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curso_edp

Course Number: MATH MM415, Spring 2008

College/University: Universidad del Valle...

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ra de valores de wavelet basica . function y=wave(x,a,b,ceros) xc= -ceros:-1 1:ceros -ceros* 1 1] ceros* 1 1]]; if ceros>1, xc= xc - 1 1] 1 1]];end y=poly(xc);y/=polyval(y,0); y=1/sqrt(abs(a))*w00((x-b)/a,y).*car(x, -ceros*a+b ceros*a+b]); function y=w00(x,p) y=polyval(p,x).*exp(-4*x.^2); Archivo: car.m Funcion : Funcion caracter stica. function y=car(x,inter) y=x>=inter(1)&x<=inter(2); Archivo:...

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de ra valores de wavelet basica . function y=wave(x,a,b,ceros) xc= -ceros:-1 1:ceros -ceros* 1 1] ceros* 1 1]]; if ceros>1, xc= xc - 1 1] 1 1]];end y=poly(xc);y/=polyval(y,0); y=1/sqrt(abs(a))*w00((x-b)/a,y).*car(x, -ceros*a+b ceros*a+b]); function y=w00(x,p) y=polyval(p,x).*exp(-4*x.^2); Archivo: car.m Funcion : Funcion caracter stica. function y=car(x,inter) y=x>=inter(1)&x<=inter(2); Archivo: bas1.m Funcion : Funcion de muestreo par en una dimension espacial. function f=bas1(f,m,im) f=f(im:m:length(f)); Archivo: bas2d.m Funcion : Funcion de muestreo par en dos dimensiones espaciales. function f=bas2d(f,m,im,n,in) f=f(im:m:size(f,1),in:n:size(f,2)); Archivo: apcoefwave.m Funcion : Calculo de los coeficientes de descomposicion en dos dimensiones espaciales. function a,p,fp]=apcoefwave2d(nivel,xab,yab,f,xp,yp,ceros) %lf=length(f); sf=size(f); %nivel0=log((lf-1)/(2*ceros))/log(2); n ivel0m=log((sf(1)-1)/(2*ceros))/log(2); nivel0n=log((sf(2)-1)/(2*ceros))/log(2); a= ]; p=a; %xt=bas1(ab,2^nivel0,1); xt=bas2d(xab,2^nivel0m,1,2^nivel0n,1); yt=bas2d(yab,2^nivel0m,1,2^nivel0n,1); %fet=bas1(f,2^nivel0,1); fet=bas2d(f,2^nivel0m,1,2^nivel0n,1); p=1; a= a,sparse(fet(:))']; p= p,length(a)+1]; fp=evuwave2d(1/2^0,xp,yp,fet,ceros); for niv=1:nivel F . V id es, Lab orato rio de C o m p u tacion , D ep artam en to d e M atem atica , Facu ltad d e C ien cias, U .N .A .H . In trod u ccion a los M etod o s C om p u tacio n ales p ara las E cu acio n es D iferen ciales en D erivad as P arciales %xt=bas1(bas1(ab,2^(nivel0-niv),1),2,2); %xt=bas2d(bas2d(xab,2^(nivel0m-niv),1,2^(nivel0m-niv),1),2,2,2,2); xt=bas2d(xab,2^(nivel0m-niv),1,2^(nivel0m-niv),1); yt=bas2d(yab,2^(nivel0m-niv),1,2^(nivel0m-niv),1); %yt=bas2d(bas2d(yab,2^(nivel0m-niv),1,2^(nivel0m-niv),1),2,2,2,2); %fet=bas1(bas1(f,2^(nivel0-niv),1),2,2); %fet=bas2d(bas2d(f,2^(nivel0m-niv),1,2^(nivel0m-niv),1),2,2,2,2); fet=bas2d(f,2^(nivel0m-niv),1,2^(nivel0m-niv),1); fat=evnwave2d(xt,yt,a,p,niv,ceros); fat=fet-fat; %fat(1:2:lf/2^(nivel0-niv))=0; fat=sparse(fat); a= a,fat(:)']; p= p,length(a)+1]; fp+=evuwave2d(1/2^niv,xp,yp,fat,ceros); end function y=evnwave2d(xn,yn,zn,p,nivel,ceros) %lx=length(xn); y=zeros(size(xn)); for k=0:(nivel-1) y+=evuwave2d(1/2^k,xn,yn,zn(p(k+1):(p(k+2)-1)),ceros); end Archivo: evdwave.m Funcion : Evaluacion de la proyeccion de nivel j. function y=evdwave(v,xn,yn,p,nivel,ceros) lx=length(xn); y=zeros(1,lx); for k=nivel(1):nivel(2) y+=evwdwave(v,1/2^k,xn,yn(p(k+1):(p(k+2)-1)),ceros); end Archivo: evuwave2d.m Funcion : Evaluacion de la proyeccion parcial de nivel k. function Wr=evuwave2d(k,xv,yv,zb,ceros) %if k==1,m=1;else m=2;end m=1; x=xv(1,:); y=yv(:,1); p=size(xv); q=sqrt(length(zb(:))); Wr=zeros(p); zb=reshape(zb,q,q); for l=0:(q-1) for n=0:(q-1) Wr+=sqrt(k)^2*zb(l+1,n+1)*wave(y,k,k*(m-1)-ceros+m*k*l,ceros)*wave(x,k,k*(m-1)ceros+m*k*n,ceros); end end Archivo:evdwave2d.m Funcion : Caclulo de la proyeccion de nivel j. function y=evdwave2d(v,xn,yn,zn,p,nivel,ceros) y=zeros(size(xn)); for k=nivel(1):nivel(2) y+=evwdwave2d(v,1/2^k,xn,yn,zn(p(k+1):(p(k+2)-1)),ceros); end Archivo: evwdwave.m Funcion : Funcion auxiliar de calculo de proyeccion . F . V id es, Lab orato rio de C o m p u tacion , D ep artam en to d e M atem atica , Facu ltad d e C ien cias, U .N .A .H . In trod u ccion a los M etod o s C om p u tacio n ales p ara las E cu acio n es D iferen ciales en D erivad as P arciales function Wr=evwdwave(v,k,xv,yb,ceros) if k==1,m=1;else m=2; end q=length(yb); p=length(xv); Wr=zeros(1,p); for l=0:(q-1) Wr+=sqrt(k)*yb(l+1)*(wave(tfa(xv+v,0,2*ceros),k,k*(m-1)ceros+m*k*l,ceros)+wave(tfa(xv-v,0,2*ceros),k,k*(m-1)-ceros+m*k*l,ceros))/2; end Archivo: evwdwave2d.m Funcion : Funcion auxiliar de calculo de proyeccion . function Wr=evwdwave2d(v,k,xv,yv,zb,ceros) %if k==1,m=1;else m=2;end m=1; x=xv(1,:); y=yv(:,1); p=size(xv); q=sqrt(length(zb(:))); Wr=zeros(p); zb=reshape(zb,q,q); for l=0:(q-1) for n=0:(q-1) Wr+=sqrt(k)^2*zb(l+1,n+1)*(wave(tfa(y+v,0,2*ceros),k,k*(m-1)ceros+m*k*l,ceros)*wave(tfa(x+v,0,2*ceros),k,k*(m-1)-ceros+m*k*n,ceros)+wave(tfa(yv,0,2*ceros),k,k*(m-1)-ceros+m*k*l,ceros)*wave(tfa(x-v,0,2*ceros),k,k*(m-1)ceros+m*k*n,ceros)+wave(tfa(y+v,0,2*ceros),k,k*(m-1)-ceros+m*k*l,ceros)*wave(tfa(xv,0,2*ceros),k,k*(m-1)-ceros+m*k*n,ceros)+wave(tfa(y-v,0,2*ceros),k,k*(m-1)ceros+m*k*l,ceros)*wave(tfa(x+v,0,2*ceros),k,k*(m-1)-ceros+m*k*n,ceros))/4; end end Archivo: show_levels_2d.m Funcion : Calcular y Graficar proyecciones a diferentes niveles para un numero par y con ra z segunda exacta de niveles. function show_levels_2d(a,p,Xm,Ym,levels) for k=1:levels Psi(:,:,k)=evdwave2d(0,Xab,Yab,a,p, 0 k],3); end figure(1); colormap bone; vc=sqrt(levels); for k=1:levels subplot(vc,vc,k); Ap=reshape (a(p(k):p(k+1)-1),sqrt(p(k+1)-p(k)),sqrt(p(k+1)-p(k))); surf(Ap); end figure(2); colormap bone; for k=1:levels subplot(vc,vc,k); surf(Xm,Ym,Psi(:,:,k)); shading interp; end figure(2); F . V id es, Lab orato rio de C o m p u tacion , D ep artam en to d e M atem atica , Facu ltad d e C ien cias, U .N .A .H . In trod u ccion a los M etod o s C om p u tacio n ales p ara las E cu acio n es D iferen ciales en D erivad as P arciales Agradecimientos Antes de concluir el presente material quiero agradecer a todas las personas que han contribuido en la realizacion de este proyecto: Dios... quien sino vos ha sido responsable el directo de todo lo que emprendo para ayudar a otros? A quien mas le debo la vida y la voluntad de levantarme y trabajar todos los d as ? Gracias amigo m o ! Te lo debo todo..., a mi familia quienes me bendicen con su amor, apoyo y comprension , a los alumnos de la primera promocion del curso cuyo manual es aqu presentado: Alex, Daniel, Manolo, Cesar y Sheila por sus valiosas contribuciones en la correccion de los listados de codigo a las que deb atender mejor, seguro que si, a mi pastor Dr. Rafael Antunez mi gran amigo y quien ha sido como un padre para mi y cuyas oraciones de bendicion a mi favor han enriquecido mi vida,a la profesora Rosibel Pacheco por su valiosa participacion y apoyo en la primera fase del curso, al profesor Venancio Carranza por su apoyo en la promocion del curso y por contribuir directamente con el mismo al facilitarme el recurso computacional y espacio f sico para realizar labores de investigacion y extension , al Dr. Jorge Destephen por ensenarme los primeros rudimentos del analisis numerico y por su motivacion para iniciar en los experimentos con entornos y lenguajes orientados a matrices, al Dr. Concepcion Ferrufino un gran amigo que compartio con la mejor de las ganas sus conocimientos de ecuaciones diferenciales parciales conmigo, al Dr. Stan Steinberg un gran amigo cuyos trabajos en discretizaciones mimeticas de ecuaciones diferenciales parciales han sido una inspiracion para mi, al Dr. Adalid Gutierrez por impartirme un curso de ecuaciones diferenciales ordinarias que desperto mi curiosidad por las mismas, al profesor Salvador Llopis (Q.E.P.D.) quien me enseno el valor de la abstraccion y el formalismo, a todos us-tedes, muchas gracias! Referencias 1. F. Vides. Introduccion al Algebra Lineal Computacional con MatLab. Verano de 2002. 2. S. Steer. SciLab Help Browser. 1995-2005. 3. S. Lipschutz. Teor a y Problemas de Algebra Lineal. Serie de Compendios Schaum Mc Graw Hill. 1988. 4. S. Lang. Introduccion al Analisis Matematico . Addison Wesley Iberoamericana. 1990. 5. D. Kinkaid y W. Cheney. Analisis Numerico las matematicas del calculo cient fico . Addison Wesley Iberoameri cana. 1990. 6. S. Nakamura. Analisis Numerico y Visualizacion Grafica con MatLab. Pearson Education. 1997. 7. C. H. Edwards y D. E. Penney. Ecuaciones Diferenciales. Prentice Hall. 4a. Edicion Esp. 2001. 8. A. Boggess y F. J. Narcowich. A First Course in Wavelets with Fourier Analysis. Prentice Hall. 2001. 9. M. R. Speigel, J. Liu y L. Abellanas. Formulas y Tablas de Matematica Aplicada. Serie de Compendios Schaum Mc Graw Hill. 2005. 10. H. F. Weinberger. Ecuaciones Diferenciales en Derivadas Parciales. Editorial Reverte . 1977. 11. A. Messiah. Mecanica Cuantica Tomo II. Editorial Tecnos. 1975. 12. M. R. Speigel. Teor a y Problemas de Variable Compleja. Serie de Compendios Schaum Mc Graw Hill. 1971. 13. M. R. Sepeigel. Teor a y Problemas de Analisis Vectorial. Serie de Compendios Schaum Mc Graw Hill. 1987. 14. D. Betounes. Partial Differential Equations for Computational Science with Maple and Vector Analysis. Springer Telos. 1998. 15. F. Vides. Los Polinomios Ortogonales y las Matematicas del Calculo Cient fico . Revista de Ciencia y Tec nolog a . Noviembre 2002. 16. F. Vides. Fundamentos de Sub-Distribuciones. Bolet n de Divulgacion Interna del Depto. de Matematica . Noviembre 2002. 17. S. Steinberg. A Discrete Calculus with Applications of High-Order Discretizations to Boundary-Value Problems. PASI/PANAM Honduras .October 5 2004. 18. R.E. Showalter. Hilbert Space Methods for Partial Differential Equations. Electronic Journal of Differential Equations. Monograph 01, 1994. 19. John P. Boyd. Chebyshev and Fourier Spectral Methods. Second Edition. University of Michigan. 1999. 20. Lloyd N. Trefethen. Spectral Methods in Matlab. Oxford University. 2004. 21. F. Vides. Modelado Numerico de Movimiento Ondulatorio en un Medio Heterogeneo No Isotropico bajo Condiciones de Estabilidad Orbital. Bolet n de Divulgacion Interna del Depto. de Matematica .Julio 2006. F . V id es, Lab orato rio de C o m p u tacion , D ep artam en to d e M atem atica , Facu ltad d e C ien cias, U .N .A .H . In trod u ccion a los M etod o s C om p u tacio n ales p ara las E cu acio n es D iferen ciales en D erivad as P arciales 22. F. Vides. Introduccion al Calculo Particular y la Dinamica de Universos Discretizables. Bolet n de Divulgacion Interna del Depto. de Matematica . Invierno de 2007. 23. A. M. Chebotarev. Lectures on Quantum Probability. M. Lomonosov, Moscow State University. Aportaciones Matematicas, Sociedad Matematica Mexicana. 2000. 24. D.H. Sattinger. Partial Differential Equations of Applied Mathematics. Lecture Notes. Department of Mathematics, Yale University. Fall, 2003. 25. A. Camacho, B. Guardian y M. Rodr guez . Solucion numerica de problemas de valores en la frontera con alta exactitud por metodos espectrales. UNAM, IPN. Ingenier a , Investigacion y Tecnolog a VI. 3. 207-217. Febrero de 2005. 26. S. A. Morris. Topology Without Tears. University of Ballarat.October 14, 2007. 27. F. Vides. Particular Analysis of Periodic Wave Packet Motion in Discretizable Manifolds under Dynamical Conditions of Orbital Stability. December 16, 2008. F . V id es, Lab orato rio de C o m p u tacion , D ep artam en to d e M atem atica , Facu ltad d e C ien cias, U .N .A .H .

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MasteringPhysics: Assignment Print ViewProblem 32.67A 63.0 -cm-diameter cyclotron uses a 490 Part A What is the maximum kinetic energy of a proton if the magnetic field strength is 0.740 ANSWER: 4.1610-13oscillating potential difference between the de
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MasteringPhysics: Assignment Print ViewAn Air-Filled Toroidal SolenoidAn air-filled toroidal solenoid has a mean radius of 15.1 (see the figure). The current flowing through it is 12.5 within the solenoid be at least 0.387 . Part A What is the least num
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MasteringPhysics: Assignment Print ViewReading Quiz 28.1Part A What quantity is represented by the symbol ANSWER: ? Current density Complex impedanceResistivity Conductivity Johnston's constantIntroduction to Electric CurrentLearning Goal: To underst
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ELECTRIC CHARGES AND FORCES25.1. Model: Use the charge model. Solve: (a) In the process of charging by rubbing, electrons are removed from one material and transferred to the other because they are relatively free to move. Protons, on the other hand, are