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:

































 

 

 

 


- .




- , : 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), .





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