ONE-DIMENSIONAL HEAT EQUATION
DIRICHLET Boundary Conditions
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiVUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEidEYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGMi9GOVEnbm9ybWFsRic= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2J1EnJiM5NjE7RicvJSVzaXplR1EjMTRGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiYtSSNtbkdGJDYlUSIyRidGMkY7RjIvRjZGOi9GPFEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGOw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiVUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEjeHhGJ0YyRjVGOEYyRjVGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjIvRjlRJ25vcm1hbEYn, t>0, 0<x<L
U=0, t>0, x=0 & x=L
U=f, t=0, 0<x<L
L:=2.0;
rho:=1/3;
SELECT INITIAL FUNCTION HERE
f(x) = 1 - H(x-L/2)
f := x -> 1 - Heaviside(x-L/2);
f(x) = x
f := x -> x;
f(x) = 1 - 2H(x-L/2)
f := x -> 1 - 2*Heaviside(x-L/4);
f(x) = 1 - H(x-L/4)
f := x -> 1 - 2*Heaviside(x-L/2);
f(x) = 1 - 2H(x-L/3) + 2H(x-2L/3)
f := x -> 1 - 2*Heaviside(x-L/3) + 2*Heaviside(x-2*L/3);
START SOLUTION HERE
plot(f(x),x=0..L,thickness=3);
Z:=(L,n,x) -> sin(n*Pi*x/L);
plot({Z(L,1,x),Z(L,2,x),Z(L,3,x)},x=0..L,thickness=3);
lambda:= (L,n) -> -n^2*Pi^2/L^2;
c:=(L,n) -> int(f(x)*Z(L,n,x),x=0..L)/int(Z(L,n,x)^2,x=0..L);
T:= (L,n,t) -> exp((rho^2)*t*lambda(L,n));
u:= (n,L,x,t) -> T(L,n,t)*Z(L,n,x);
U:= (m,L,x,t)-> sum(c(L,n)*u(n,L,x,t),n=1..m);
with(plots):animate({U(20,L,x,t),2,-2},x=0..L,t=0..8,frames=100,color=black,thickness=3);
IMPLICIT Boundary Conditions
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiVUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEidEYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGMi9GOVEnbm9ybWFsRic= = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2J1EnJiM5NjE7RicvJSVzaXplR1EjMTRGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiYtSSNtbkdGJDYlUSIyRidGMkY7RjIvRjZGOi9GPFEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGOw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiVUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEjeHhGJ0YyRjVGOEYyRjVGOC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjIvRjlRJ25vcm1hbEYn, t>0, 0<x<L
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiVUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEieEYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGMi9GOVEnbm9ybWFsRic==0, t>0, x=0 & LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiVUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNictRi82JlEieEYnRjJGNUY4RjJGNS8lMGZvbnRfc3R5bGVfbmFtZUdRJ05vcm1hbEYnRjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YyRkAvRjlRJ25vcm1hbEYn =U, t>0, x=L
U=f, t=0, 0<x<L
SELECT INITIAL FUNCTION HERE
(We are assuming here that L=1 and \317\201=1)
f(x) = 1 - H(x-1/2)
restart:
f := x -> (1 - Heaviside(x-1/2));
f(x) = x
f := x -> x;
f(x) = 1 - 2H(x-1/2)
f := x -> 1 - 2*Heaviside(x-1/2);
f(x) = 1 - H(x-1/4)
f := x -> 1 - 2*Heaviside(x-1/4);
f(x) = 1 - 2H(x-1/3) + 2H(x-2/3)
f := x -> 1 - 2*Heaviside(x-1/3) + 2*Heaviside(x-2/3);
START SOLUTION HERE
plot(f(x),x=0..1,thickness=3);
delta0 := fsolve(s*tanh(s)=1, s=.001..100);
lambda0 := delta0^2;
Z0 := x -> cosh(delta0*x);
plot(Z0(x),x=0..1,thickness=3);
for k from 1 to 50 do delta(k):= fsolve(s*sin(s)=-cos(s),s=(k-1)*Pi..k*Pi); od;
lambda:= n -> -delta(n)^2;
Z := (n,x) -> cos(delta(n)*x);
c0:= int(f(x)*Z0(x),x=0..1)/int(Z0(x)^2,x=0..1);
c:=n -> int(f(x)*Z(n,x),x=0..1)/int(Z(n,x)^2,x=0..1);
c(1);
c(2);
c(3);
plot({Z0(x),Z(1,x),Z(2,x),Z(3,x)},x=0..1,thickness=3);
T:= (n,t) -> exp(t*lambda(n));
u:= (n,x,t) -> T(n,t)*Z(n,x);
U:= (m,x,t)-> c0*exp(t*lambda0)*Z0(x) + sum(c(n)*u(n,x,t),n=1..m);
with(plots):animate({U(10,x,t),10,-10},x=0..1,t=0..2,frames=200,color=black,thickness=3);