u
t
u
rr
-1
r
u
r
u(r,0)f(r)
u(a,t)0, t≥0
f(r)(r)
J
0
x
01
x
01
av1
u(r)R(r)T(t)
R(r)(t)(r)T(t)+(r)T(t)
′
T
′′
R
-1
r
′
R
R(r)T(t)
′
T
T(t)
′′
R
R(r)
-1
r
′
R
R(r)
r·(r)+(r)+λ·rR(r)0, 0<r<1(1)
′′
R
′
R
wr
xk·r
x+bar+x0(2).
′′
R
′
R
R
(2)
barR(x)(kr)
J
0
J
0
u(a,t)0,t≥0
R(a)0
J
0
k·a
k
m
x
0m
′
W
W(t)(3)
c
1
-λ·t
e
W(t)
c
1
-·t
2
k
n
e
u(r,t)··(r)
∞
∑
n1
d
m
-·t
2
k
n
e
J
0
k
m
d
m
2r·f(r)·(r)dr
a
∫
0
J
0
k
m
2
a
2
J
1
x
m
k
m
x
0m
a
We are going to find the solution according to the following data : where x01 is the first positive root of the zeroth order Bessel function (x) and .
u(ρ,t)
f(ρ)(ρ),
J
0
x
01
J
0
aν1
In[]:=
Clear["Global`*"]
In[]:=
a=1;
In[]:=
x01=N@BesselJZero[0,1];
In[]:=
f[r_]:=BesselJ[0,x01*r]
In[]:=
k1:=
x01
a
DuetotheorthogonalityoftheBesselfunctions,thenumeratorofiszeroexceptfor.Soisjust.
d
m
m1
u
u(r,t)exp-(r/a)
d
1
x
01
2
a
J
0
x
01
In[]:=
d1=*
2r*BesselJ[0,k1*r]*f[r]r
a
∫
0
2
a
2
BesselJ[1,x01]
Out[]=
1.
In[]:=
u[r_,t_]:=E^(-k1^2*t)*BesselJ[0,k1*r];
The 3D plot of the solution to this problem shows the temperature distribution in the disk as a function of radial distance ρ and time t.
The radial axis (ρ) would extend from the center of the disk (ρ=0) to the edge of the disk (ρ=a). The time axis (t) would start at t=0 and extend to some maximum time of interest. The vertical axis would represent the temperature u(ρ, t).
At t=0, the temperature distribution would match the initial condition f(ρ). As time progresses, you would see the temperature distribution evolve according to the heat equation.
The plot would show how heat diffuses from hotter regions to cooler regions over time. If the initial temperature distribution has a peak at some radius, you would see this peak spread out over time, as heat diffuses from the peak to the cooler regions of the disk.
The 3D plot provides a visual representation of the solution to the heat equation, showing how the temperature distribution in the disk evolves over time.
The radial axis (ρ) would extend from the center of the disk (ρ=0) to the edge of the disk (ρ=a). The time axis (t) would start at t=0 and extend to some maximum time of interest. The vertical axis would represent the temperature u(ρ, t).
At t=0, the temperature distribution would match the initial condition f(ρ). As time progresses, you would see the temperature distribution evolve according to the heat equation.
The plot would show how heat diffuses from hotter regions to cooler regions over time. If the initial temperature distribution has a peak at some radius, you would see this peak spread out over time, as heat diffuses from the peak to the cooler regions of the disk.
The 3D plot provides a visual representation of the solution to the heat equation, showing how the temperature distribution in the disk evolves over time.
In[]:=
Manipulate[ParametricPlot3D[{r*Cos[θ],r*Sin[θ],u[r,t]},{r,0,1},{θ,0,2Pi},PlotRange->{-1.2,1.2},BoundaryStyle->Directive[Red,Thick],ColorFunction->"SolarColors",Mesh->True],{t,0,10,0.0001},SaveDefinitions->True]
Out[]=
In[]:=
graphs=TableParametricPlot3D[{r*Cos[θ],r*Sin[θ],u[r,t]},{r,0,1},{θ,0,2Pi},PlotRange{-1.2,1.2},BoundaryStyleDirective[Red,Thick],ColorFunction"SolarColors"],t,0,2.5,
2.5
8
Out[]=
,
,
,
,
,
,
,
,
We simulate the phenomenon of heat diffusion in the disk, using the Animate function in combination with ParametricPlot3D for .
aν1
In[]:=
Animate[ParametricPlot3D[{r*Cos[θ],r*Sin[θ],u[r,t]},{r,0,1},{θ,0,2Pi},PlotRange{-1.2,1.2},BoundaryStyleDirective[Red,Thick],ColorFunction"SolarColors",MeshTrue],{t,0,3.5,0.0001},AnimationRunning->False,SaveDefinitions->True]
Out[]=