In[]:=
(*model3*)
In[]:=
eq31=D[u[x,t],t]D[u[x,t],{x,2}]
Out[]=
(0,1)
u
(2,0)
u
In[]:=
eq32=D[u[x,t],x]+λu[x,t]F/.x->0
Out[]=
λu[0,t]+[0,t]F
(1,0)
u
In[]:=
eq33=D[u[x,t],x]0/.x1
Out[]=
(1,0)
u
In[]:=
eq34=u[x,t]u0/.t0
Out[]=
u[x,0]u0
In[]:=
(*Laplacetransform*)
In[]:=
eq36=LaplaceTransform[eq31,t,s]
Out[]=
sLaplaceTransform[u[x,t],t,s]-u[x,0]LaplaceTransform[x,t],t,s
(2,0)
u
In[]:=
eq37=LaplaceTransform[eq32,t,s]
Out[]=
λLaplaceTransform[u[0,t],t,s]+LaplaceTransform[0,t],t,s
(1,0)
u
F
s
In[]:=
eq38=LaplaceTransform[eq33,t,s]
Out[]=
LaplaceTransform[1,t],t,s0
(1,0)
u
In[]:=
eq39=LaplaceTransform[eq34,t,s]
Out[]=
u[x,0]u0
s
In[]:=
eq41=su1[x,s]-u0D[u1[x,s],{x,2}]
Out[]=
-u0+su1[x,s][x,s]
(2,0)
u1
In[]:=
eq42=λu1[x,s]+D[u1[x,s],x]==/.x0
F
s
Out[]=
λu1[0,s]+[0,s]
(1,0)
u1
F
s
In[]:=
eq43=D[u1[x,s],x]==0/.x1
Out[]=
(1,0)
u1
In[]:=
(*solutionaftertheLaplacetransform*)
In[]:=
eq44=DSolve[{eq41,eq42,eq43},u1[x,s],{x,s}]//Simplify//Normal
Out[]=
u1[x,s]u0(u0((-F+u0λ)+(-F+u0λ)
-
s
x
s
(2+x)
s
-λ)-s
x
s
+λ)+2
s
2
s
x
-1+-1+sλ
2
s
3/2
s
2
s
In[]:=
eq45=u1[x,s]/.eq44[[1]]//Simplify//Normal
Out[]=
-
s
x
s
(2+x)
s
-λ)-s
x
s
+λ)+2
s
2
s
x
-1+-1+sλ
2
s
3/2
s
2
s
In[]:=
eq46=CollectExpand[eq45],,,,Simplify
s
x
2
s
-s
x
-
s
x+s
(2+x)
Out[]=
-
s
x+s
(2+x)
s
-λ)-1+-1+sλ
2
s
3/2
s
2
s
u0(
s
+λ)-1+-1+sλ
2
s
3/2
s
2
s
s
x
-1+-1+sλ
2
s
3/2
s
2
s
2
s
-s
x
-1+-1+sλ
2
s
3/2
s
2
s
In[]:=
(*InverseLaplacetransform*)
In[]:=
eq51=f[x_,t_,F_,u0_,λ_]:=InverseLaplaceTransform[eq46,s,N@t,Method"Crump"]
In[]:=
eq52=ListLinePlot[Table[{t,f[1/2,t,1,2,3]},{t,0.001,4,1/30}],PlotRangeAll](*for:x=1/2,F=1,u0=2,λ=3*)
Out[]=
$Aborted
eq53=data=Flatten[Table[{x,t,f[x,t,1,2,3]},{x,-1,1,1/5},{t,0.001,3,1/5}],1];
eq54=ListPlot3D[data,InterpolationOrder3,AxesLabel{"x","t"}](*for:F=1,u0=2,λ=3*)