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Ƽ

ĤäƤߤɤ񤤤Ƥ ݡȲȤƻȤäƤߤΤɤ

ε

  • f(x) = x^2 - 2 ζץ񤤤ưƤߤ褦 ʤ餫 ޢ2 Ǥ뤬ΤϤΤɤ餫ɤ ˡ¿Τǡʲ󤷤ƤɬפϤʤʣäƤۤ
    • ʬˡǡ
    • Ϥߤˡǡ
    • ˡǡ
    • ̾ˡ(򾯤񤭴ʤȤʤ)
    • Newton ˡǡ
  • f(x) = e^{-x} - x ζƱͤ˵Ƥߤ褦
  • f(x) = 3x^2 + 1 + (log (Pi - x))^2/(Pi^4) ˤĤƶ󥸤Ƥߤ褦 ֤󡢤ä񤷤
  • ϢΩǤ롢
       2x^2 + y^2 - 1 = 0,
       x - (3)y = 0,
    ζƤߤ褦 ֤ Newton ˡ
  • f(x) = e^{-x} - x ζ(ʰ)ۥȥԡˡǵƤߤ褦
  • f(x) = x^3 - 3x + 3 ζ򡢥ۥȥԡˡǵƤߤ褦

ϢΩ켡ε

  • LU ʬȤäϢΩ켡εԤץ񤤤Ƥߤ褦
    ʤΤΥץ񤯤ϡƼ׻ȤС٥ȥȥ٥ȥѤ׻롼񤤤ƤΤ񤤤ۤڤ
  • Ʊͤ CG ˡȤץ񤤤Ƥߤ褦

ʬε

  • du/dt = u(1-u) Ȥñζץ񤤤ưƤߤ褦 ˡȤƤϰʲͤʤΤ뤬Runge-Kutta ˡǥץबȤС줫¿нǽ
    • Euler ˡ,
    • Runge-Kutta ˡ,
    • ¿ʳˡ,
  • ϢΩʬ
       du/dt = (2-v)u,
       dv/dt = (2u - 3)v,
    ζץ񤤤ơưƤߤ褦 βϻȯŸ (u,v) ʿ̤Ʊ뤬֤ɤ줯餤Ƹ뤫ץդƳΤƤߤ褦

ʬε

  • ޤϡȤƤϾʬˤʤäƤޤͤ׻˴٤ˡȡ u = u(x) Ф 1
       u_{xx} = -C, (C > 0, const.) in [0,L],
       u(0) = 0, u(L) = 1,
    ζ򡢰ʲˡʤɤǷ׻Ƥߤ褦
    ǽʤС̩(Ƿ׻ǽ)ӤƤߤƤ⤤
    • ʬˡ,
    • ͭˡ
  • ˡ˥ץȤơǮȻζƤߤ褦 Ūˤȡ u = u(x) ФȯŸǡ
       u_t = u_{xx} in [0,L],
       u(0) = 0, u(L) = 1,
    ζ򡢰ʲˡʤɤǷ׻Ƥߤ褦
    ʤˡιͤ˴ŤơΤߤ򲼵ˡΥơ Runge-Kutta ˡʤɤǽƤ⤤
    ǽʤС̩(Ƿ׻ǽ)ӤƤߤƤ⤤
    • ʬˡ,
    • ͭˡ

¤¸Ͳˡ

ֲΤ ͲǤƸפȤŪŪȤͲˡ̤˹¤¸ͲˡȸƤ֤ιͤˤĤƤⲼΤ褦˻Ƥߤ褦

  • Greenspan 󾧤Newton ưФ륨ͥ륮¸ˡ˴𤤤ơ ñҤεưɽ(㤨в˵)ζᡢͥ륮Τ¸Ƥ뤫ǧƤߤ褦
       d^2 w/dt^2 = - a sin(w),
       w = w(t) ϿҤαľγ١ a := g/l, g:ϲ®١l: ҤλĹ.
  • ǮȻФơֶֶоΤʺʬˡ(Crank-Nicolson ȸƤФ) u^2 dx (u_x)^2 dx ֤ȤȤ˸ֻפƸ뤳ȤΤƤ롥 ºݤˡץ񤤤ưƳǧƤߤ褦
    ʤ Crank-Nicolson ϰʲΤȤꡥ
       { u_k^(n+1) - u_k^(n) }/t = { u_{k-1}^(n+1) - 2u_k^(n+1) + u_{k+1}^(n+1) + u_{k-1}^(n) - 2u_k^(n) + u_{k+1}^(n)} /(2x^2), for n = 0,1,2,..., k = 1,2,...,N-1,
       u_0^(n) = 0, u_N^(n) = 1,
       u_k^(n) u(kx, nt) ζ, N = L/x.