In[]:=
Clear["Global`*"]​​c=4;​​a=1;​​b=1;
In[]:=
f[x_,y_]=x*y*(1-x)*(1-y)​​g[x_,y_]:=0
Out[]=
(1-x)x(1-y)y
In[]:=
​​μ[n_]=
2
n*π
a
;​​κ[m_]=
2
m*π
b
;​​λ[n_,m_]=μ[n]+κ[m];​​ω[n_,m_]=c*
λ[n,m]
//Simplify​​
Out[]=
4
2
m
+
2
n
π
In[]:=
U[n_,m_]=Sin
n*π*x
a
*Sin
m*π*y
b

Out[]=
Sin[nπx]Sin[mπy]
In[]:=
A[n_,m_]=FullSimplify
4
a*b
b
∫
0
a
∫
0
f[x,y]*U[n,m]xy,n∈Integers&&m∈Integers
Out[]=
16(-1+
m
(-1)
)(-1+
n
(-1)
)
3
m
3
n
6
π
In[]:=
B[n_,m_]=
4
a*b*w[n,m]
b
∫
0
a
∫
0
g[x,y]*U[n,m]xy
Out[]=
0
In[]:=
T[n_,m_]=A[n,m]*Cos[ω[n,m]*t]+B[n,m]*Sin[ω[n,m]*t]
Out[]=
16(-1+
m
(-1)
)(-1+
n
(-1)
)Cos4
2
m
+
2
n
πt
3
m
3
n
6
π
In[]:=
u[x_,y_,t_,l_,k_]:=
l
∑
n=1
k
∑
m=1
Sin[nπx]Sin[mπy]*
16(-1+
m
(-1)
)(-1+
n
(-1)
)Cos4
2
m
+
2
n
πt
3
m
3
n
6
π
In[]:=
u[x,y,t,1,1]//TraditionalForm​​u[x,y,t,1,2]//TraditionalForm​​u[x,y,t,2,2]//TraditionalForm​​u[x,y,t,3,3]//TraditionalForm
Out[]//TraditionalForm=
64cos(4
2
πt)sin(πx)sin(πy)
6
π
Out[]//TraditionalForm=
64cos(4
2
πt)sin(πx)sin(πy)
6
π
Out[]//TraditionalForm=
64cos(4
2
πt)sin(πx)sin(πy)
6
π
Out[]//TraditionalForm=
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[]:=
graphs=TableParametricPlot3D[{x,y,u[x,y,t,3,3]},{x,0,2},{y,0,3},AxesFalse,BoxedFalse,BoxRatios{1,1,1},ImageSize600,BoundaryStyleDirective[Red,Thick],ColorFunction"MintColors",PlotRangeAll],t,0,2,
1
4
;​​p1=Partition[graphs,3];​​Show[GraphicsGrid[p1]]
Out[]=
In[]:=
u[x_,y_,t_,l_,k_]:=
l
∑
n=1
k
∑
m=1
Sin[nπx]Sin[mπy]*
16(-1+
m
(-1)
)(-1+
n
(-1)
)Cos4
2
m
+
2
n
πt
3
m
3
n
6
π
​​Manipulate[ParametricPlot3D[{x,y,u[x,y,t,3,3]},{x,0,2},{y,0,3},PlotRange{{0,2},{0,3},{-7,7}},BoundaryStyleDirective[Blue,Thick],ColorFunction"SolarColors",MeshTrue,BoxRatios{1,1,1}],{t,0,10,0.0001}]
Out[]=
​
t