{"id":323,"date":"2012-06-25T17:28:33","date_gmt":"2012-06-25T08:28:33","guid":{"rendered":"http:\/\/www.bi.appi.keio.ac.jp\/?p=323"},"modified":"2012-10-24T10:06:40","modified_gmt":"2012-10-24T01:06:40","slug":"%e7%89%b9%e6%80%a7%e6%96%b9%e7%a8%8b%e5%bc%8f%e3%81%a8%e9%87%8d%e6%a0%b9","status":"publish","type":"post","link":"https:\/\/www.bi.appi.keio.ac.jp\/?p=323","title":{"rendered":"\u7279\u6027\u65b9\u7a0b\u5f0f\u3068\u91cd\u6839"},"content":{"rendered":"

\"\"<\/a>\u4eca\u65e5\u306e\u8b1b\u7fa9\u3067\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u7279\u6027\u6839\u3068\u4e00\u822c\u89e3\u306e\u8a71\u3092\u3057\u305f\u3089\uff0c\u91cd\u6839\u3092\u3082\u3064\u5834\u5408\u306b\u3064\u3044\u3066\uff0c\u9ad8\u6821\u3067\u7fd2\u3046\u96a3\u63a53\u9805\u6f38\u5316\u5f0f\u3067\u91cd\u6839\u3092\u6301\u3064\u5834\u5408\u306e\u89e3\u304d\u65b9\u3068\u540c\u3058\u8003\u3048\u65b9\u3067\u5fae\u5206\u65b9\u7a0b\u5f0f\u3082\u8aac\u660e\u3067\u304d\u308b\u3053\u3068\u304c\u5206\u304b\u3063\u305f\u3068\u8b1b\u7fa9\u306e\u5f8c\u3067\u6559\u3048\u3066\u304f\u308c\u307e\u3057\u305f\uff0e\u4ee5\u4e0b\u306b\u307e\u3068\u3081\u3066\u307f\u307e\u3059\uff0e<\/p>\n

\u96a3\u63a53\u9805\u6f38\u5316\u5f0f\u3067\u7279\u6027\u65b9\u7a0b\u5f0f\u304c\u91cd\u6839\u3092\u3082\u3064\u5834\u5408<\/h2>\n

[latex]y(k+2)=2ay(k+1)-a^2y(k)[\/latex]
\n\u3092\u5909\u5f62\u3057\u3066\uff0c\u5dee\u304c\u7b49\u6bd4\u6570\u5217\u306e\u5f62\u306b\u306a\u308b\u3088\u3046\u306b\u3059\u308b\uff0e<\/p>\n

[latex]y(k+2)-ay(k+1)=a(y(k+1)-ay(k))[\/latex]<\/p>\n

\u53f3\u8fba\u306f
\n[latex]a^{k+1}(y(1)-ay(0))[\/latex]
\n\u3068\u306a\u308b\uff0e<\/p>\n

\u4e21\u8fba\u3092[latex]a^{k+1}[\/latex]\u3067\u5272\u308b\uff0e<\/p>\n

[latex]
\n\\displaystyle\\frac{y(k+2)}{a^{k+1}}-\\frac{y(k+1)}{a^k}=y(1)-ay(0)
\n[\/latex]
\n\u3068\u306a\u308a\uff0c[latex]\\displaystyle\\frac{y(k+1)}{a^k}[\/latex]\u306f\uff0c\u521d\u9805[latex]y(0)\/a^{-1}[\/latex]\uff0c\u516c\u5dee[latex]y(1)-ay(0)[\/latex]\u306e\u7b49\u5dee\u6570\u5217\u306b\u306a\u308b\uff0e<\/p>\n

\u3057\u305f\u304c\u3063\u3066
\n[latex]
\n\\displaystyle\\frac{y(k)}{a^{k-1}}=\\frac{y(0)}{a^{-1}} + (k-1)(y(1)-ay(0))
\n[\/latex]
\n\u3068\u306a\u308b\uff0e<\/p>\n

\u4e21\u8fba\u3092[latex]a^{k-1}[\/latex]\u500d\u3057\u3066
\n[latex]
\ny(k)=a^ky(0)+a^{k-1}(k-1)(y(1)-ay(0))
\n[\/latex]
\n\u306b\u306a\u308b\uff0e<\/p>\n

\u96a3\u63a53\u9805\u6f38\u5316\u5f0f\u3067\u306f\uff0c[latex]y(k)[\/latex]\u3092\u6c42\u3081\u308b\u3053\u3068\u306b\u306a\u308b\u304c\uff0c\u7dda\u5f62\u5dee\u5206\u65b9\u7a0b\u5f0f\u3068\u307f\u305f\u5834\u5408\u306b\u306f\uff0c\u57fa\u5e95\u306b[latex]a^k[\/latex]\u3068[latex]ka^k[\/latex]\u3092\u7528\u3044\u308c\u3070\u3088\u3044\u3053\u3068\u304c\u5206\u304b\u308b\uff0e<\/p>\n

\u5fae\u5206\u65b9\u7a0b\u5f0f<\/h2>\n

\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u7279\u6027\u6839\u304c\u91cd\u6839\u306e\u5834\u5408\uff0c
\n[latex]
\n\\displaystyle\\frac{d^2y(t)}{dt^2} -2a\\frac{dy(t)}{dt} + a^2y(t)=0
\n[\/latex]
\n\u3092\u5909\u5f62\u3057\u3066
\n[latex]
\n\\displaystyle\\frac{d^2y(t)}{dt^2} – a\\frac{dy(t)}{dt} = a\\left(\\frac{dy(t)}{dt} – ay(t)\\right)
\n[\/latex]
\n\u3068\u306a\u308b\uff0e
\n\u3053\u308c\u306f\uff0c
\n[latex]
\n\\displaystyle\\frac{d}{dt}\\left(\\frac{dy(t)}{dt}-ay(t)\\right)= a\\left(\\frac{dy(t)}{dt} – ay(t)\\right)
\n[\/latex]
\n\u306a\u306e\u3067\uff0c
\n[latex]
\n\\displaystyle\\frac{dy(t)}{dt}-ay(t) = e^{at}\\left(\\frac{dy(0)}{dt} – ay(0)\\right)
\n[\/latex]
\n\u306b\u306a\u308b\uff0e
\n[latex]e^{at}[\/latex]\u3067\u4e21\u8fba\u3092\u5272\u308b\u3068\uff0c
\n[latex]
\ne^{-at}\\displaystyle\\frac{dy(t)}{dt}-ae^{-at}y(t) = \\frac{dy(0)}{dt} -ay(0)
\n[\/latex]
\n\u306a\u306e\u3067\uff0c\u5de6\u8fba\u3092\u6b21\u306e\u3088\u3046\u306b\u66f8\u304d\u63db\u3048\u3066
\n[latex]
\n\\displaystyle\\frac{d}{dt}\\left(e^{at}y(t)\\right) = \\frac{dy(0)}{dt} -ay(0)
\n[\/latex]
\n\u306b\u306a\u308b\uff0e
\n\u7a4d\u5206\u3057\u3066\uff0c
\n[latex]
\ne^{-at}y(t) = \\left(\\displaystyle\\frac{dy(0)}{dt} -ay(0)\\right)t + c
\n[\/latex]
\n\u306b\u306a\u308a\uff08[latex]c[\/latex]\u306f\u7a4d\u5206\u5b9a\u6570\uff09\uff0c
\n[latex]
\ny(t)=\\left(\\displaystyle\\frac{dy(0)}{dt} -ay(0)\\right)te^{at} + ce^{at}
\n[\/latex]
\n\u306b\u306a\u308b\uff0c
\n\u3064\u307e\u308a\uff0c\u57fa\u5e95\u306f[latex]e^{at}[\/latex]\u3068[latex]te^{at}[\/latex]\u306b\u306a\u308b\uff0e<\/p>\n

\u5b9f\u969b\u554f\u984c\u3068\u3057\u3066\uff0c\u91cd\u6839\u306e\u3068\u304d\u306e\u57fa\u5e95\u3092\u5c0e\u51fa\u3059\u308b\u6642\u9593\u306f\u306a\u3044\u3067\u3057\u3087\u3046\u304b\u3089\uff0c\u91cd\u6839\u306e\u3068\u304d\u306e\u57fa\u5e95\u3092\u899a\u3048\u3066\u304a\u304f\u3068\u3088\u3044\u3067\u3057\u3087\u3046\uff0e\u307e\u305f\uff0c\u8b1b\u7fa9\u3067\u8aac\u660e\u3057\u305f\u5b9a\u6570\u5909\u5316\u6cd5\u3092\u7406\u89e3\u3057\u3066\u304a\u304f\u3068\u3088\u3044\u3068\u601d\u3044\u307e\u3059\uff0e<\/p>\n","protected":false},"excerpt":{"rendered":"

\u4eca\u65e5\u306e\u8b1b\u7fa9\u3067\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u7279\u6027\u6839\u3068\u4e00\u822c\u89e3\u306e\u8a71\u3092\u3057\u305f\u3089\uff0c\u91cd\u6839\u3092\u3082\u3064\u5834\u5408\u306b\u3064\u3044\u3066\uff0c\u9ad8\u6821\u3067\u7fd2\u3046\u96a3\u63a53\u9805\u6f38\u5316\u5f0f\u3067\u91cd\u6839\u3092\u6301\u3064\u5834\u5408\u306e\u89e3\u304d\u65b9\u3068\u540c\u3058\u8003\u3048\u65b9\u3067\u5fae\u5206\u65b9\u7a0b\u5f0f\u3082\u8aac\u660e\u3067\u304d\u308b\u3053\u3068\u304c\u5206\u304b\u3063\u305f\u3068\u8b1b\u7fa9\u306e\u5f8c\u3067\u6559\u3048\u3066\u304f\u308c\u307e\u3057\u305f\uff0e\u4ee5\u4e0b\u306b\u307e\u3068\u3081\u3066\u307f … “\u7279\u6027\u65b9\u7a0b\u5f0f\u3068\u91cd\u6839” \u306e<\/span>\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[],"class_list":["post-323","post","type-post","status-publish","format-standard","hentry","category-9"],"_links":{"self":[{"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/posts\/323","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=323"}],"version-history":[{"count":38,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/posts\/323\/revisions"}],"predecessor-version":[{"id":450,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/posts\/323\/revisions\/450"}],"wp:attachment":[{"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=323"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=323"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=323"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}