Funciones en Matlab 2

7/23/2019 Funciones en Matlab 2

1/21

function [Ra] =RaicesBesselMatriz(N,M,epsi)

Ra=zeros(N,M);

for n=1:N

[R]=RaicesBessel(n,M,epsi);

Ra(n,:)=[R];

end

end

DETERMINAR O ON!TR"IR #A! $"NIONE! DE BE!!E#

function [%] = felipe( f,p,N,# )

%Desarrolla la funcion f(x) en serie de ourier Bessel a !ra"#s de p,

%ci$ B suandos& 'a funcin es! dada en *,'&

x=[*:*&1:(+&pi&N-+*)];

.=/(x) $essel0(p,x);

.1=.(x); =1;

M=len!.(x);

for n=1:M21

if (.1(n)&.1(n-1)3*) 4()=[x(n)]; B()=[x(n-1)];

=-1;

end

end

R=len!.(4);

for n=1:R

iz=[4(n)]; de=[B(n)];

5.ile (de2iz)6*&******1

ed=(iz-de)&7+;

if .(iz)&.(ed)3* de=ed;

else

iz=ed;

end

end

raiz(n)=ed;

end

for n=1:N

f1=/(x) x&f(x)&$essel0(p,raiz(n)&x&7');

c(n)=in!eral(f1,*,'); c(n)=+&c(n)&7(('&$essel0(p-1,raiz(n)))&8+);

end

9=/(x) *&x;

for n=1:N

9=/(x) 9(x)-c(n)&$essel0(p,raiz(n)&x&7');

nd

O&IONA# =[*: *&*1:'];

1=f(x); %f1=f(u)

91=9(u);

iure, plo! (u,91,


2/21

ARA A#"#AR #O! M !"MANDO! ORDEN &

function [u] = !erie$ourierBessel(f,p,M,epsi)

[R]=BesselRaices(p,M,epsi);

for =1:M

f1=/(x) x&f(x)&$essel0(p,R()&x);

>()=in!eral(f1,*,1);

>()=>()&+&7($essel0(p-1,R())&8+);end

u=/(x) *&x;

for =1:M

u=/(x) u(x)->()&$essel0(p,R()&x);

end

x=[*:*&*1:1];

fun1=f(x); u1=u(x);

?ure(),plo!(x,fun1,@r@),.old on,rid on, plo!(x,u1,@$@)

end

f='() *./01 M=*- epsi=+/+++++* p=-

&ARA ENONTRAR #O! N"MERO"! M RAIE! N N"MERO DE RAIE! 2 &

E# 3RADO DE BE!!E#

function [R]=BesselRaices(p,M,epsi)

x=[*:*&1:(pi&M)&+];

'=len!.(x); r=*;

for l=1:'21

if $essel0(p,x(l))&$essel0(p,x(l-1))3*

r=r-1;

4(r)=[x(l)]; B(r)=[x(l-1)]; end

end

for =1:M

iz=4(); de=B();

5.ile(de2iz)6epsi;

ed=(de-iz)&7+;

if $essel0(p,iz)&$essel0(p,ed)3*

de=ed;

else

iz=ed; end

end

R()=ed;

end

end

p=pi N=*- epsi==+/+++++*

[R]=BesselRaices(p,M,epsi)

function [%,R] =Bessel4uan(f,p,M,#,epsi)

[R]=BesselRaices(p,M,epsi);for n=1:M

f1=/(x) x&f(x)&$essel0(p,R(n)&(x&7'));

>(n)=in!eral(f1,*,');


3/21

d='&$essel0(p-1,R(n));

d=d&8+;

d=d&7+;

>(n)=>(n)&7d;

end

9=/(x) *&x;

for n=1:M

9=/(x) 9(x)->(n)&$essel0(p,R(n)&(x&7'));end

u=[*:*&1:'];

91=9(u);

f1=f(u);

?ure(),plo!(u,91,@r@),.old on,rid on,plo!(u,f1,@$@);

end

$='(5) ep(sin(cos())) #=*- M=*- p=- epsi=+/++++*

[%,R] =Bessel4uan(f,p,M,#,epsi)

function [u] =On6a7o8o9enea(f,9,a,#,N)for n=1:N

f1=/(x) f(x)&sin(n&pi&x&7');

4(n)=in!eral(f1,*,')&+&7';

1=/(x) (x)&sin(n&pi&x&7');

B(n)=+&in!eral(1,*,')&7(n&pi&a);

end

u=/(x,!) *&x-*&!;

for n=1:N

u=/(x,!) u(x,!)-(4(n)&cos(a&n&pi&!&7')

-B(n)&sin(a&n&pi&!&7'))&sin(n&pi&x&7');end

x=[*:*&1:']; !=[*:1:1***];

M=len!.(!);

for =1:M

u1=u(x,!());

plo!(x,u1,@r@,@'ine9id!.@,A), .old on, rid on, .old o,axis([* ' 21 1])

pause(*&1)

end

end

function [u,A+,A,] = $ourierBessel:6(f,#,N,M,epsi)

[R]=RaicesBessel(*,M,epsi);

[Ra]=RaicesBesselMa!riz(N,M,epsi);

4*=zeros(1,M); 4=zeros(N,M); >=zeros(N,M);for =1:M

f1=/(?,r) r&f(r,?)&$essel0(*,R()&r&7');

d=('&$essel0(1,R()))&8+;


4/21

d=d&pi&7+;

4*()=in!eral+(f1,2pi,pi,*,');

4*()=4*()&7d;

end

for n=1:N

for =1:M

fcos=/(?,r) r&f(?,r)&$essel0(n,Ra(n,)&r&7')&cos(n&?);

fsin=/(?,r) r&f(?,r)&$essel0(n,Ra(n,)&r&7')&sin(n&?); 4(n,)=in!eral+(fcos,2pi,pi,*,');

>(n,)=in!eral+(fsin,2pi,pi,*,');

d=('&$essel0(n-1,Ra(n,)))&8+;

d=d&pi&7+;

4(n,)=4(n,)&7d; >(n,)=>(n,)&7d;

end

end

u=/(?,r) *&r-?&*;

for =1:M

u=/(?,r) u(?,r)-4*()&$essel0(*,R()&r&7');end

u=/(?,r) +&u(?,r);

for n=1:N

for =1:M

u=/(?,r) u(?,r)-4(n,)&$essel0(n,Ra(n,)&r&7')&cos(n&?);

u=/(?,r) u(?,r)->(n,)&$essel0(n,Ra(n,)&r&7')&sin(n&?);

end

end

C=[*:*&1:']; i=[*:*&1:+&pi];

[i,C]=es.rid(i,C);[i,C]=pol+car!(i,C);

u1=u(i,C);

?ure(),surf(i,C,u1),.old on,rid on

end

function [;] =]=ourierBessel+d(f,',N,M,epsi);

5=/(?,r,!) r&*-?&*-!&*;

for =1:M

5=/(?,r,!) 5(?,r,!)-4*()&cos(a&R()&!&7')&$essel0(*,R()&r&7');end

5=/(r,?,!) 5(r,?,!)&+;

for n=1:N


5/21

for =1:M

5=/(?,r,!) 5(r,?,!)

-4(n,)&cos(a&Ra(n,)&!&7')&$essel0(n,Ra(n,)&r&7')&cos(?&n);

5=/(?,r,!) 5(r,?,!)

->(n,)&cos(a&Ra(n,)&!&7')&$essel0(n,Ra(n,)&r&7')&sin(?&n);

end

end

s=[*:*&1:']; i=[*:*&*1:+&pi];[i,s]=es.rid(i,s);

[i,s]=pol+car!(i,s);

!=[*:1:**]; E=len!.(!);

for n=1:E

51=5(i,s,!(n));

surf(i,s,51);.old on, rid on,axis([2+&pi,+&pi,2+&',+&',21,1]),"ie5(),;

pause(*&A)

.old o;

end

end

function [u]=Ta8orircular:(f,a,#,N,M,epsi)

[R]=RaicesBessel(*,M,epsi);


4*=zeros(1,M); B*=zeros(1,M); 4=zeros(N,M); B=zeros(N,M);

>=zeros(N,M); D=zeros(N,M);

for =1:M

f1=/(?,r) r&f(?,r)&$essel0(*,R()&r&7');

4*()=in!eral+(f1,2pi,pi,*,');

d=pi&('&$essel0(1,R()))&8+; 4*()=4*()&7d;

end

for n=1:N

for =1:M

f1=/(?,r) r&f(?,r)&$essel0(n,Ra(n,)&r&7')&cos(n&?);

f+=/(?,r) r&f(?,r)&$essel0(n,Ra(n,)&r&7')&sin(n&?);

4(n,)=in!eral+(f1,2pi,pi,*,');

>(n,)=in!eral+(f+,2pi,pi,*,');

d=pi&('&$essel0(n-1,Ra(n,)))&8+;

4(n,)=4(n,)&+&7d; >(n,)=>(n,)&+&7d; end

end

u=/(?,r,!) ?&*-r&*-!&*;

for =1:M

u=/(?,r,!) u(?,r,!)- (4*()&cos(a&R()&!&7')

-B*()&sin(a&R()&!&7'))&$essel0(*,R()&r&7');

end

for n=1:N

for =1:M

u=/(?,r,!) u(?,r,!)-(4(n,)&cos(a&Ra(n,)&!&7')-B(n,)&sin(a&Ra(n,)&!&7'))&$essel0(n,Ra(n,)&r&7')&cos(n&?);

u=/(?,r,!) u(?,r,!)-(>(n,)&cos(a&Ra(n,)&!&7')

-D(n,)&sin(a&Ra(n,)&!&7'))&$essel0(n,Ra(n,)&r&7')&sin(n&?);


6/21

end

end

!=[*:1:+**]; ?=[*:*&1:(+&pi)]; r=[*:*&*A:'];

[?,r]=es.rid(?,r);

[?,r]=pol+car!(?,r);

C=len!.(!);

?ure(),.old on,rid on

for s=1:C u1=u(?,r,!(s));

surf(?,r,u1), "ie5(2F&A,*);axis([2pi pi 2' ' 2*& *&]);

pause(*&*1);

.old o

end

end

function [u] =Ta8or!in>(f,9,a,#,N,epsi)

[R]=RaicesBessel(*,N,epsi);

4=zeros(N,1); B=zeros(N,1);for n=1:N

f1=/(r) r&f(r)&$essel0(*,R(n)&r&7');

f+=/(r) r&(r)&$essel0(*,R(n)&r&7');

4(n)=in!eral(f1,*,');

B(n)=in!eral(f+,*,');

d=('&$essel0(1,R(n)))&8+;

d1=d&a&R(n);

4(n)=4(n)&+&7d;

B(n)=B(n)&+&'&7d1;

endu=/(?,r,!) r&*-*&!-*&?;

for n=1:N

u=/(?,r,!) u(?,r,!)-(4(n)&cos(a&R(n)&!&7')

-B(n)&sin(a&R(n)&!&7'))&$essel0(*,R(n)&r&7');

end

u=/(?,r,!)-u(?,r,!)-*&sin(?);

r=[*:*&*1:']; ?=[2pi:*&1:pi-1]; !=[*:1:1***];

[?,r]=es.rid(?,r);


?ure();M=len!.(!);

for =1:M

u1=u(?,r,!());

surf(?,r,u1), .old on, rid on;axis([2pi pi 2' ' 2*& *&]),"ie5(2*,F&A);

pause(*&*1);

.old o;

end

end

function [u]=Ta8orircular(f,9,a,#,N,M,epsi)[R]=RaicesBessel(*,M,epsi);


4*=zeros(1,M); B*=zeros(1,M); 4=zeros(N,M); B=zeros(N,M);


7/21

>=zeros(N,M); D=zeros(N,M);

for =1:M

f1=/(?,r) r&f(?,r)&$essel0(*,R()&r&7');

1=/(?,r) r&(?,r)&$essel0(*,R()&r&7');

4*()=in!eral+(f1,2pi,pi,*,');

B*()=in!eral+(1,2pi,pi,*,');

d=pi&('&$essel0(1,R()))&8+;

d1=d&a&R()&7'; 4*()=4*()&7d; B*()=B*()&7d1;

end

for n=1:N

for =1:M

f1=/(?,r) r&f(?,r)&$essel0(n,Ra(n,)&r&7')&cos(n&?);

f+=/(?,r) r&f(?,r)&$essel0(n,Ra(n,)&r&7')&sin(n&?);

1=/(?,r) r&(?,r)&$essel0(n,Ra(n,)&r&7')&cos(n&?);

+=/(?,r) r&(?,r)&$essel0(n,Ra(n,)&r&7')&sin(n&?);

4(n,)=in!eral+(f1,2pi,pi,*,');

>(n,)=in!eral+(f+,2pi,pi,*,'); B(n,)=in!eral+(1,2pi,pi,*,');

D(n,)=in!eral+(+,2pi,pi,*,');

d=pi&('&$essel0(n-1,Ra(n,)))&8+;

d1=d&a&Ra(n,);

4(n,)=4(n,)&+&7d; >(n,)=>(n,)&+&7d;

B(n,)=B(n,)&+&'&7d1; D(n,)=D(n,)&+&'&7d1;

end

end

u=/(?,r,!) ?&*-r&*-!&*;

for =1:M u=/(?,r,!) u(?,r,!)- (4*()&cos(a&R()&!&7')

-B*()&sin(a&R()&!&7'))&$essel0(*,R()&r&7');

end

for n=1:N

for =1:M

u=/(?,r,!) u(?,r,!)-(4(n,)&cos(a&Ra(n,)&!&7')

-B(n,)&sin(a&Ra(n,)&!&7'))&$essel0(n,Ra(n,)&r&7')&cos(n&?);

u=/(?,r,!) u(?,r,!)-(>(n,)&cos(a&Ra(n,)&!&7')

-D(n,)&sin(a&Ra(n,)&!&7'))&$essel0(n,Ra(n,)&r&7')&sin(n&?);

endend

!=[*:1:1***]; ?=[*:*&*1:(+&pi)]; r=[*:*&1:']; lf=len!.(?); lr=len!.(r);

G=lf&lr;

[?,r]=es.rid(?,r);


C=len!.(!);

?ure(),.old on,rid on

for s=1:C

u1=u(?,r,!(s));

surf(?,r,u1), "ie5(*,H*);axis([2+&pi +&pi 2+&' +&' 2+ +]); pause(*&*1);

.old o;

end


8/21

end

function [z] =#aplasian*D(,?,a,,@)

M=len!.(I);

=di(I,+);

z=zeros(M,1);

z(1)=[a];

z(M)=[$];d=ones(M2+,1);

d*=2+&d;

B=[d d* d];

$=[21 * 1];

>=spdias(B,$,M2+,M2+);

D=(.&8+)&di(I,+);

D(1)=[D(1)2z(1)];

D(M2+)=[D(M2+)2z(M)];

J=in"(>)D@;

for =+:M21 z()=[J(21)];

end

?ure(),plo!(x,I,@r@),.old on, rid on,plo!(x,z,@$@);

end

function [] = #aplasianMatriz(A,@)

[M,N]=size(4);

>=zeros(M2+,N2+);

for =+:M21

for n=+:N21

>(21,n21)=[4(-1,n)2K&4(,n)-4(21,n)-4(,n-1)-4(,n21)]&7(.&8+);

end

end

end

METODO DE DI$ERENIA! $INITA! &ARA DARNO! #A OORDENADA DE

#O!


9/21

d=[dp;*]; dM=[*;dp];

d1=d; dM1=dM;

for =+:

d1=[d1;d];

dM1=[dM;dM1];

end

B=[d d1 do dM1 d];

$=[2K 21 * 1 K];>=spdias(B,$,1+,1+);

D=zeros(1+,1);

D(1)=[(.&8+)&1(+,+)2u(1,+)2u(+,1)];

D(+)=[(.&8+)&1(+,)2u(1,)];

D()=[(.&8+)&1(+,K)2u(1,K)];

D(K)=[(.&8+)&1(+,A)2u(1,A)2u(+,L)];

D(A)=[(.&8+)&1(,+)2u(,1)];

D(L)=[(.&8+)&1(,)];

D(F)=[(.&8+)&1(,K)];

D()=[(.&8+)&1(,A)2u(,L)];D(H)=[(.&8+)&1(K,+)2u(A,+)2u(K,1)];

D(1*)=[(.&8+)&1(K,)2u(A,)];

D(11)=[(.&8+)&1(K,K)2u(A,K)];

D(1+)=[(.&8+)&1(K,A)2u(A,A)2u(K,L)];

O=in"(>)&D;

end

function [u] =#aplace($,3,a,,c,6,@)

x=[a:.:$]; I=[c:.:d];M=len!.(I); N=len!.(x);

[x,I]=es.rid(x,I); 1=(x,I);

1=(x,I);

u=zeros(M,N);

u(1,:)=[1(1,:)]; u(M,:)=[1(M,:)];

u(:,1)=[1(:,1)]; u(:,N)=[1(:,N)];

s=(M2+)&(N2+);

d=ones(s,1);

d*=2K&d;

dp=ones(N2,1); d=[dp;*]; d1=d;d=ones(N2,1); dM=[*;d]; dM1=dM;

for =1:M2

d1=[d1;d];

dM1=[dM1;dM];

end

B=[d d1 d* dM1 d];

$=[2(N2+) 21 * 1 (N2+)];

>=spdias(B,$,s,s);

D=zeros(s,1);

r=1;D((N2+)&(r21)-1)=[(.&8+)&1(+,+)2u(+,1)2u(1,+)];

D((N2+)&(r21)-(N2+))=[(.&8+)&1(+,N21)2u(+,N)2u(1,N21)];

for n=+:N2


10/21

D((N2+)&(r21)-n)=[(.&8+)&1(+,n-1)2u(1,n-1)];

end

r=(M2+);

D((N2+)&(r21)-1)=[(.&8+)&1(M21,+)2u(M21,1)2u(M,+)];

D((N2+)&(r21)-(N2+))=[(.&8+)&1(M21,N21)2u(M21,N)2u(M,N21)];

for n=+:N2

D((N2+)&(r21)-n)=[(.&8+)&1(M21,n-1)2u(M,n-1)];

endfor r=+:M2

D((N2+)&(r21)-1)=[(.&8+)&1(r-1,+)2u(r-1,1)];

D((N2+)&(r21)-(N2+))=[(.&8+)&1(r-1,N21)2u(r-1,N)];

for n=+:N2

D((N2+)&(r21)-n)=[(.&8+)&1(r-1,n-1)];

end

end

O=in"(>)D;

for r=1:M2+

for n=1:N2+ u(r-1,n-1)=[O((N2+)&(r21)-n)];

end

end

surf(x,I,u);

end

function [#A] =#aplasiano:D(A)

[M,N]=size(4);

'4=zeros(M2+,N2+);

for =+:M21 for n=+:N21

'4(21,n21)=[4(21,n)2K&4(,n)-4(-1,n)-4(,n21)-4(,n-1)];

end

end

end

function [u] = Euacion#aplaceMatriz(A,Bi,Bs,#i,#6)

[M,N]=size(4);

['4]='aplasiano+D(4);

1=zeros(M,N);1(+:M21,+:N21)=['4];

u=zeros(M,N); .=1;

u(1,:)=[Bi]; u(M,:)=[Bs]; u(:,1)=['i]; u(:,N)=['d];

s=(M2+)&(N2+);

d=ones(s,1);

d*=2K&d;

dp=ones(N2,1); d=[dp;*]; d1=d;

d=ones(N2,1); dM=[*;d]; dM1=dM;

for =1:M2

d1=[d1;d]; dM1=[dM1;dM];

end

B=[d d1 d* dM1 d];


11/21

$=[2(N2+) 21 * 1 (N2+)];

>=spdias(B,$,s,s);

D=zeros(s,1);

r=1;

D((N2+)&(r21)-1)=[(.&8+)&1(+,+)2u(+,1)2u(1,+)];

D((N2+)&(r21)-(N2+))=[(.&8+)&1(+,N21)2u(+,N)2u(1,N21)];

for n=+:N2

D((N2+)&(r21)-n)=[(.&8+)&1(+,n-1)2u(1,n-1)];end

r=(M2+);

D((N2+)&(r21)-1)=[(.&8+)&1(M21,+)2u(M21,1)2u(M,+)];

D((N2+)&(r21)-(N2+))=[(.&8+)&1(M21,N21)2u(M21,N)2u(M,N21)];

for n=+:N2

D((N2+)&(r21)-n)=[(.&8+)&1(M21,n-1)2u(M,n-1)];

end

for r=+:M2

D((N2+)&(r21)-1)=[(.&8+)&1(r-1,+)2u(r-1,1)];

D((N2+)&(r21)-(N2+))=[(.&8+)&1(r-1,N21)2u(r-1,N)];for n=+:N2

D((N2+)&(r21)-n)=[(.&8+)&1(r-1,n-1)];

end

end

O=in"(>)D;

for r=1:M2+

for n=1:N2+

u(r-1,n-1)=[O((N2+)&(r21)-n)];

end

endend

function [u] =&oissonEcuacion($,3,a,,c,6,@)

x=[a:.:$]; I=[c:.:d]; N=len!.(x); M=len!.(I);

[x,I]=es.rid(x,I);

1=(x,I); 1=(x,I);

u=zeros(M,N);

u(1,:)=[1(1,:)]; u(M,:)=[1(M,:)];

u(:,1)=[1(:,1)]; u(:,N)=[1(:,N)];

s=(M2+)&(N2+);d=ones(s,1);

d*=2K&d;

dp=ones(N2,1); d=[dp;*]; dM=[*;dp];

d1=d; dM1=dM;

for =:M21

d1=[d1;d]; dM1=[dM;dM1];

end

B=[d d1 d* dM1 d];

$=[2(N2+) 21 * 1 (N2+)];

>=spdias(B,$,s,s);D=zeros(s,1);

D(1)=[(.&8+)&1(+,+)2u(1,+)2u(+,1)];

D(N2+)=[(.&8+)&1(+,N21)2u(1,N21)2u(+,N)];


12/21

for n=+:N2

D(n)=[(.&8+)&1(+,n-1)2u(1,n-1)];

end

D((N2+)&(M2)-1)=[(.&8+)&1(M21,+)2u(M,+)2u(M21,1)];

D((N2+)&(M2)-N2+)=[(.&8+)&1(M21,N21)2u(M,N21)2u(M21,N)];

for n=+:N2

D((N2+)&(M2)-n)=[(.&8+)&1(M21,n-1)2u(M,n-1)];

endfor =+:M2

D((N2+)&(21)-1)=[(.&8+)&1(-1,+)2u(-1,1)];

D((N2+)&(21)-N2+)=[(.&8+)&1(-1,N21)2u(-1,N)];

for n=+:N2

D((N2+)&(21)-n)=[(.&8+)&1(-1,n-1)];

end

end

O=in"(>)D;

for =1:M2+

for n=1:N2+ u(-1,n-1)=[O((N2+)&(21)-n)];

end

end

?ure()

surf(x,I,u);

end

function [r] =RaicesBessel(p,N)

x=[*:*&1:(+&pi&N-+*)];

M=len!.(x);f=/(x) $essel0(p,x);

f1=f(x); =1;

for n=1:M21

if (f1(n)&f1(n-1)3*)

4()=[x(n)]; B()=[x(n-1)];

=-1;

end

end

'=len!.(4);

for l=1:' iz=4(l); de=B(l);

5.ile (de2iz)6*&******1

ed=(de-iz)&7+;

if (f(iz)&f(ed)3*)==1

de=ed;

else

iz=ed;

end

end

r(l)=ed;end

end


13/21

function [u] =Ta8orOscilante(f,9,a,#,N)

[r]=RaicesBessel(*,N);

r*=r;

for =1:N

f1=/(r,?) r&f(r,?)&$essel0(*,r()&r&7');

1=/(r,?) r&(r,?)&$essel0(*,r()&r&7')

4*()=in!eral+(f1,*,',2pi,pi);

4*()=4*()&7(pi&('&$essel0(1,r())&8+)); B*()=in!eral+(1,*,',2pi,pi);

B*()=B*()&7((a&pi&'&r())&($essel0(1,r())&8+));

end

for n=1:N

[r]=RaicesBessel(n,N)

r=r(1:N);

s(n,:)=[r];

end

for n=1:N

for =1:N .=/(r,?) r&f(r,?)&cos(n&?)&$essel0(n,s(n,)&r&7');

4(n,)=in!eral+(.,*,',2pi,pi);

4(n,)=+&4(n,)&7(pi&('&$essel0(n-1,r(n,)))&8+);

P=/(r,?) r&f(r,?)&sin(n&?)&$essel0(n,s(n,)&r&7');

>(n,)=in!eral+(P,*,',2pi,pi);

>(n,)=+&>(n,)&7(pi&('&$essel0(n-1,r(n,)))&8+);

d=/(r,?) r&(r,?)&cos(n&?)&$essel0(n,s(n,)&r&7');

B(n,)=in!eral+(d,*,',2pi,pi);

B(n,)=+&B(n,)&7(pi&('&$essel0(n-1,r(n,)))&8+);

B(n,)=B(n,)&a&7s(n,); e=/(r,?) r&(r,?)&sin(n&?)&$essel0(n,s(n,)&r&7');

D(n,)=in!eral+(d,*,',2pi,pi);

D(n,)=+&D(n,)&7(pi&('&$essel0(n-1,r(n,))&8+));

D(n,)=D(n,)&a&7s(n,);

end

end

u=/(r,?,!) r&*-?&*-!&*;

for =1:N

u=/(r,?,!) u(r,?,!)-(4*()&cos(a&r*()&!&7')

-B*()&sin(a&r*()&!&7'))&$essel0(*,r*()&r&7');end

for n=1:N

for =1:N

u=/(r,?,!)-(4(n,)&cos(a&s(n,)&!&7')

-B(n,)&sin(a&s(n,)&!&7'))&$essel0(n,s(n,)&r&7')&cos(n&?);

u=/(r,?,!)-(>(n,)&cos(a&s(n,)&!&7')

-D(n,)&sin(a&s(n,)&!&7'))&$essel0(n,s(n,)&r&7')&sin(n&?);

end

end

r=[*:*&1:'];?=[*:*&1:+&pi];

!=[*:*&1:1*];

[r,?]=es.rid(r,?);


14/21

[r,?]=car!opol(r,?);

E=len!.(!);

for n=1:E

u1=u(r,?,!(n));

?ure();surf(r,?,u1),.old on,rid on,.old o

pause(*&A)

end

end

function [u]=On6aDif$inita(f,9,#,a,@,T)

x=[*:.:']; N=len!.(x);

G=.&7(+&a);

!=[*:G:E]; M=len!.(!);

u=zeros(M,N);

f1=f(x); 1=(x);

u(:,1)=[*]; u(:,N)=[*];

u(1,:)=[f1(:)];

for n=+:N21 u(+,n)=u(1,n)-G&1(n);

end

for =+:M21

for n=+:N21

u(-1,n)=[+&u(,n)2u(21,n)-(u(,n-1)2+&u(,n)-u(,n21))&7K];

end

end

?ure();

for =1:M

plo!(x,u(,:),@r@,@'ine9id!.@,),.old on,rid on; pause(*&1);

.old o

end

end

function [u] = Eplicita7iperolica(f,9,a,#,@,T)

x=[*:.:']; G=.&7(+&a); N=len!.(x);

!=[*:G:E]; M=len!.(!);

u=zeros(M,N); f1=f(x), 1=(x);

u(1,:)=[f1(:)]; u(:,1)=[*]; u(:,N)=[*];for n=+:N21

u(+,n)=u(1,n)-G&1(n);

end

for =+:M

for n=+:N21

u(-1,n)=[+&u(,n)2u(21,n)-(u(,n-1)2+&u(,n)-u(,n21))&7K];

end

end

?ure(), .old on, rid on

for =1:M plo!(x,u(,:));

pause(*&1);

.old o


15/21

end

end

function ["] =7iperolica:DDif$($,3,a,#,D,@,T)

x=[*:.:']; N=len!.(x);I=[*:.:D]; M=len!.(I);

G=.&7(+&a); !=[*:G:E]; Q=len!.(!);

[x,I]=es.rid(x,I);

1=(x,I); 1=(x,I);

=zeros(M,N,Q);

(1,:,:)=[*]; (M,:,:)=[*];

(:,1,:)=[*]; (:,N,:)=[*];

for =+:M21

for n=+:N21

(,n,1)=[1(,n)]; end

end

for =+:M21

for n=+:N21

(,n,+)=(,n,1)-G&1(,n);

end

end

for o=+:Q21;

for =+:M21

for n=+:N21 (,n,o-1)=[+&(,n,o)2(,n,o21)-((-1,n,o)2+&(,n,o)-(2

1,n,o))&7+-((,n-1,o)2+&(,n,o)-(,n21,o))&7+];

end

end

end

?ure();

for o=1:Q

surf(x,I,(:,:,o)), .old on,rid on,axis([* ' * D 2+ +]);

pause(*&*1);

.old oend

end

&oisson

function [] = #aplasianMatriz(A,@)

[M,N]=size(4);

>=zeros(M2+,N2+);

for =+:M21

for n=+:N21

>(21,n21)=[4(-1,n)2K&4(,n)-4(21,n)-4(,n-1)-4(,n21)]&7(.&8+); end

end

end


16/21

function [u] =#aplace($,3,a,,c,6,@)

x=[a:.:$]; I=[c:.:d];

M=len!.(I); N=len!.(x);

[x,I]=es.rid(x,I); 1=(x,I);

1=(x,I);u=zeros(M,N);

u(1,:)=[1(1,:)]; u(M,:)=[1(M,:)];

u(:,1)=[1(:,1)]; u(:,N)=[1(:,N)];

s=(M2+)&(N2+);

d=ones(s,1);

d*=2K&d;



for =1:M2

d1=[d1;d]; dM1=[dM1;dM];

end

B=[d d1 d* dM1 d];

$=[2(N2+) 21 * 1 (N2+)];

>=spdias(B,$,s,s);

D=zeros(s,1);

r=1;

D((N2+)&(r21)-1)=[(.&8+)&1(+,+)2u(+,1)2u(1,+)];

D((N2+)&(r21)-(N2+))=[(.&8+)&1(+,N21)2u(+,N)2u(1,N21)];

for n=+:N2 D((N2+)&(r21)-n)=[(.&8+)&1(+,n-1)2u(1,n-1)];

end

r=(M2+);

D((N2+)&(r21)-1)=[(.&8+)&1(M21,+)2u(M21,1)2u(M,+)];

D((N2+)&(r21)-(N2+))=[(.&8+)&1(M21,N21)2u(M21,N)2u(M,N21)];

for n=+:N2

D((N2+)&(r21)-n)=[(.&8+)&1(M21,n-1)2u(M,n-1)];

end

for r=+:M2

D((N2+)&(r21)-1)=[(.&8+)&1(r-1,+)2u(r-1,1)];D((N2+)&(r21)-(N2+))=[(.&8+)&1(r-1,N21)2u(r-1,N)];

for n=+:N2

D((N2+)&(r21)-n)=[(.&8+)&1(r-1,n-1)];

end

end

O=in"(>)D;

for r=1:M2+

for n=1:N2+

u(r-1,n-1)=[O((N2+)&(r21)-n)];

endend

surf(x,I,u);

end


17/21

function [#A] =#aplasiano:D(A)

[M,N]=size(4);

'4=zeros(M2+,N2+);

for =+:M21

for n=+:N21

'4(21,n21)=[4(21,n)2K&4(,n)-4(-1,n)-4(,n21)-4(,n-1)];

end

endend

function [u] = Euacion#aplaceMatriz(A,Bi,Bs,#i,#6)

['4]='aplasiano+D(4);

1=zeros(M,N);

1(+:M21,+:N21)=['4];

u=zeros(M,N); .=1;

u(1,:)=[Bi]; u(M,:)=[Bs]; u(:,1)=['i]; u(:,N)=['d];

s=(M2+)&(N2+);

d=ones(s,1);d*=2K&d;



for =1:M2

d1=[d1;d];

dM1=[dM1;dM];

end

B=[d d1 d* dM1 d];

$=[2(N2+) 21 * 1 (N2+)];

>=spdias(B,$,s,s);D=zeros(s,1);

r=1;

D((N2+)&(r21)-1)=[(.&8+)&1(+,+)2u(+,1)2u(1,+)];

D((N2+)&(r21)-(N2+))=[(.&8+)&1(+,N21)2u(+,N)2u(1,N21)];

for n=+:N2

D((N2+)&(r21)-n)=[(.&8+)&1(+,n-1)2u(1,n-1)];

end

r=(M2+);

D((N2+)&(r21)-1)=[(.&8+)&1(M21,+)2u(M21,1)2u(M,+)];

D((N2+)&(r21)-(N2+))=[(.&8+)&1(M21,N21)2u(M21,N)2u(M,N21)];for n=+:N2

D((N2+)&(r21)-n)=[(.&8+)&1(M21,n-1)2u(M,n-1)];

end

for r=+:M2

D((N2+)&(r21)-1)=[(.&8+)&1(r-1,+)2u(r-1,1)];

D((N2+)&(r21)-(N2+))=[(.&8+)&1(r-1,N21)2u(r-1,N)];

for n=+:N2

D((N2+)&(r21)-n)=[(.&8+)&1(r-1,n-1)];

end

endO=in"(>)D;

for r=1:M2+

for n=1:N2+


18/21

u(r-1,n-1)=[O((N2+)&(r21)-n)];

end

end

end

function [A] =Bor6a6a(A,r)

[M,N]=size(4);

4$=ones(M,N);p=1&7r; a=[p:p:1];

for n=1:r

4$(n,n:N2(n21))=[a(n)];

4$(M2(n21),n:N2(n21))=[a(n)];

4$(n:M2(n21),n)=[a(n)];

4$(n:M2(n21),N2(n21))=[a(n)];

end

end

function [u] = $oto8ontaCe(A,B,r,Bi,Bs,#i,#6)['a]='aplasiano+D(4); ['$]='aplasiano+D(B);

['a$]=Bordada('a,r); ['$$]=Bordada('$,r);

[,]=size('a$);

>=ones(,); D=>2'a$;

4f='a&'a$-'$&D;

[M,N]=size(4);

1=zeros(M,N);

1(+:M21,+:N21)=[4f];

u=zeros(M,N); .=1;

u(1,:)=[Bi]; u(M,:)=[Bs]; u(:,1)=['i]; u(:,N)=['d];s=(M2+)&(N2+);

d=ones(s,1);

d*=2K&d;



for =1:M2

d1=[d1;d];

dM1=[dM1;dM];

end

B=[d d1 d* dM1 d];$=[2(N2+) 21 * 1 (N2+)];

>=spdias(B,$,s,s);

D=zeros(s,1);

r=1;

D((N2+)&(r21)-1)=[(.&8+)&1(+,+)2u(+,1)2u(1,+)];

D((N2+)&(r21)-(N2+))=[(.&8+)&1(+,N21)2u(+,N)2u(1,N21)];

for n=+:N2

D((N2+)&(r21)-n)=[(.&8+)&1(+,n-1)2u(1,n-1)];

end

r=(M2+);D((N2+)&(r21)-1)=[(.&8+)&1(M21,+)2u(M21,1)2u(M,+)];

D((N2+)&(r21)-(N2+))=[(.&8+)&1(M21,N21)2u(M21,N)2u(M,N21)];

for n=+:N2


19/21

D((N2+)&(r21)-n)=[(.&8+)&1(M21,n-1)2u(M,n-1)];

end

for r=+:M2

D((N2+)&(r21)-1)=[(.&8+)&1(r-1,+)2u(r-1,1)];

D((N2+)&(r21)-(N2+))=[(.&8+)&1(r-1,N21)2u(r-1,N)];

for n=+:N2

D((N2+)&(r21)-n)=[(.&8+)&1(r-1,n-1)];

endend

O=in"(>)D;

for r=1:M2+

for n=1:N2+

u(r-1,n-1)=[O((N2+)&(r21)-n)];

end

end

end

function [u*] = Ecuacionalor(f,9,s,a,l,@,T)x=[*:.:l]; N=len!.(x);

G=(.&7(a)&8+)&7K;

!=[*:G:E]; M=len!.(!);

f1=f(x);

1=(!); s1=s(!);

u1=zeros(M,N);

u1(1,:)=[f1(:)];

u1(:,1)=[1(:)];

u1(:,N)=[s1(:)];

for =1:M21 for n=+:N21

u1(-1)=[u1(,n)&7+-(u1(,n-1)-u1(,n21))&7K];

end

end

?ure();

4=ax(ax(u1)); >=in(in(u1));

for =1:M

B=zeros(1*,N);

for r=1:1*

B(r,:)=[u1(,:)]; end

is.o5(B,[> 4]), colorap(5in!er), .old on, rid on

pause (*&1);

.old o

end

end

a=*+ #=:+ @=+,- T=-+

f='() (/(#.))/0:

9='(t) +/t

s='(t) +/t[u*] = Ecuacionalor(f,9,s,a,l,@,T)

function [u] = 8e8rana6iferencias(f,9,A,B,,D,a,@,i,T) poisson


20/21

x=[4:.,B];

M=len!.(x);

I=[>:i:D];

N=len!.(I);

G=(.-i)&7(+&a);

!=[*:G:E];

'=len!.(!);

[x,I]=es.rid(x,I);1=f(x,I);

1=(x,I);

u=Oeros(M,N,');

u(1,:,:)=[*];

u(M,:,:)=[*];

u(:,1,:)=[*];

u(:,N,:)=[*];

u(:,:,1)=[1(:,:)];

for =+:M21

for n=+:N21 u(,n,+)=u(,n,1)-G&1(,n);

end

end

for l=+:'21

for =+:M21

for n=+:N21

u(,n,l-1)=+&u(,n,l)2u(,n,l21)-[[(u(,n-1,l)2+&u(,n,l)-u(,n2

1,l))&i&8+]-[(u(-1,n,l)2+&u(,n,l)-u(21,n,l))&.&8+]]&7K;

end

endend

?ure()

for l=1:'

surf(x,I,u(:,:,l)),.old on, rid on, pause(*&+),.old o

end

end

Realizar la funcin !erie 6e $ourier Bessel en funcin 6e un ran9o 6e

(+,#)

func!ion [9,R] = Besse0Se"in(f,p,M,',epsi)

[R]=BesselRaices(p,M,epsi);

for n=1:M

f1=/(x) x&f(x)&$essel0(p,R(n)&(x&7'));

>(n)=in!eral(f1,*,');

d='&$essel0(p-1,R(n));

d=d&8+; d=d&7+;

>(n)=>(n)&7d;

end


21/21

9=/(x) *&x;

for n=1:M

9=/(x) 9(x)->(n)&$essel0(p,R(n)&(x&7'));

end

u=[*:*&1:'];

91=9(u);

f1=f(u);

?ure(),plo!(u,91,@r@), .old on, rid on, plo!(u,f1,@$@);end

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Funciones en Matlab 2