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}] PDE[[1]]/(X[x] Y[y] T[t]) PDE[[2]]/(X[x] Y[y] T[t]) // Expand ODEt = T'[t] == -(k1^2 + k2^2) T[t] ODEx = X''[x] == -k1^2 X[x] ODEy = Y''[y] == -k2^2 Y[y] solX = DSolve[ODEx, X[x], x] (X[x] /. solX[[1]]) /. x -> 0 (X[x] /. solX[[1]]) /. x -> a k1 = (n Pi)/a solY = DSolve[ODEy, Y[y], y] (Y[y] /. solY[[1]]) /. y -> 0 (Y[y] /. solY[[1]]) /. y -> b k2 = (m Pi)/b DSolve[ODEt, T[t], t] Sin[ k1 x] Sin[k2 y] 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] 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}]] 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] 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] Εξειδικεύουμε a = 1 b = 2 f[x_, y_] := Sin[3 Pi x] Sin[Pi y] f[x, y] c[n, m] uAp[x, y, t, 5, 5] // FullSimplify cTable = Table[{n, m, c[n, m]}, {n, 1, 10}, {m, 1, 10}]; TableForm[Flatten[cTable, 1], TableHeadings -> {None, {"n", "m", "c[n,m]"}}] uAp[x, y, t, 3, 2] uSol[x_, y_, t_] := E^(-10 π^2 t) Sin[3 π x] Sin[π y] uSol[x, y, t] 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"}]