ), ut

t =0

2. (n=2)

2u =u + x 3 - 3xy 2 , u a. 2 t 2u = 3u + x 3 + y 3 , u b. 2 t 2u = a 2 u , u c. 2 t
t =0

t =0

= e x cos y , ut

t =0

= e x sin y;

t =0

= x 2 , ut = r4;

t =0

= y2;

= r 4 , ut

t =0

2u = a 2 u + r 2 e t , u d. 2 t

t =0

= 0, ut

t =0

= 0; = ( x , y ).

2u = a 2 u + f ( x , y , t ) , u e. 2 t
3. (n=1)

t =0

= ( x, y ), ut

t =0

u 2u = 4 2 + t + et , u a. t x

t =0

= 2;
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Maple

b. c.

u 2 u = 2 + 3t 2 , u t x

t =0

= sin x; = e- x ;
2

u 2 u = + sin t , u t x 2

t =0

2 u 2u =a + f ( x , t ), u d. t x 2

t =0

= ( x ).

4. (n=2) a. b.

u =u + et , u t

t =0

= cos x cos y;
= xye
- x2 - y
2

u =u + cos t , u t u =u, u t
t =0

t =0

;

c. 2 d.

= cos xy;

u = a 2 u + f ( x, y, t ), u t

t =0

= ( x , y ).

5.
u 2u - a 2 2 = 0, 0 < x < l , t > 0, t x

u ( x, 0) = x (l - 0) u (0, t ) = (t ), u (l , t ) = (t ). 6.
2u 2u = a 2 2 + f ( x, t ), 0 < x < l , t < + t 2 x u u (0, t ) = µ1 (t ), ( l , t ) = µ 2 ( t ), x x u( x, 0) = ( x ), u ( x , 0) = ( x ) . t

7.
2u 2u = a 2 2 + f ( x, t ), 0 < x, t < + t 2 x u (0, t ) = µ (t ), x u( x, 0) = ( x ), u ( x , 0) = ( x ) . t

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Maple

8.
u 2u = a 2 2 + f ( x, t ), 0 < x < l , 0 < t < + t x u (l , t ) = µ2 (t ) u(0, t ) = µ1 (t ), x u ( x , 0) = ( x ) .

9.

2 u 2u =a + f ( x, t ), 0 < x, t < + t x 2 u(0, t ) = µ (t ), u( x, 0) = ( x ).

10.
u = 0, a < r < b, (0 a < b + ) u r = f a ( ),
r =a

u r

= f b ( ).
r =b

11.
2u 2u + = 0, 0 < x < a, 0 < y < b, x 2 y 2 u x = f 0 ( y ), u
x =0 x =a

= f a ( y ),

u

y =0

= g 0 ( x ),

u y

= g b ( x ).
y =a

12.
0, - 1 x 0, f ( x) = 2 x , 0 x l


y ''+ y = 0, y ( - l ) = y ( l ), y '( - l ) = y '( l ), x [ - l , l ].

13. f ( x ) = x 2
d dx 2 dy 2 x + x y = 0, dx y (0) = O (1), y '( a ) = 0, x [0, a ].

14. f ( x ) = c 2 - x 2
d 2 dy 2 x + x y = 0, dx dx
y ''+ y = 0, y ( a ) = 0,

y (0) = O (1),

y '( a ) + hy ( a ) = 0,

x [0, a ], h > 0.

15. ­ :
y '(b) = 0, x [a , b].
http://ani.cs.msu.su 15

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Maple

16. , =0 , =l . . 5. . . , ­ . : ; , .

. . , u = u1 + u2 , , . . , . . z . ; =0 T0 , = ­ . .

. .

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. .

Maple

2T T kt , > 0, - = 0, 0 < x < a, = 2 x c T
x =0

= T0 , T

x =a

= 0,

T

=0

= ( x ),

( x ) - .

T = T1 ( x ) + T2 ( x, ) . T1 ( x ) , . T1 ( x ) :
2T1 = 0, T1 x 2
x =0

= T0 , T1

x =a

x = 0. T1 ( x ) = T0 1 - a

,

. T2 ( x ) :
2T2 T - = 0, x 2 T2 x =0 = T0 , T2
x

x =a

= 0,

T

=0

= ( x ) - T1 ( x ).

( x ) - T1 ( x ) = ( x ) - T0 1 - - . a . . 1. .
1 T r r r r T r0 = O (1) - T Q =- , 0 < r < a, > 0, k
r =a

,T

= 0,

T

=0

= ( r ),

( r ) - .

2. .
1 u r r r r u
r 0

2 q( r ) T 1u - 2 2 =- sin( wt ), 0 < r < a, > 0, v 2 = 0 T0 v t u = , u r =a = 0, = w( r ), u t = 0 = ( r ), t t =0

T0 - , - , ( r ), ( r ) ( ).

. .

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Maple

3. .
2u 2u + = 0, 0 < x < a , 0 < y < b, x 2 y 2 u
x =0

= f 0 ( y ), u

x =a

= f a ( y ),

u

y =0

= 0 ( x ), u

y =a

= b ( x ).


2u 2u 2u u u -6 + 10 2 + - 3 = 0. 2 x xy y x y

.
a1 2u 2u 2u u u + a2 + a3 2 + a4 + a5 + a6u + a7 = 0. 2 x xy y x y

> a:=1, -6, 10, 1, -3, 0, 0; a:=1,-6,10,1,-3,0,0 > equ:=a[l]*diff(u(x,y),x,x)+a[2]* diff(u(x,y),x,y)+a[3]* diff(u(x,y),y,y)+ a[4]* diff(u(x,y),x)+ a[5]* diff(u(x,y), y)+ a[6]*u+a[7]=;
2 2 2 u( x, y ) + 10 2 u( x, y ) + u( x, y ) - 3 u( x, y ) = 0 equ:= 2 u( x, y ) - 6 y x xy y x

> eq:=lhs(equ);
u( x, y ) + 10 2 u( x, y ) + u( x, y ) - 3 u( x, y ) eq:= 2 u( x, y ) - 6 y x xy y x
2





2





2









>A:=linalq[matrix](2,2,[coeff(eq, diff(u(x,y),x,x)), coeff(eq, diff(u(x,y),x,y))/2, coeff(eq, diff(u(x,y),x,y))/2, coeff(eq, diff(u(x,y),y,y))]); >Delta:=simplify(linalg[det](A));
1 -3 A := - 3 10 := 1

, - .

. .

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Maple

>A[1,1]*z^2-2*A[1,2]*z+A[2,2]=0; > res1:=solve(A[1,1] *z^2-2*A[1,2]*z+A[2,2],z);
z 2 + 6 z + 10 = 0 res1:= -3 + I , -3 - I

>res2:={seq(dsolve(diff(y(x),x)=res1[i], y(x), i=1,2)} res2:={y(x)=-(3-I)x+_C1, y(x)=-(3+I)x+_C1} , . >res2:=subs(y(x)=y, res2); res2:={y(x)=-(3-I)x+_C1, y(x)=-(3+I)x+_C1} >{seq(solve(res2[i],-C1), i=1..nops(res2))}; {y+3x+xI,y+3x-xI} itr:= {xi=coeff(%[1],I),eta=%[1]-coeff(%[1],I)*I};
itr := { = y + 3, = x}

>tr:=solve(itr,{x,y}); PDEtools[dchange](tr, eq, itr, [eta, xi],simplify)=0
itr := { y = - 3, x = }

2 2 u( , ) + 2 u( , ) + u( , ) = 0 2

. 1. . Maple. , 2002. 2. . . . . . . , 1977. 3. . . . . , 1981. 4. . . . , 2004 5. . . . . , 1982. 6. .. . Maple. , 2006. 7. . . , . . . Mathcad 12, MATLAB 7, Maple 9. : , 2006, 496.

. .

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Maple

8. .. . . MAPLE: . . , 2006. . · · http://writers.fultus.com/aladjev/book01.html - http://www.cybertester.com/ - Cyber Tester, LLC,

Maple. - . . LIFT CYCLE, . , · , Maple. http://maple.bug-list.org/ - Maple, . GEMM http://www.CAS-testing.org/, Cyber Tester, LLC. · · · · · · http://users.kaluga.ru/math/ " ", . http://www.math.rsu.ru/mexmat/kvm/MME/courses/prog/ - http://www.geocities.com/optomaplev - Maplehttp://www-europe.mathworks.com ( http://www.mathworks.com) - http://www.maplesoft.com - Waterloo Maple (Maple) http://www.wolfram.com - Wolfram Research (Mathematica) ( Maple MatLab). , . MathWorks (Matlab)

. .

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