equ:=
> eq:=lhs(equ);
eq:=
> A:= linalg[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));
, .
.
>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);
> res2:={seq(dsolve(diff(y(x),x)=res1[i],y(x)),i=1..2)};
, . .
> res2:=subs(d(x)=d,res2);
> {seq(solve(res2[i],_C1),i=1..nops(res2))};
> itr:= {xi=coeff(%[1],I),eta=%[1]-coeff(%[1],I)*I};
>tr:=solve(itr,{x,y});PDEtools[dchange](tr,eq,itr,[eta,xi], simplify)=0;
2.8
Maple. .
> a[1]:= 1;a[2]:=-2;a[3]:=-3;a[4]:=0;a[5]:=0;a[6]:=0;a[7]:=0;
> eq:=diff(u(x,y),x,x)-2*diff(u(x,y),x,y)-3*diff(u(x,y),y,y)=0;
eq:=
Maple - mapde(eq,canom). . , .
> with(PDEtools):
> mapde(eq,canom);
&where
> op(%);
,
> pdsolve(%[1]);
> sol:=u(x,y)=subs(%%[2],rhs(%));
.
> simplify(subs(sol,eq));
> simplify(lhs(%));
2.9 .
Maple. .
> a[1]:= 1;a[2]:=-4;a[3]:=-5;a[4]:=0;a[5]:=0;a[6]:=0;a[7]:=0;
> eq:=diff(u(x,y),x,x)-2*diff(u(x,y),x,y)-3*diff(u(x,y),y,y)=0;
eq:=
Maple - mapde(eq,canom). . , .
> with(PDEtools):
> mapde(eq,canom);
&where
> op(%);
,
> pdsolve(%[1]);
> sol:=u(x,y)=subs(%%[2],rhs(%));
.
> simplify(subs(sol,eq));
> simplify(lhs(%));
2.4 ( Ē)
:
- , ;
- ;
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- , , .
³, .
, , , .
:
. (2.26)
(2.26)
. (2.27)
(2.27)
. (2.28)
(2.4), :
, (2.29)
. (2.30)
(2.30) ,
. (2.31)
(2.31) ,
. (3.7)
. (2.32)
(2.33), (2.28) (2.32)
. (2.34)
, , .
, (2.35)
. (2.36)
г (2.36)
. (2.37)
(2.37) 0 ,
, (2.38)
. (2.39)
, (2.39) :
. (2.40)
, (2.35) (2.40),
. (2.41)
(2.41)
. (2.42)
(2.28)
,
(2.43)
.
ϳ (2.43) (2.34)
. (2.44)
(2.44) Ē.
,
(2.45)
, , (2.46)
, , Ē
. (2.47)
2.10 Ē , , .
(2.44)
.
.
2.11 Ē , , .
(2.47).
.
,
.
2.12 , , . Ē.
Ē .
> Int(cos(z),z=x-at..x+at)=int(cos(x),x=x-at..x+at);
2.13 , , . Ē.
, .
Maple.
> a:=1;f(xi,eta):=6;
> with(student):
> 1/2*a*Doubleint(f(xi,eta),xi=x-t+eta..x+t-eta,eta=0..t);
> z:=value(%);
> psi(xi):=4*x;a:=1;
>d:=1/2*a*Int(psi(xi),xi=x-a*t..x+a*t)=1/2*a*int(psi(xi),xi=x-a*t..x +a*t);
,
2.14 , , , , , . Ē.
|
|
.
> Eqn:=diff(u(x,t),t$2)-a^2*diff(u(x,t),x$2)=0;
pdsolve() .
> pdsolve(Eqn);
.. , Eqn . ³, , . . u () x t.
> u:=unapply(rhs(%),x,t);
, .
: , (D[2](u)).
> D[2](u)(x,0)=0;
.
> dsolve(%,_F1(x));
, _1 . , _F1 F.
> _F1:=F;
> _F2:=x->F(-x);
³, :
> u(x,t);
> u(x,0);
> f:=x->sin(x);
> F:=1/2*f;
> a:=1;
> u(x,t);
animate() (. 2.1-2.2).
>plots[animate](u(x,t),x=-10..10,t=0..15, view=-2..2, scaling= unconstrained, numpoints=100,titlefont=[HELVETICA,BOLD,12]);
. t=1c.
. t=4c.
1. ?
2. .
3. .
4. .
5. .
6. .
7. .
8. .
9. .
10. .
11. , () .
12. ?
13. Ē.
14. Ē , .
15. .
16. .
17. ?
18. ? .
19. ? ? .
20. ? ? .
21. ? ? .
2.1
, .
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15 16. | 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. |
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