Μερικές Διαφορικές Εξισώσεις

Εξίσωση θερμότητας ($\partial_{t} u-\sigma \nabla^2 u=0$)

Με τρεις μεταβλητές

Clear["Global`*"]
u[x, y, t] = X[x] Y[y] T[t]
PDE = D[u[x, y, t], t] == 
  D[u[x, y, t], {x, 2}] + D[u[x, y, t], {y, 2}]

\(T[t] X[x] Y[y]\)

\[X[x] Y[y] (T')[t]=T[t] Y[y] (X'')[x]+T[t] X[x] (Y'')[y]\]

PDE[[1]]/(X[x] Y[y] T[t])
PDE[[2]]/(X[x] Y[y] T[t]) // Expand

\frac{(T')[t]}{T[t]}

\frac{(X'')[x]}{X[x]}+\frac{(Y'')[y]}{Y[y]}

ODEt = T'[t] == -(k1^2 + k2^2) T[t]
ODEx = X''[x] == -k1^2 X[x]
ODEy = Y''[y] == -k2^2 Y[y]

(T')[t]=(-({k1}^{2})-{k2}^{2}) T[t]

(X'')[x]=-({k1}^{2}) X[x]

(Y'')[y]=-({k2}^{2}) Y[y]

solX = DSolve[ODEx, X[x], x]
(X[x] /. solX[[1]]) /. x -> 0
(X[x] /. solX[[1]]) /. x -> a

{{X[x]\to C_{1} \cos(k1 x)+C_{2} \sin(k1 x)}}

C_{1}

C_{1} \cos(a k1)+C_{2} \sin(a k1)

k1 = (n Pi)/a

\frac{n \pi }{a}

solY = DSolve[ODEy, Y[y], y]
(Y[y] /. solY[[1]]) /. y -> 0
(Y[y] /. solY[[1]]) /. y -> b

{{Y[y]\to C_{1} \cos(k2 y)+C_{2} \sin(k2 y)}}

C_{1}

C_{1} \cos(b k2)+C_{2} \sin(b k2)

k2 = (m Pi)/b

\frac{m \pi }{b}

DSolve[ODEt, T[t], t]

{{T[t]\to (E^{-\frac{(m^{2}) ({\pi }^{2}) t}{b^{2}}-\frac{(n^{2}) ({\pi }^{2}) t}{a^{2}}}) C_{1}}}

Sin[ k1 x]
Sin[k2 y]

\sin(\frac{n \pi x}{a})

\sin(\frac{m \pi y}{b})

uTerm[x_, y_, t_, n_, m_] := 
 E^(-((m^2 π^2 t)/b^2) - (n^2 π^2 t)/a^2)  Sin[(n π x)/a] Sin[(m π y)/b]
 uTerm[x,y,t,n,m]

(E^{-\frac{(m^{2}) ({\pi }^{2}) t}{b^{2}}-\frac{(n^{2}) ({\pi }^{2}) t}{a^{2}}}) \sin(\frac{n \pi x}{a}) \sin(\frac{m \pi y}{b})

Θέλουμε $\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}c_{n.m}\operatorname{uTerm}(x,y,t,n,m)=u(x,y,t)$, άρα $\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}c_{n.m}\sin(\frac{n \pi x}{a})\sin(\frac{m \pi y}{b})=f(x,y)$.

Assuming[Element[n, Integers] && Element[m, Integers],  Integrate[  Integrate[Sin[(n Pi x)/a]^2 Sin[(m Pi y)/b]^2, {x, 0, a}], {y, 0, b}]]

(\frac{1}{4}) a b

c[n_, m_] := 
 Integrate[
  Integrate[f[x, y] Sin[(n Pi x)/a] Sin[(m Pi y)/b], {x, 0, a}], {y, 0, 
   b}]/((a b)/4)
c[n, m]

\frac{4 (\int_{0}^{b} \int_{0}^{a} f(x,y) \sin(\frac{n \pi x}{a}) \sin(\frac{m \pi y}{b}) \,dx \,dy)}{a b}

uAp[x_, y_, t_, n0_, m0_] := 
 Sum[c[n, m] uTerm[x, y, t, n, m], {n, 1, n0}, {m, 1, m0}]
uAp[x, y, t, 3, 4]

\frac{4 (E^{-\frac{({\pi }^{2}) t}{a^{2}}-\frac{({\pi }^{2}) t}{b^{2}}}) (\int_{0}^{b} \int_{0}^{a} f(x,y) \sin(\frac{\pi x}{a}) \sin(\frac{\pi y}{b}) \,dx \,dy) \sin(\frac{\pi x}{a}) \sin(\frac{\pi y}{b})}{a b}+\frac{4 (E^{-\frac{4 ({\pi }^{2}) t}{a^{2}}-\frac{({\pi }^{2}) t}{b^{2}}}) (\int_{0}^{b} \int_{0}^{a} f(x,y) \sin(\frac{2 \pi x}{a}) \sin(\frac{\pi y}{b}) \,dx \,dy) \sin(\frac{2 \pi x}{a}) \sin(\frac{\pi y}{b})}{a b}+\frac{4 (E^{-\frac{9 ({\pi }^{2}) t}{a^{2}}-\frac{({\pi }^{2}) t}{b^{2}}}) (\int_{0}^{b} \int_{0}^{a} f(x,y) \sin(\frac{3 \pi x}{a}) \sin(\frac{\pi y}{b}) \,dx \,dy) \sin(\frac{3 \pi x}{a}) \sin(\frac{\pi y}{b})}{a b}+\frac{4 (E^{-\frac{({\pi }^{2}) t}{a^{2}}-\frac{4 ({\pi }^{2}) t}{b^{2}}}) (\int_{0}^{b} \int_{0}^{a} f(x,y) \sin(\frac{\pi x}{a}) \sin(\frac{2 \pi y}{b}) \,dx \,dy) \sin(\frac{\pi x}{a}) \sin(\frac{2 \pi y}{b})}{a b}+\frac{4 (E^{-\frac{4 ({\pi }^{2}) t}{a^{2}}-\frac{4 ({\pi }^{2}) t}{b^{2}}}) (\int_{0}^{b} \int_{0}^{a} f(x,y) \sin(\frac{2 \pi x}{a}) \sin(\frac{2 \pi y}{b}) \,dx \,dy) \sin(\frac{2 \pi x}{a}) \sin(\frac{2 \pi y}{b})}{a b}+\frac{4 (E^{-\frac{9 ({\pi }^{2}) t}{a^{2}}-\frac{4 ({\pi }^{2}) t}{b^{2}}}) (\int_{0}^{b} \int_{0}^{a} f(x,y) \sin(\frac{3 \pi x}{a}) \sin(\frac{2 \pi y}{b}) \,dx \,dy) \sin(\frac{3 \pi x}{a}) \sin(\frac{2 \pi y}{b})}{a b}+\frac{4 (E^{-\frac{({\pi }^{2}) t}{a^{2}}-\frac{9 ({\pi }^{2}) t}{b^{2}}}) (\int_{0}^{b} \int_{0}^{a} f(x,y) \sin(\frac{\pi x}{a}) \sin(\frac{3 \pi y}{b}) \,dx \,dy) \sin(\frac{\pi x}{a}) \sin(\frac{3 \pi y}{b})}{a b}+\frac{4 (E^{-\frac{4 ({\pi }^{2}) t}{a^{2}}-\frac{9 ({\pi }^{2}) t}{b^{2}}}) (\int_{0}^{b} \int_{0}^{a} f(x,y) \sin(\frac{2 \pi x}{a}) \sin(\frac{3 \pi y}{b}) \,dx \,dy) \sin(\frac{2 \pi x}{a}) \sin(\frac{3 \pi y}{b})}{a b}+\frac{4 (E^{-\frac{9 ({\pi }^{2}) t}{a^{2}}-\frac{9 ({\pi }^{2}) t}{b^{2}}}) (\int_{0}^{b} \int_{0}^{a} f(x,y) \sin(\frac{3 \pi x}{a}) \sin(\frac{3 \pi y}{b}) \,dx \,dy) \sin(\frac{3 \pi x}{a}) \sin(\frac{3 \pi y}{b})}{a b}+\frac{4 (E^{-\frac{({\pi }^{2}) t}{a^{2}}-\frac{16 ({\pi }^{2}) t}{b^{2}}}) (\int_{0}^{b} \int_{0}^{a} f(x,y) \sin(\frac{\pi x}{a}) \sin(\frac{4 \pi y}{b}) \,dx \,dy) \sin(\frac{\pi x}{a}) \sin(\frac{4 \pi y}{b})}{a b}+\frac{4 (E^{-\frac{4 ({\pi }^{2}) t}{a^{2}}-\frac{16 ({\pi }^{2}) t}{b^{2}}}) (\int_{0}^{b} \int_{0}^{a} f(x,y) \sin(\frac{2 \pi x}{a}) \sin(\frac{4 \pi y}{b}) \,dx \,dy) \sin(\frac{2 \pi x}{a}) \sin(\frac{4 \pi y}{b})}{a b}+\frac{4 (E^{-\frac{9 ({\pi }^{2}) t}{a^{2}}-\frac{16 ({\pi }^{2}) t}{b^{2}}}) (\int_{0}^{b} \int_{0}^{a} f(x,y) \sin(\frac{3 \pi x}{a}) \sin(\frac{4 \pi y}{b}) \,dx \,dy) \sin(\frac{3 \pi x}{a}) \sin(\frac{4 \pi y}{b})}{a b}

Εξειδικεύουμε

a = 1
b = 2
f[x_, y_] := Sin[3 Pi x] Sin[Pi y]
f[x, y]

1

2

\sin(3 \pi x) \sin(\pi y)

c[n, m]

-\frac{24 \sin(m \pi ) \sin(n \pi )}{(-4+m^{2}) (-9+n^{2}) ({\pi }^{2})}

uAp[x, y, t, 5, 5] // FullSimplify

(E^{-10 ({\pi }^{2}) t}) \sin(3 \pi x) \sin(\pi y)

cTable = Table[{n, m, c[n, m]}, {n, 1, 10}, {m, 1, 10}];

TableForm[Flatten[cTable, 1], 
 TableHeadings -> {None, {"n", "m",  "c[n,m]"}}]

\begin{pmatrix} "n" & "m" & "c(n,m)" \\ 1 & 1 & 0 \\ 1 & 2 & 0 \\ 1 & 3 & 0 \\ 1 & 4 & 0 \\ 1 & 5 & 0 \\ 1 & 6 & 0 \\ 1 & 7 & 0 \\ 1 & 8 & 0 \\ 1 & 9 & 0 \\ 1 & 10 & 0 \\ 2 & 1 & 0 \\ 2 & 2 & 0 \\ 2 & 3 & 0 \\ 2 & 4 & 0 \\ 2 & 5 & 0 \\ 2 & 6 & 0 \\ 2 & 7 & 0 \\ 2 & 8 & 0 \\ 2 & 9 & 0 \\ 2 & 10 & 0 \\ 3 & 1 & 0 \\ 3 & 2 & 1 \\ 3 & 3 & 0 \\ 3 & 4 & 0 \\ 3 & 5 & 0 \\ 3 & 6 & 0 \\ 3 & 7 & 0 \\ 3 & 8 & 0 \\ 3 & 9 & 0 \\ 3 & 10 & 0 \\ 4 & 1 & 0 \\ 4 & 2 & 0 \\ 4 & 3 & 0 \\ 4 & 4 & 0 \\ 4 & 5 & 0 \\ 4 & 6 & 0 \\ 4 & 7 & 0 \\ 4 & 8 & 0 \\ 4 & 9 & 0 \\ 4 & 10 & 0 \\ 5 & 1 & 0 \\ 5 & 2 & 0 \\ 5 & 3 & 0 \\ 5 & 4 & 0 \\ 5 & 5 & 0 \\ 5 & 6 & 0 \\ 5 & 7 & 0 \\ 5 & 8 & 0 \\ 5 & 9 & 0 \\ 5 & 10 & 0 \\ 6 & 1 & 0 \\ 6 & 2 & 0 \\ 6 & 3 & 0 \\ 6 & 4 & 0 \\ 6 & 5 & 0 \\ 6 & 6 & 0 \\ 6 & 7 & 0 \\ 6 & 8 & 0 \\ 6 & 9 & 0 \\ 6 & 10 & 0 \\ 7 & 1 & 0 \\ 7 & 2 & 0 \\ 7 & 3 & 0 \\ 7 & 4 & 0 \\ 7 & 5 & 0 \\ 7 & 6 & 0 \\ 7 & 7 & 0 \\ 7 & 8 & 0 \\ 7 & 9 & 0 \\ 7 & 10 & 0 \\ 8 & 1 & 0 \\ 8 & 2 & 0 \\ 8 & 3 & 0 \\ 8 & 4 & 0 \\ 8 & 5 & 0 \\ 8 & 6 & 0 \\ 8 & 7 & 0 \\ 8 & 8 & 0 \\ 8 & 9 & 0 \\ 8 & 10 & 0 \\ 9 & 1 & 0 \\ 9 & 2 & 0 \\ 9 & 3 & 0 \\ 9 & 4 & 0 \\ 9 & 5 & 0 \\ 9 & 6 & 0 \\ 9 & 7 & 0 \\ 9 & 8 & 0 \\ 9 & 9 & 0 \\ 9 & 10 & 0 \\ 10 & 1 & 0 \\ 10 & 2 & 0 \\ 10 & 3 & 0 \\ 10 & 4 & 0 \\ 10 & 5 & 0 \\ 10 & 6 & 0 \\ 10 & 7 & 0 \\ 10 & 8 & 0 \\ 10 & 9 & 0 \\ 10 & 10 & 0 \end{pmatrix}

uAp[x, y, t, 3, 2]

(E^{-10 ({\pi }^{2}) t}) \sin(3 \pi x) \sin(\pi y)

uSol[x_, y_, t_] := E^(-10 π^2 t) Sin[3 π x] Sin[π y]
uSol[x, y, t]

(E^{-10 ({\pi }^{2}) t}) \sin(3 \pi x) \sin(\pi y)

DensityPlot[uSol[x, y, 0], {x, 0, a}, {y, 0, b}, PlotPoints -> 100, 
 PlotLegends -> {"x","y"}]
DensityPlot[uSol[x, y, 0.3], {x, 0, a}, {y, 0, b}, PlotPoints -> 100, 
 PlotLegends -> {"x","y"}]