Μερικές Διαφορικές Εξισώσεις

Εξίσωση κύματος ($\partial_{tt} u-c^2 \nabla^2 u=0$)

Τρεις μεταβλητές (πεπερασμένο χωρίο)

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

u^{(0,0,2)}(x,y,t)=u^{(0,2,0)}(x,y,t)+u^{(2,0,0)}(x,y,t)

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

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

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]}

ODEx = X''[x] == -kx^2 X[x]
ODEy = Y''[y] == -ky^2 Y[y]

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

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

solX = DSolve[ODEx, X[x], x]
solY = DSolve[ODEy, Y[y], y]

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

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

(X[x] /. solX[[1]]) /. x -> 0
(X[x] /. solX[[1]]) /. x -> 1

C_{1}

C_{1} \cos(kx)+C_{2} \sin(kx)

kx = n Pi
Sin[kx x]

n \pi

\sin(n \pi x)

(Y[y] /. solY[[1]]) /. y -> 0
(Y[y] /. solY[[1]]) /. y -> 3

C_{1}

C_{1} \cos(3 ky)+C_{2} \sin(3 ky)

ky = (m Pi)/3
Sin[ky y]

(\frac{1}{3}) m \pi

\sin((\frac{1}{3}) m \pi y)

ODEt = T''[t] == -(kx^2 + ky^2) T[t]
solt = DSolve[ODEt, T[t], t]

(T'')[t]=(-\frac{1}{9} (m^{2}) ({\pi }^{2})-(n^{2}) ({\pi }^{2})) T[t]

{{T[t]\to (E^{(\frac{1}{3}) (\sqrt{-(m^{2})-9 (n^{2})}) \pi t}) C_{1}+(E^{-\frac{1}{3} (\sqrt{-(m^{2})-9 (n^{2})}) \pi t}) C_{2}}}

Θα προσπαθήσουμε να αντικαταστήσουμε τα αρνητικά υπόριζα στις εκθετικές με τριγωνομετρικές συναρτήσεις.

solT = DSolveValue[ODEt, T[t], t]
solT = Expand[Assuming[kx^2 + ky^2 > 0, Simplify[solT]]]

(E^{(\frac{1}{3}) (\sqrt{-(m^{2})-9 (n^{2})}) \pi t}) C_{1}+(E^{-\frac{1}{3} (\sqrt{-(m^{2})-9 (n^{2})}) \pi t}) C_{2}

(E^{(\frac{1}{3}) I (\sqrt{m^{2}+9 (n^{2})}) \pi t}) C_{1}+(E^{-\frac{1}{3} I (\sqrt{m^{2}+9 (n^{2})}) \pi t}) C_{2}

solT = ExpToTrig[solT]

C_{1} \cos((\frac{1}{3}) (\sqrt{m^{2}+9 (n^{2})}) \pi t)+C_{2} \cos((\frac{1}{3}) (\sqrt{m^{2}+9 (n^{2})}) \pi t)+I C_{1} \sin((\frac{1}{3}) (\sqrt{m^{2}+9 (n^{2})}) \pi t)-I C_{2} \sin((\frac{1}{3}) (\sqrt{m^{2}+9 (n^{2})}) \pi t)

solT = Simplify[solT, {C[1] + I*C[1] == C[3], C[2] - I*C[2] == C[4]}]

(C_{1}+C_{2}) \cos((\frac{1}{3}) (\sqrt{m^{2}+9 (n^{2})}) \pi t)+I (C_{1}-C_{2}) \sin((\frac{1}{3}) (\sqrt{m^{2}+9 (n^{2})}) \pi t)

solT /. {(C[1] + C[2]) -> C[3], I (C[1] - C[2]) -> C[4]}

C_{3} \cos((\frac{1}{3}) (\sqrt{m^{2}+9 (n^{2})}) \pi t)+C_{4} \sin((\frac{1}{3}) (\sqrt{m^{2}+9 (n^{2})}) \pi t)

$u(x,y,t)=\sum _{n=1}^{\infty } \sin \left(\frac{\pi m y}{3}\right) \sin (\pi n x) \left(a_{n,m} \cos \left(\frac{1}{3} \pi t \sqrt{m^2+9 n^2}\right)+b_{n,m} \sin \left(\frac{1}{3} \pi t \sqrt{m^2+9 n^2}\right)\right)$. Άρα:

$\sum _{n=1}^{\infty } \sin \left(\frac{\pi m y}{3}\right) \sin (\pi n x) a_{n,m}=xy(1-x)(1-y)$
$b_{n,m}$, λόγω της $T'(0)=0$.

a[n_, m_] := 
 Integrate[  Sin[n π x] Sin[(m π y)/3] x y (1 - x) (3 - y), {x, 0, 1}, {y, 0,  3}]/Integrate[Sin[n π x]^2 Sin[(m π y)/3]^2, {x, 0, 1}, {y, 0, 3}]
Assuming[Element[n, Integers] && Element[m, Integers], a[n, m]]

\frac{144 (-1+{(-1)}^{m}) (-1+{(-1)}^{n})}{(m^{3}) (n^{3}) ({\pi }^{6})}

uAp[x_, y_, t_, n0_, m0_] := 
 Sum[(144 (-1 + (-1)^m) (-1 + (-1)^n))/(m^3 n^3 π^6)
    Cos[1/3 Sqrt[m^2 + 9 n^2] π t] Sin[n π x] Sin[(m π y)/3], {n, 1, 
   n0}, {m, 1, m0}]

uAp[x, y, t, 3, 3] // TrigReduce

-\frac{16 \cos((\sqrt{10}) \pi t-3 \pi x-\pi y)}{81 ({\pi }^{6})}-\frac{16 \cos((\sqrt{2}) \pi t-\pi x-\pi y)}{3 ({\pi }^{6})}+\frac{16 \cos((\sqrt{2}) \pi t+\pi x-\pi y)}{3 ({\pi }^{6})}+\frac{16 \cos((\sqrt{10}) \pi t+3 \pi x-\pi y)}{81 ({\pi }^{6})}-\frac{16 \cos((\frac{1}{3}) (\sqrt{82}) \pi t-3 \pi x-(\frac{1}{3}) \pi y)}{3 ({\pi }^{6})}-\frac{144 \cos((\frac{1}{3}) (\sqrt{10}) \pi t-\pi x-(\frac{1}{3}) \pi y)}{{\pi }^{6}}+\frac{144 \cos((\frac{1}{3}) (\sqrt{10}) \pi t+\pi x-(\frac{1}{3}) \pi y)}{{\pi }^{6}}+\frac{16 \cos((\frac{1}{3}) (\sqrt{82}) \pi t+3 \pi x-(\frac{1}{3}) \pi y)}{3 ({\pi }^{6})}+\frac{16 \cos((\frac{1}{3}) (\sqrt{82}) \pi t-3 \pi x+(\frac{1}{3}) \pi y)}{3 ({\pi }^{6})}+\frac{144 \cos((\frac{1}{3}) (\sqrt{10}) \pi t-\pi x+(\frac{1}{3}) \pi y)}{{\pi }^{6}}-\frac{144 \cos((\frac{1}{3}) (\sqrt{10}) \pi t+\pi x+(\frac{1}{3}) \pi y)}{{\pi }^{6}}-\frac{16 \cos((\frac{1}{3}) (\sqrt{82}) \pi t+3 \pi x+(\frac{1}{3}) \pi y)}{3 ({\pi }^{6})}+\frac{16 \cos((\sqrt{10}) \pi t-3 \pi x+\pi y)}{81 ({\pi }^{6})}+\frac{16 \cos((\sqrt{2}) \pi t-\pi x+\pi y)}{3 ({\pi }^{6})}-\frac{16 \cos((\sqrt{2}) \pi t+\pi x+\pi y)}{3 ({\pi }^{6})}-\frac{16 \cos((\sqrt{10}) \pi t+3 \pi x+\pi y)}{81 ({\pi }^{6})}

Plot3D[x y (1 - x) (3 - y), {x, 0, 1}, {y, 0, 3}]

Table[Plot3D[uAp[x, y, 0.1 n, 10, 10], {x, 0, 1}, {y, 0, 3}, 
  PlotLabel -> 0.1 n], {n, 0, 10}]