Worksheet: Turkey.mw 

Cooking a Spherical Turkey: 

The Heat Equation in Spherical Coordinates 

This worksheet examines a series solution that arises in a separation of variables solution for the heat equation in spherical coordinates : 

=(+ )   0 < r < a      

with boundary conditions 

u(a,t) =    u(0,t) bounded 

with initial condition 

u(r,0)=     0 < x< a  

>	restart:with(plots):

 

>	a:=1;kappa:=2/100;T0:=75;T1:=350;

 

Let's approximate the initial condition with the eigenfunctions that came from separation of variables. 

>	f:=n->sin(n*Pi*r/a)/r;

 

>	c:=n->int(1*f(n)*r^2,r=0..a)/int((f(n))^2*r^2,r=0..a);

 

>	seq(c[n]=c(n),n=1..10);

 

>

M:=10;

 

>	u:=(r,t)->T1+(T0-T1)*sum(c(m)*f(m)*exp(-kappa*m^2*Pi^2*t/a^2),m=1..M);

 

>

u(r,t);

 

>	plot({u(r,0),75},r=0..a,thickness=2);

 

Note the Gibb's phenomena at the endpoints -- convergence is non-uniform at t=0, but uniform for any positive t. 

>	animate(u(r,t),r=0..a,t=0..5,thickness=2,frames=50);

 

>	Uc:=simplify(expand(limit(u(r,t),r=0)));

 

>	plot({Uc,150},t=0..5,thickness=2);

 

>	fsolve(Uc=150,t=4..6);

 

It takes approximately 5 hours to cook the turkey!!