Zadanie:
- Homogénna RVT \(u_t - a^2 u_{xx}= 0\) na intervale \((0,1)\)
- V krajných bodoch nula: \(u(0,t)=\sin(t), u(1,t)=\cos(t)\)
- Začiatočná podmienka: \(u(x,0) = x\)
Zo slajdov:
Transformácia \(u(x,t) = v(x,t) + w(x,t)\), kde \(w(x,t)\) je lineárna funkcia premennej \(x\) s rovnakými okrajovými podmienkami ako \(u(x,t)\): \[w(x,t) = \sin(t) + [\cos(t) - \sin(t)] x \]
Potom: \[u_t = v_t + \cos(t) + [-\sin(t) - \cos(t)] x, u_{xx} = v_{xx}\] z čoho dostaneme ODR pre \(v\): \[v_t - a^2 v_{xx} = -\cos(t) + [\sin(t) + \cos(t)]x\]
Začiatočná podmienka pre \(v\) je: \[x = u(x,0) = v(x,0)+ w(x,0) = v(x,0) + \sin(0) + [\cos(0)-\sin(0)]x \] \[\Rightarrow v(x,0)=0\]
Nulové okrajové podmienky: \(v(0,t)=v(1,t)=0\)
Teda \(v\) rieši nehomogénnu RVT s nulovými okrajovými podmienkami \(\rightarrow\) typ II. (teda vieme ju riešiť)
Riešenie PDR pre \(v(x,t)\)
- Rozvoj začiatočnej podmienky \[\alpha_k(0)=0\]
- Rozvoj pravej strany - z videa + Wolfram Alpha:

\[f_k(t) = B_k \sin(t) + (B_k-A_k) \cos(t)\]
- Z videa + doplnené Cramerovo pravidlo pre sústavu rovníc:

riesenie <- function(x,t){ # x = cislo (nie vektor)
N <- 20
k <- 1:N
a <- 1
Ak <- (2 - 2*(-1)^k)/(k * pi)
Bk <- 2 * (-1)^(k+1)/(k * pi)
Mk <- (-(Bk-Ak) - Bk * k^2 * pi^2 * a^2)/(-1 - k^4*pi^4*a^4)
Nk <- (Bk - k^2 * a^2 * pi^2 * (Bk - Ak))/(-1 - k^4 * pi^4 * a^4)
Ck <- (-Nk)
alpha_k <- Ck * exp(-k^2 * a^2 * pi^2*t) + Mk * sin(t) + Nk * cos(t)
v <- sum(alpha_k * sin(k*pi*x))
u <- v + sin(t) + x * (cos(t) - sin(t))
return(u)
}
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