u
t
v*(
u
rr
+
-1
r
u
r
), 0≤r<a,t>0
​​
u(r,0)f(r)
u(a,t)0, t≥0
Weknowthat
f(r)
J
0
(
x
01
r)
,where
x
01
isthefirstpositiverootoftheBesselFunctionand
av1
.​​Nowwesolvetheproblemwiththemethodseparationofvariables.​​Nowwesolvetheproblemwiththemethodseparationofvariables.​​Let
u(r)R(r)T(t)
.​​Then,
R(r)
′
T
(t)
′′
R
(r)T(t)+
-1
r
′
R
(r)T(t)
​​Wedividebothsideswith
R(r)T(t)
​​Thus,​​​​
′
T
(t)
T(t)

′′
R
(r)
R(r)
+
-1
r
′
R
(r)
R(r)
-λ
​​​​Sowehavethefollowingtwoequations​​​​
r·
′′
R
(r)+
′
R
(r)+λ·rR(r)0, 0<r<1(1)
​​ItisSturmLiouvilleproblem,fromwhichwecanconclude​​theweightfunctionis
wr
.Wechangevariables
xk·r
thenweobtain​​
x
′′
R
+bar
′
R
+x
R
0(2).
​​Theequation
(2)
hasassolutiontheBesselfunction
barR
J
0
(x)
J
0
(kr)
.​​​​Theboundarycondition
u(a,t)0,t≥0
become
R(a)0
,so
J
0
(ka)0
.​​Thatmeans
k·a
mustbetherootoftheBesselfunctionwithzeroorder.​​​​
k
m
·a
x
0m
, m1,2,....
​​​​Now
′
W
(t)+λW(t)0
hasgeneralsolution​​​​
W(t)
c
1
-λ·t
e
(3)
​​​​Thenwehave
W(t)
c
1
-
2
k
n
·t
e
​​​​Nowfromsuperpositionthegeneralsolutiontotheproblemwillbe​​​​
u(r,t)
∞
∑
n1
d
m
·
-
2
k
n
·t
e
·
J
0
(
k
m
r)
andthecoefficientwillbe​​
d
m

2
a
∫
0
r·f(r)·
J
0
(
k
m
r)dr
2
a
2
J
1
(
x
m
)
,where
k
m

x
0m
a
​​NowIamtryingtowritetheMathematicacode
In[]:=
Clear["Global`*"]
In[]:=
a=1;
In[]:=
x01=N@BesselJZero[0,1];
In[]:=
f[r_]:=BesselJ[0,x01*r]
In[]:=
k1:=
x01
a
Due to the orthogonality of the Bessel functions, the numerator of
d
m
is zero except for
m1
. So
u
is just $
u(r,t)
d
1
exp-
x
01

2
a

J
0
(
x
01
r/a)
$ .
In[]:=
d1=
2
a
∫
0
r*BesselJ[0,k1*r]*f[r]r
2
a
*
2
BesselJ[1,x01]
Out[]=
1.
In[]:=
u[r_,t_]:=E^(-k1^2*t)*BesselJ[0,k1*r];
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}]
Out[]=
​
t
In[]:=
graphs=Table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"],t,0,2.5,
2.5
8

Out[]=

,
,
,
,
,
,
,
,
