- , : x - , t (. 4.3).
. 4.3. .
:
; , , (4.55)
. (4.56)
- , .
- (4.54):
, , (4.57)
(4.58)
, .
(4.54) , , :
; ; ; . (4.59)
C (4.59) (4.57) :
; (4.60)
; (4.61)
, ; (4.62)
; (4.63)
. (4.64)
, , (4.60) (4.61):
;
, (4.65)
- (4.60)-(4.64)
(4.66)
. (4.67)
, , .. ( ).
(4.66) :
. (4.68)
, (4.69)
,
, . (4.70)
, ( ) , . - , .
(4.69) (4.68), :
. (4.71)
.
; ; ; , .
- .
M- ( , ):
function blow_f
g=input('g=');
s=input('s=');
n=input('n=');
L=300;P=300;
h=L/(n+1);
alfa=10^8*(100+g+s);
x=0:h:L;
a0=6*eye(n);a0(1,1)=5;a0(n,n)=5;
a1=ones(n-1,1);
a2=ones(n-2,1);
A=a0-4*(diag(a1,-1)+diag(a1,1))+diag(a2,-2)+diag(a2,2)
A=alfa*A/h^4;
F=zeros(n,1);F((n+1)/2)=P;
sq_A=sqrtm(A);
fJ=sqrt(eig(A));
t0=pi/(4*fJ(n));
tmax=125*t0;
nt=3;
t=[t0,tmax/2,tmax];
res=zeros(nt,n+2);
fprintf('\n Y(x,t)\n')
for i=1:nt
Y_t=inv(sq_A)*funm(sq_A*t(i),'sin')*F;
res(i,2:n+1)=Y_t';
fprintf('Y(%6.4f):',t(i)),fprintf('%8.4f',res(i,:)),fprintf('\n')
end
hold on
plot(x,res(1,:),'.-')
plot(x,res((nt+1)/2,:),'o-.r')
plot(x,res(nt,:),'*:g')
grid on
s1=sprintf('t=%6.4f',t(1));
s2=sprintf('t=%6.4f',t((nt+1)/2));
s3=sprintf('t=%6.4f',t(nt));
legend(s1,s2,s3,0)
title(sprintf('Y(x,t)=-inv(sqrt(A))*sin(sqrt(A)t))*F\n%s %s %s', s1,s2,s3))
|
|
:
g=3
s=12
n=7
A =
5 -4 1 0 0 0 0
-4 6 -4 1 0 0 0
1 -4 6 -4 1 0 0
0 1 -4 6 -4 1 0
0 0 1 -4 6 -4 1
0 0 0 1 -4 6 -4
0 0 0 0 1 -4 5
Y(x,t)
Y(0.0027): 0.0000 -0.0001 -0.0053 0.0217 0.7703 0.0217 -0.0053 -0.0001 0.0000
Y(0.1673): 0.0000 2.2085 4.2797 5.8565 5.5146 5.8565 4.2797 2.2085 0.0000
Y(0.3346): 0.0000 -1.3399 -3.5149 -3.9787 -4.1294 -3.9787 -3.5149 -1.3399 0.0000
>>
. A :
Ø sqrtm(A),
Ø funm(sqrtm(A),sin).
Function | Syntax for Evaluating Function at Matrix A |
exp | funm (A, @exp) |
log | funm(A, @log) |
sin | funm(A, @sin) |
cos | funm(A, @cos) |
sinh | funm(A, @sinh) |
cosh | funm(A, @cosh) |
, , expm(A), logm(A) sqrtm(A), .