In[]:=
Clear["Global`*"]c=4;L=1;f[x_,y_]:=x*y*(1-x)*(1-y)g[x_,y_]:=0μ[n_]=;κ[m_]=;λ[n_,m_]=μ[n]+κ[m];ω[n_,m_]=c**SinA[n_,m_]:=f[x,y]*U[n,m]xyB[n_,m_]:=*ω[n,m]g[x,y]*U[n,m]xyT[n_,m_]:=A[n,m]*Cos[ω[n,m]*t]+B[n,m]*Sin[ω[n,m]*t]u[x_,y_,t_,N_]:=Sum[A[n,m]*Sin[n*Pi*x/L]*Sin[m*Pi*y/L]*Cos[ω[n,m]*t],{n,1,N},{m,1,N}](*Theexpressionforu1belowreferstothecaseofnonzeroinitialvelocityg(x,y)*)u1[x_,y_,t_,N_]:=Sum[Sin[n*Pi*x/L]*Sin[m*Pi*y/L]*(A[n,m]*Cos[ω[n,m]*t]+B[n,m]*Sin[ω[n,m]*t]),{n,1,N},{m,1,N}]
2
(n*π)
2
(m*π)
λ[n,m]
//SimplifyU[n_,m_]=Sinn*π*x
L
m*π*y
L
4
2
L
L
∫
0
L
∫
0
4
2
L
L
∫
0
L
∫
0
Out[]=
4+π
2
m
2
n
Out[]=
Sin[nπx]Sin[mπy]
In[]:=
B[n,m]
Out[]=
0
In[]:=
A[n,m]
Out[]=
4(-2+2Cos[mπ]+mπSin[mπ])(-2+2Cos[nπ]+nπSin[nπ])
3
m
3
n
6
π
In[]:=
A[3,m]//Simplify
Out[]=
-
16(-2+2Cos[mπ]+mπSin[mπ])
27
3
m
6
π
In[]:=
A[n,1]//Simplify
Out[]=
-
16(-2+2Cos[nπ]+nπSin[nπ])
3
n
6
π
In[]:=
A[3,1]
Out[]=
64
27
6
π
In[]:=
U[x_,y_,t_]=u[x,y,t,3]
Out[]=
64Cos[4
2
πt]Sin[πx]Sin[πy]6
π
64Cos[4
10
πt]Sin[3πx]Sin[πy]27
6
π
64Cos[4
10
πt]Sin[πx]Sin[3πy]27
6
π
64Cos[12
2
πt]Sin[3πx]Sin[3πy]729
6
π
In[]:=
u[x_,y_,t_]:=+++Manipulate[Plot3D[u[x,y,t],{x,0,1},{y,0,1},PlotRange{-1.5,1.5},PlotPoints50],{t,0.001,20}]
64Cos[4
2
πt]Sin[πx]Sin[πy]6
π
64Cos[4
10
πt]Sin[3πx]Sin[πy]27
6
π
64Cos[4
10
πt]Sin[πx]Sin[3πy]27
6
π
64Cos[12
2
πt]Sin[3πx]Sin[3πy]729
6
π
Out[]=
