Example 7.14.mw

> restart:
 

> with(plots):
 

> eq:=diff(u(x,y),y$2)=-diff(u(x,y),x$2)-1;
 

diff(diff(u(x, y), y), y) = `+`(`-`(diff(diff(u(x, y), x), x)), `-`(1)) (1)
 

> bc1:=u(x,y)=0;
 

u(x, y) = 0 (2)
 

> bc2:=u(x,y)=0;
 

u(x, y) = 0 (3)
 

> bc3:=u(x,y)=0;
 

u(x, y) = 0 (4)
 

> bc4:=u(x,y)=0;
 

u(x, y) = 0 (5)
 

> eq:=expand(subs(u(x,y)=g(x,y)+w(x),eq));
 

diff(diff(g(x, y), y), y) = `+`(`-`(diff(diff(g(x, y), x), x)), `-`(diff(diff(w(x), x), x)), `-`(1)) (6)
 

> eq_w:=diff(w(x),`$`(x,2))+1=0;
 

`+`(diff(diff(w(x), x), x), 1) = 0 (7)
 

> w(x):=rhs(dsolve({eq_w,w(0)=0,w(1)=0},w(x)));
 

`+`(`-`(`*`(`/`(1, 2), `*`(`^`(x, 2)))), `*`(`/`(1, 2), `*`(x))) (8)
 

> eqg:=diff(g(x,y),`$`(y,2)) = -diff(g(x,y),`$`(x,2));
 

diff(diff(g(x, y), y), y) = `+`(`-`(diff(diff(g(x, y), x), x))) (9)
 

> Eq:=subs(g(x,y)=X(x)*Y(y),eqg):
 

> Eq:=expand(Eq/X(x)/Y(y)):
 

> Eq_Y:=lhs(Eq)=lambda^2:
 

> Eq_Y:=eval(Eq_Y*Y(y)):
 

> dsolve(Eq_Y,Y(y)):
 

> Y(y):=C[1]*sinh(lambda*y)+C[2]*cosh(lambda*y):
 

> Bc3:=Y(y)=0;
 

`+`(`*`(C[1], `*`(sinh(`*`(lambda, `*`(y))))), `*`(C[2], `*`(cosh(`*`(lambda, `*`(y)))))) = 0 (10)
 

> Eq_Bc3:=eval(subs(y=0,Bc3)):
 

> C[2]:=solve(Eq_Bc3):
 

> Y(y):=eval(Y(y));
 

`*`(C[1], `*`(sinh(`*`(lambda, `*`(y))))) (11)
 

> Eq_X:=rhs(Eq)=lambda^2:
 

> Eq_X:=expand(Eq_X*X(x)):
 

> dsolve({Eq_X},X(x)):
 

> X(x):=c[1]*sin(lambda*x)+c[2]*cos(lambda*x):
 

> Bc1:=X(x)=0;
 

`+`(`*`(c[1], `*`(sin(`*`(lambda, `*`(x))))), `*`(c[2], `*`(cos(`*`(lambda, `*`(x)))))) = 0 (12)
 

> Bc2:=X(x)=0;
 

`+`(`*`(c[1], `*`(sin(`*`(lambda, `*`(x))))), `*`(c[2], `*`(cos(`*`(lambda, `*`(x)))))) = 0 (13)
 

> Eq_Bc1:=eval(subs(x=0,Bc1)):
 

> c[2]:=solve(Eq_Bc1,c[2]);
 

0 (14)
 

> Eq_Bc2:=eval(subs(x=1,Bc2));
 

`*`(c[1], `*`(sin(lambda))) = 0 (15)
 

> Eq_Eig:=sin(lambda)=0;
 

sin(lambda) = 0 (16)
 

> _EnvAllSolutions := true:
 

> solve(Eq_Eig,lambda);
 

`*`(Pi, `*`(_Z1)) (17)
 

> G:=eval(X(x)*Y(y));
 

`*`(c[1], `*`(sin(`*`(lambda, `*`(x))), `*`(C[1], `*`(sinh(`*`(lambda, `*`(y))))))) (18)
 

> Gn:=subs(c[1]=A[n]/C[1],lambda=lambda[n],G):
 

> u(x,y):=Sum(Gn,n=1..infinity)+w(x):
 

> u(x,y):=subs(lambda[n]=n*Pi,u(x,y));
 

`+`(Sum(`*`(A[n], `*`(sin(`*`(n, `*`(Pi, `*`(x)))), `*`(sinh(`*`(n, `*`(Pi, `*`(y))))))), n = 1 .. infinity), `-`(`*`(`/`(1, 2), `*`(`^`(x, 2)))), `*`(`/`(1, 2), `*`(x))) (19)
 

> eq_An:=eval(subs(y=1,u(x,y)))=rhs(bc4);
 

`+`(Sum(`*`(A[n], `*`(sin(`*`(n, `*`(Pi, `*`(x)))), `*`(sinh(`*`(n, `*`(Pi)))))), n = 1 .. infinity), `-`(`*`(`/`(1, 2), `*`(`^`(x, 2)))), `*`(`/`(1, 2), `*`(x))) = 0 (20)
 

> phi[n]:=sin(n*Pi*x);
 

sin(`*`(n, `*`(Pi, `*`(x)))) (21)
 

> r(x):=1;
 

1 (22)
 

> I1:=int(phi[n]^2*r(x),x=0..1):
 

> I2:=int((0-w(x))*phi[n]*r(x),x=0..1):
 

> vars:={sin(n*Pi)=0}:
 

> I1:=subs(vars,I1):
 

> I2:=subs(vars,I2):
 

> A[n]:=I2/I1/sinh(n*Pi);
 

`/`(`*`(piecewise(n = 0, 0, `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(2), `*`(2, `*`(cos(`*`(n, `*`(Pi)))))))), `*`(`^`(n, 3), `*`(`^`(Pi, 3))))))), `*`(piecewise(n = 0, 0, `/`(1, 2)), `*`(sinh(`*`(n, `*`(Pi... (23)
 

> u(x,y):=eval(u(x,y));
 

`+`(Sum(`/`(`*`(piecewise(n = 0, 0, `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(2), `*`(2, `*`(cos(`*`(n, `*`(Pi)))))))), `*`(`^`(n, 3), `*`(`^`(Pi, 3)))))), `*`(sin(`*`(n, `*`(Pi, `*`(x)))), `*`(sinh(`*`(n, `... (24)
 

> u(x,y):=subs(infinity=N,u(x,y)):
 

> ua:=subs(N=20,u(x,y)):
 

> uu:=piecewise(y<0.9999,ua,y>0.9999,0);
 

piecewise(`<`(y, .9999), `+`(Sum(`/`(`*`(piecewise(n = 0, 0, `+`(`/`(`*`(`/`(1, 2), `*`(`+`(`-`(2), `*`(2, `*`(cos(`*`(n, `*`(Pi)))))))), `*`(`^`(n, 3), `*`(`^`(Pi, 3)))))), `*`(sin(`*`(n, `*`(Pi, `*`... (25)
 

> plot3d(evalf(uu),x=0..1,y=1..0,axes=boxed,title="Figure Exp. 7.33.",labels=[x,y,"u"],orientation=[-120,60]);
 

Plot
 

> plot([subs(y=0,uu),subs(y=0.6,uu),subs(y=0.8,uu),subs(y=0.9,uu),subs(y=0.95,uu),subs(y=1,uu)],x=0..1,axes=boxed,title="Figure Exp. 7.34.",thickness=5,labels=[x,"u"],legend=["y=0","y=0.6","y=0.8","y=0.9","y=0.95","y=1.0"]);
 

Plot_2d
 

>