/* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/
/* [ Created with wxMaxima version 13.04.2 ] */

/* [wxMaxima: title   start ]
Problema

   [wxMaxima: title   end   ] */

/* [wxMaxima: comment start ]
Viga simplemente apoyada de longitud H y momentos concentrados M en los extremos. Propiedades E, I. 
Condiciones de contorno y(0)=0, y(H)=0. La ecuación diferencial es E*I*d2y/dx2-M(x)=0
   [wxMaxima: comment end   ] */

/* [wxMaxima: input   start ] */
/* Función de prueba y derivadas*/
y(x):=A*sin(%pi*x/H);
dy(x):=diff(y(x),x);
ddy(x):=diff(dy(x),x);
ddy(x);
dy(x);
assume(H>0);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: title   start ]
Solución exacta
   [wxMaxima: title   end   ] */

/* [wxMaxima: input   start ] */
yexa(x):=M*x/(2*E*I)*(x-H);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: title   start ]
Variacional
   [wxMaxima: title   end   ] */

/* [wxMaxima: input   start ] */
/* Integrando formulación variacional*/
Vr(x):=E*I/2*(dy(x))^2+M*y(x);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* Principio variacional*/
PI(A):=integrate(Vr(x),x,0,H);
PI(A);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* Minimización respecto a coeficientes*/
dVar(A):=diff(PI(A),A);
dVar(A);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/*Resolver A*/
Avar:solve(dVar(A)=0,A);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
Avar:ev(A,Avar[1]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
Avar;
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* Aproximación variacional*/
yvar(x):=Avar*sin(%pi*x/H);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: title   start ]
Residuo
   [wxMaxima: title   end   ] */

/* [wxMaxima: input   start ] */
R(x):=E*I*ddy(x)-M;
R(x);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: title   start ]
Colocación
   [wxMaxima: title   end   ] */

/* [wxMaxima: input   start ] */
Sub(A):=subst(H/2, x, R(x));
Sub(A);
Acol:solve(Sub(A)=0,A);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
Acol:ev(A,Acol[1]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* Aproximación colocación*/
ycol(x):=Acol*sin(%pi*x/H);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: title   start ]
Subdominio
   [wxMaxima: title   end   ] */

/* [wxMaxima: input   start ] */
Subdom(A):=integrate(R(x),x,0,H);
Subdom(A);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
Asub:solve(Subdom(A)=0,A);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
Asub:ev(A,Asub[1]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* Aproximación subdominio*/
ysub(x):=Asub*sin(%pi*x/H);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: title   start ]
Galerkin
   [wxMaxima: title   end   ] */

/* [wxMaxima: input   start ] */
Ga(A):=integrate(sin(%pi*x/H)*R(x),x,0,H);
Aga:solve(Ga(A)=0,A);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
Aga:ev(A,Aga[1]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* Aproximación Galerkin*/
ygal(x):=Aga*sin(%pi*x/H);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: title   start ]
Minimos cuadrados

   [wxMaxima: title   end   ] */

/* [wxMaxima: input   start ] */
Er(A):=integrate(R(x)^2,x,0,H);Er(A);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
dEr(A):=diff(Er(A),A);dEr(A);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
Alsq:solve(dEr(A)=0,A);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
Alsq:ev(A,Alsq[1]);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
/* Aproximación M.C.*/
ylsq(x):=Alsq*sin(%pi*x/H);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: title   start ]
Evaluar funciones
   [wxMaxima: title   end   ] */

/* [wxMaxima: input   start ] */
E:2e6;M:1;I:5e-6;H:2;
plot2d ([yexa(x),yvar(x),ycol(x),ysub(x),ygal(x),ylsq(x)], [x, 0, 2],
[legend,"exacta","variacional","colocación","sub","gal","ls"],
[style,lines])$
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
yvar(x);
/* [wxMaxima: input   end   ] */

/* [wxMaxima: input   start ] */
ygal(x);
/* [wxMaxima: input   end   ] */

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