{"id":3019,"date":"2020-06-25T20:34:30","date_gmt":"2020-06-25T11:34:30","guid":{"rendered":"https:\/\/www.ktech.biz\/jp\/?p=3019"},"modified":"2020-08-10T23:52:26","modified_gmt":"2020-08-10T14:52:26","slug":"10-ode-3","status":"publish","type":"post","link":"https:\/\/www.ktech.biz\/jp\/num\/10-ode-3\/","title":{"rendered":"10. \u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f (3) \u89e3\u6cd5\u306e\u5b89\u5b9a\u6027, \u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f, \u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f"},"content":{"rendered":"

\u76ee\u6b21<\/h3>\n

\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f (1)<\/a>: \u6982\u8981, \uff11\u6bb5\u6cd5, \u591a\u6bb5\u6cd5, \u4f7f\u308f\u308c\u306a\u304f\u306a\u3063\u305f\u89e3\u6cd5
\n\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f (2)<\/a>: \u8aa4\u5dee\u306e\u63a8\u5b9a, \u88dc\u5916\u6cd5, \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u81ea\u52d5\u8abf\u7bc0, \u5bc6\u51fa\u529b, 2\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f, \u9045\u5ef6\u5fae\u5206\u65b9\u7a0b\u5f0f
\n\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f (3): \u89e3\u6cd5\u306e\u5b89\u5b9a\u6027, \u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f, \u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f
\n\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f (4)<\/a>: \u5b9f\u7528\u30eb\u30fc\u30c1\u30f3\u306e\u30d9\u30f3\u30c1\u30de\u30fc\u30af (\u5b9f\u7528\u30eb\u30fc\u30c1\u30f3\u306e\u9078\u629e, \u30b9\u30c6\u30a3\u30d5\u3067\u306f\u306a\u3044\u554f\u984c, \u30b9\u30c6\u30a3\u30d5\u306a\u554f\u984c, \u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f)<\/p>\n

\u89e3\u6cd5\u306e\u5b89\u5b9a\u6027<\/span>\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f<\/span>\u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f<\/span><\/div>
\n

\u89e3\u6cd5\u306e\u5b89\u5b9a\u6027<\/h2>\n

\u521d\u671f\u5024\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u308b\u9593\u306b\u6570\u5024\u89e3\u304c\u7570\u5e38\u306a\u5024\u306b\u306a\u3063\u305f\u308a\u767a\u6563\u3057\u305f\u308a\u3059\u308b\u3053\u3068\u304c\u3042\u308b. \u305d\u306e\u539f\u56e0\u306f, \u5fae\u5206\u65b9\u7a0b\u5f0f\u81ea\u4f53\u306e\u554f\u984c\u3067\u3042\u308b\u5834\u5408\u3068, \u6570\u5024\u89e3\u6cd5\u306e\u554f\u984c\u3067\u3042\u308b\u5834\u5408\u304c\u3042\u308b.<\/p>\n

\u5fae\u5206\u65b9\u7a0b\u5f0f\u81ea\u4f53\u306e\u5b89\u5b9a\u6027<\/h3>\n

\u300c\u8aa4\u5dee\u306e\u63a8\u5b9a\u300d\u306e\u8aac\u660e\u56f3\u306b\u793a\u3055\u308c\u308b\u3088\u3046\u306b, \u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u89e3\u66f2\u7dda\u304c\u6642\u9593\u304c\u9032\u3080\u306b\u3064\u308c\u3066\u96e2\u308c\u3066\u3044\u304f\u5834\u5408, \u521d\u671f\u5024\u306e\u308f\u305a\u304b\u306a\u5dee\u304c\u5927\u304d\u304f\u62e1\u5927\u3059\u308b\u73fe\u8c61\u304c\u751f\u3058\u308b\u3053\u3068\u304c\u3042\u308b. \u3053\u308c\u306f\u4e0d\u5b89\u5b9a\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4f8b\u3067\u3042\u308b.<\/p>\n

\u9006\u306b, \u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u89e3\u66f2\u7dda\u304c\u6642\u9593\u304c\u9032\u3080\u306b\u3064\u308c\u3066\u8fd1\u3065\u3044\u3066\u3044\u304f\u5834\u5408, \u3080\u3057\u308d\u8aa4\u5dee\u304c\u7e2e\u5c0f\u3057\u3066\u3044\u304f\u306e\u3067\u5b89\u5b9a\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4f8b\u3067\u3042\u308b.<\/p>\n

\u7e70\u308a\u8fd4\u3059\u3068, \u95a2\u6570\u304c\u3069\u3061\u3089\u306e\u50be\u5411\u3092\u793a\u3059\u304b\u306f\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e \u2202f\/\u2202y \u306e\u7b26\u53f7\u306b\u3088\u308a\u6c7a\u307e\u308b.<\/p>\n

\u6570\u5024\u89e3\u6cd5\u306e\u5b89\u5b9a\u6027<\/h3>\n

\u5b89\u5b9a\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u3042\u3063\u3066\u3082, \u4f7f\u7528\u3059\u308b\u6570\u5024\u89e3\u6cd5\u306b\u3088\u3063\u3066\u4e0d\u5b89\u5b9a\u73fe\u8c61\u3092\u793a\u3059\u3053\u3068\u304c\u3042\u308b. \u3053\u308c\u306f\u6570\u5024\u89e3\u6cd5\u306e\u5b89\u5b9a\u6027\u306e\u554f\u984c\u3067\u3042\u308b.<\/p>\n

\u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u5236\u7d04<\/h4>\n

\u3044\u307e, \u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u5c0e\u95a2\u6570 \u2202f\/\u2202y \u304c\u4e00\u5b9a\u5024 \u03bb \u3067, \u03bb < 0 \u3068\u3059\u308b(\u3053\u308c\u306f\u5b89\u5b9a\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u3067\u3042\u308b). \u307e\u305f, \u5404\u30b9\u30c6\u30c3\u30d7\u3067\u52a0\u3048\u3089\u308c\u308b\u8aa4\u5dee\u306f\u30b9\u30c6\u30c3\u30d7\u306b\u3088\u3089\u305a\u4e00\u5b9a\u3067\u3042\u308b\u3068\u3059\u308b. \u305d\u306e\u5834\u5408, \u6570\u5024\u89e3\u6cd5\u306e\u5b89\u5b9a\u6027\u306f\u6b21\u306e\u7dda\u5f62\u30c6\u30b9\u30c8\u554f\u984c\u3092\u8abf\u3079\u308b\u3053\u3068\u3067\u308f\u304b\u308b.<\/p>\n

  \n  dy\/dt = \u03bby (t > 0, \u03bb(\u5b9a\u6570) < 0), y(0) = y0<\/sub>\n\n<\/pre>\n
 
\n\u3053\u306e\u554f\u984c\u306e\u89e3\u306f,<\/p>\n
  \n  y(t) = y0<\/sub>exp(\u03bbt)\n\n<\/pre>\n
 
\n\u3068\u306a\u308b\u304b\u3089, t \u2192 \u221e \u306e\u3068\u304d y(t) \u2192 0 \u3068\u306a\u308b.<\/p>\n
(1) \u30aa\u30a4\u30e9\u30fc\u6cd5<\/p>\n
\u4e0a\u306e\u30c6\u30b9\u30c8\u554f\u984c\u3092\u30aa\u30a4\u30e9\u30fc\u6cd5\u3067\u89e3\u304f\u3068,<\/p>\n
  \n  yn<\/sub> = y0<\/sub>(1 + h\u03bb)n<\/sup>\n\n<\/pre>\n
 
\n\u3068\u306a\u308b\u304b\u3089, n \u2192 \u221e \u306e\u3068\u304d yn<\/sub> \u2192 0 \u3068\u306a\u3089\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044. \u305d\u306e\u305f\u3081\u306b\u306f,<\/p>\n
  \n  |1 + h\u03bb| < 1\n\n<\/pre>\n
 
\n\u3067\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044. \u3053\u308c\u3092\u5b89\u5b9a\u6761\u4ef6\u3068\u3044\u3046.<\/p>\n
\u03bb\u3092\u8907\u7d20\u6570\u3068\u3057\u3066 z = h\u03bb \u3068\u304a\u304d, \u5b89\u5b9a\u6761\u4ef6 |R(z)| < 1 (\u305f\u3060\u3057, R(z) = 1 + z) \u3092\u8907\u7d20\u5e73\u9762\u4e0a\u306b\u56f3\u793a\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
\"ode_euler\"<\/p>\n
\u8272\u4ed8\u304d\u90e8\u5206 { z \u2208 C; |R(z)| < 1 } \u3092\u7d76\u5bfe\u5b89\u5b9a\u9818\u57df\u3068\u3044\u3046. \u03bb\u304c\u5b9f\u6570\u3067\u3042\u308c\u3070 -2 < z = h\u03bb < 0 \u3067\u3042\u308b. \u533a\u9593 [-2, 0] \u3092\u7d76\u5bfe\u5b89\u5b9a\u533a\u9593\u3068\u3044\u3046. \u3053\u306e\u3068\u304d, \u30b9\u30c6\u30c3\u30d7\u5e45\u306f 0 < h < -2\/\u03bb \u3068\u3057\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044.<\/p>\n
(2) \u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5<\/p>\n
\u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5\u3067\u306fR(z)\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
  \n  R(z) = 1\/(1 - z)\n\n<\/pre>\n
 
\n\u7d76\u5bfe\u5b89\u5b9a\u9818\u57df\u3092\u56f3\u793a\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
\"ode_beuler\"<\/p>\n
\u3053\u306e\u3088\u3046\u306b, \u5de6\u534a\u5e73\u9762\u304c\u7d76\u5bfe\u5b89\u5b9a\u9818\u57df\u306b\u542b\u307e\u308c\u308b\u3068\u304d\u306bA\u5b89\u5b9a\u3067\u3042\u308b\u3068\u3044\u3044, \u30b9\u30c6\u30c3\u30d7\u5e45 h \u306b\u5bfe\u3059\u308b\u5236\u9650\u304c\u306a\u304f\u306a\u308b.<\/p>\n
(3) \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/p>\n
s\u6bb5p\u6b21\u306eRK\u6cd5\u3067\u306f R(z) \u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
  \n  R(z) = 1 + z + z2<\/sup>\/2! + ... + zp<\/sup>\/p! + \u03b3p+1<\/sub>zp+1<\/sup> + ... + \u03b3s<\/sub>zs<\/sup>\n\n<\/pre>\n
 
\n\u3053\u3053\u3067, \u03b3p+1<\/sub>, … \u306fRK\u6cd5\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u306b\u3088\u308a\u6c7a\u307e\u308b.<\/p>\n
p = s = 2, 3, 4 \u306b\u3064\u3044\u3066\u7d76\u5bfe\u5b89\u5b9a\u9818\u57df\u3092\u56f3\u793a\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
\"ode_rk\"<\/p>\n
\u6b21\u6570\u304c\u4e0a\u304c\u308b\u3068\u5b89\u5b9a\u9818\u57df\u3082\u5e83\u304f\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u7d76\u5bfe\u5b89\u5b9a\u533a\u9593\u306f\u305d\u308c\u305e\u308c, [-2, 0], [-2.513, 0], [-2.785, 0] \u3067\u3042\u308b. \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u967d\u89e3\u6cd5\u306f A\u5b89\u5b9a\u3067\u306f\u306a\u3044.<\/p>\n
(4) \u591a\u6bb5\u6cd5 (\u4e88\u6e2c\u5b50\u30fb\u4fee\u6b63\u5b50\u6cd5)<\/p>\n
\u591a\u6bb5\u6cd5\u3082\u967d\u89e3\u6cd5\u306f A\u5b89\u5b9a\u3067\u306f\u306a\u3044.<\/p>\n
\u591a\u6bb5\u6cd5\u306e\u9670\u89e3\u6cd5\u3067\u306f A\u5b89\u5b9a\u306a\u89e3\u6cd5\u306f2\u6b21\u4ee5\u4e0b\u3067\u3042\u308b (A\u5b89\u5b9a\u306a\u591a\u6bb5\u6cd5\u3067\u6700\u3082\u9ad8\u6b21\u306a\u3082\u306e\u306f\u53f0\u5f62\u5247\u3067\u3042\u308b).<\/p>\n
(5) \u53f0\u5f62\u5247 (\u9670\u89e3\u6cd5, \u30a2\u30c0\u30e0\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5\u306e k=1)<\/p>\n
\u53f0\u5f62\u5247\u3067\u306fR(z)\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
  \n  R(z) = (1 + z\/2)\/(1 - z\/2)\n\n<\/pre>\n
 
\n\u3053\u306e\u7d76\u5bfe\u5b89\u5b9a\u9818\u57df\u306f\u5de6\u534a\u5e73\u9762\u5168\u90e8\u3068\u306a\u308a, \u53f0\u5f62\u5247\u306fA\u5b89\u5b9a\u3067\u3042\u308b.<\/p>\n
\u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u5236\u7d04 (\u9023\u7acb\u65b9\u7a0b\u5f0f\u306e\u5834\u5408)<\/h4>\n
\u9023\u7acb\u65b9\u7a0b\u5f0f\u306e\u5834\u5408, \u7dda\u5f62\u30c6\u30b9\u30c8\u554f\u984c\u3067 y \u3092\u30d9\u30af\u30c8\u30eb\u306b\u62e1\u5f35\u3057\u3066\u6271\u3046.<\/p>\n
  \n  dy<\/strong>\/dt = Jy<\/strong>, y<\/strong>(0) = y0<\/sub><\/strong>\n\n<\/pre>\n
 
\nJ \u306f y<\/strong> \u306e\u5143\u6570\u3068\u540c\u3058\u6b21\u5143\u6570\u306e\u6b63\u65b9\u884c\u5217\u3068\u306a\u308b. \u5b89\u5b9a\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u3067\u3042\u308b\u305f\u3081\u306e\u6761\u4ef6\u306f, J \u306e\u3059\u3079\u3066\u306e\u56fa\u6709\u5024 \u03bbi<\/sub> \u306e\u5b9f\u90e8\u304c\u8ca0\u3067\u3042\u308b\u3053\u3068\u3067\u3042\u308b.<\/p>\n
  \n  Re(\u03bbi<\/sub>) < 0 (\u3059\u3079\u3066\u306e i \u306b\u3064\u3044\u3066)\n\n<\/pre>\n
 
\n\u307e\u305f, z = h\u03bbi<\/sub> \u3068\u3057\u3066, \u8907\u7d20\u5e73\u9762\u4e0a\u3067 z \u304c\u7d76\u5bfe\u5b89\u5b9a\u9818\u57df\u306b\u542b\u307e\u308c\u3066\u3044\u308c\u3070\u305d\u306e h \u306b\u3064\u3044\u3066\u5b89\u5b9a\u3067\u3042\u308b.<\/p>\n
\u6570\u5024\u5b9f\u9a13 (20)<\/h3>\n
<\/span>\u30aa\u30a4\u30e9\u30fc\u6cd5\u3068\u30b9\u30c6\u30c3\u30d7\u5e45<\/div>
\n\u6b21\u306e\u521d\u671f\u5024\u554f\u984c\u3092\u30aa\u30a4\u30e9\u30fc\u6cd5\u306b\u3088\u308a\u89e3\u304f.<\/p>\n
  \n  y' = -10y, y(0) = 1\n\n<\/pre>\n
 
\n\u3053\u306e\u65b9\u7a0b\u5f0f\u306e\u89e3\u306f, y(t) = exp(-10t) \u3067\u3042\u308b. \u30b9\u30c6\u30c3\u30d7\u5e45\u3092 0.09, 0.19, 0.2, 0.21 \u3068\u5909\u3048\u3066\u8a08\u7b97\u3057\u3066\u307f\u305f\u7d50\u679c, \u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f.<\/p>\n
\"ode_test-8\"<\/p>\n
h = 0.2 \u3092\u5883\u306b, \u3053\u308c\u3088\u308a\u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u5927\u304d\u304f\u3059\u308b\u3068\u767a\u6563\u3057\u305f. 0.2 \u306f\u4e0a\u306e(1)\u3067\u8aac\u660e\u3057\u305f\u7d76\u5bfe\u5b89\u5b9a\u533a\u9593\u306e\u4e0a\u9650\u306b\u3042\u305f\u308a, \u3053\u308c\u3092\u8d85\u3048\u308b\u3068\u4e0d\u5b89\u5b9a\u306b\u306a\u308b\u3053\u3068\u304c\u78ba\u304b\u3081\u3089\u308c\u305f. \u306a\u304a, 0.1 < h < 0.2 \u3067\u306f\u632f\u52d5\u3057\u306a\u304c\u3089\u53ce\u675f\u3059\u308b\u304c, 0.1 < h \u3067\u306f\u632f\u52d5\u3057\u306a\u3044.<\/p>\n
t = 0 \u306e\u8fd1\u304f\u306e\u5024\u304c\u6025\u6fc0\u306b\u5909\u5316\u3059\u308b\u90e8\u5206\u3092\u907f\u3051\u3066, \u5024\u304c\u304b\u306a\u308a\u843d\u3061\u7740\u3044\u305f t = 0.6 \u304b\u3089\u8a08\u7b97\u3092\u59cb\u3081\u3066\u307f\u308b. \u3059\u306a\u308f\u3061, \u521d\u671f\u6761\u4ef6\u3092 y(0.6) = exp(6) \u3068\u5909\u3048\u3066\u8a08\u7b97\u3057\u3066\u307f\u308b\u3068, \u7d50\u679c\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f.<\/p>\n
\"ode_test-8_2\"<\/p>\n
\u540c\u3058\u3088\u3046\u306a\u50be\u5411\u3092\u793a\u3057, \u767a\u6563\u304c\u6b62\u307e\u3089\u306a\u3044\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n
A\u5b89\u5b9a\u3067\u306a\u3044\u89e3\u6cd5\u3092\u4f7f\u3046\u9650\u308a, \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u5024\u306b\u306f\u4e0a\u9650\u304c\u3042\u308b.
\n<\/div><\/div>\n
\u6ce8 – \u300c\u4f7f\u308f\u308c\u306a\u304f\u306a\u3063\u305f\u89e3\u6cd5\u300d\u3067\u8aac\u660e\u3057\u305f\u30df\u30eb\u30f3\u6cd5\u306e\u4e0d\u5b89\u5b9a\u6027\u306f\u3053\u3053\u3067\u8aac\u660e\u3057\u305f\u5b89\u5b9a\u6027\u3068\u306f\u5225\u306e\u554f\u984c\u3067, \u30b9\u30c6\u30c3\u30d7\u5e45\u3092\u5c0f\u3055\u304f\u3057\u3066\u3082\u4e0d\u5b89\u5b9a\u6027\u306f\u304a\u3055\u307e\u3089\u306a\u3044.<\/p>\n<\/div>\n
\n
\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f<\/h2>\n
\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f\u306e\u4f8b<\/h3>\n
\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f(\u786c\u3044\u65b9\u7a0b\u5f0f\u3068\u3082\u3044\u3046)\u306f\u5316\u5b66\u53cd\u5fdc\u3084\u96fb\u6c17\u56de\u8def\u306a\u3069\u306e\u554f\u984c\u3067\u73fe\u308c\u308b. \u975e\u5e38\u306b\u6025\u306a\u5909\u5316\u3092\u3059\u308b\u89e3\u3092\u6301\u3061, \u901a\u5e38\u306e\u89e3\u6cd5\u3067\u306f\u89e3\u304d\u306b\u304f\u3044\u3082\u306e\u3092\u3044\u3046.<\/p>\n
\u6b21\u306e\u521d\u671f\u5024\u554f\u984c\u306f\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f\u306e\u4f8b\u3067\u3042\u308b.<\/p>\n
  \n  dy\/dt = ( -2    1    ) y + ( -cos(t)             )\n          ( 1998 -1999 )     ( 1999cos(t) - sin(t) )\n\n  y(0) = ( 1 )\n         ( 2 )\n\n<\/pre>\n
 
\n\u4e00\u822c\u89e3\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<\/p>\n
  \n  y(t) = C1<\/sub>exp(-t) ( 1 ) + C2<\/sub>exp(-2000t) (   1   ) + (   0    )\n                   ( 1 )                 ( -1998 )   ( cos(t) )\n\n<\/pre>\n
 
\n\u4e0a\u306e\u65b9\u7a0b\u5f0f\u306e y \u306e\u4fc2\u6570\u884c\u5217\u306e\u56fa\u6709\u5024\u306f\u5b9f\u6570\u3067, -1 \u304a\u3088\u3073 -2000 \u3067\u3042\u308b. \u5f93\u3063\u3066, \u300c\u89e3\u6cd5\u306e\u5b89\u5b9a\u6027\u300d\u306e\u8b70\u8ad6\u304b\u3089, \u30aa\u30a4\u30e9\u30fc\u6cd5\u306a\u3089\u3070\u30b9\u30c6\u30c3\u30d7\u5e45 h < 0.001, 4\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306a\u3089\u3070 h < 0.00139 \u3068\u3057\u306a\u3051\u308c\u3070\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u306a\u3044\u306f\u305a\u3067\u3042\u308b. \u5b9f\u969b\u306b\u8a66\u3057\u3066\u307f\u308b\u3068, h \u3092\u3053\u308c\u3089\u306e\u5236\u7d04\u6761\u4ef6\u3088\u308a\u3082\u5927\u304d\u304f\u3059\u308b\u3068\u767a\u6563\u3057\u3066\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u306a\u3044\u3053\u3068\u304c\u78ba\u304b\u3081\u3089\u308c\u305f.<\/p>\n
XLPack\u306b\u53ce\u9332\u3055\u308c\u3066\u3044\u308b\u901a\u5e38\u306e\u30b9\u30c6\u30c3\u30d7\u5e45\u81ea\u52d5\u8abf\u7bc0\u6a5f\u80fd\u4ed8\u304d\u306e\u30eb\u30fc\u30c1\u30f3 Derkf, Dverk, Deabm \u3067\u540c\u3058\u4f8b\u984c\u3092 Tol = 0.01 \u3068\u3057\u3066\u89e3\u3044\u3066\u307f\u305f. \u305d\u308c\u305e\u308c\u975e\u5e38\u306b\u5c0f\u3055\u306a\u30b9\u30c6\u30c3\u30d7\u5e45\u304c\u4f7f\u308f\u308c\u306a\u304c\u3089\u3082\u89e3\u304c\u6c42\u3081\u3089\u308c\u305f. \u5f8c\u3067\u793a\u3059BDF\u6cd5\u306b\u6bd4\u3079\u308b\u3068100\u500d\u304f\u3089\u3044\u306e\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u3092\u8981\u3057\u305f.<\/p>\n
\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f\u306e\u5b9a\u7fa9<\/h3>\n
\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f\u306f, \u4e0a\u306e\u4f8b\u306e\u3088\u3046\u306b\u5b89\u5b9a\u6027\u306e\u554f\u984c\u306b\u3088\u308a\u901a\u5e38\u306e\u89e3\u6cd5\u3067\u306f\u30b9\u30c6\u30c3\u30d7\u5e45\u306b\u5236\u7d04\u304c\u3042\u308b\u305f\u3081\u306b, \u8a08\u7b97\u306b\u975e\u5e38\u306b\u6642\u9593\u304c\u304b\u304b\u3063\u305f\u308a, \u6761\u4ef6\u306b\u3088\u3063\u3066\u306f\u632f\u52d5\u3084\u767a\u6563\u3092\u3057\u305f\u308a\u3059\u308b\u3082\u306e\u3067\u3042\u308b. \u4e0a\u306e\u4f8b\u306e\u3088\u3046\u306b, \u4e00\u898b\u6025\u306a\u5909\u5316\u3092\u3059\u308b\u3088\u3046\u306b\u306f\u898b\u3048\u306a\u3044\u3082\u306e\u3067\u3042\u3063\u3066\u3082, \u4e00\u822c\u89e3\u306b\u6307\u6570\u9805\u3092\u542b\u3093\u3067\u3044\u305f\u308a\u3059\u308b\u3068\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f\u3068\u306a\u308b.<\/p>\n
\u6b21\u306e\u3088\u3046\u306a\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u8003\u3048\u308b.<\/p>\n
  \n  dy<\/strong>\/dt = Ay<\/strong> + \u03c6(t)\n  y<\/strong>(0) = y0<\/sub><\/strong>\n\n<\/pre>\n
 
\n\u884c\u5217A\u306e\u56fa\u6709\u5024\u03bbi<\/sub>\u306f\u3059\u3079\u3066\u7570\u306a\u308a\u305d\u306e\u5b9f\u6570\u90e8\u306f\u8ca0\u3067\u3042\u308b\u3068\u3059\u308b.<\/p>\n
\u786c\u5ea6\u6bd4(stiffness ratio)\u3092\u6b21\u306e\u3088\u3046\u306b, \u5b9f\u6570\u90e8\u304c\u6700\u5927\u306e\u56fa\u6709\u5024\u03bbmax<\/sub>\u3068\u5b9f\u6570\u90e8\u304c\u6700\u5c0f\u306e\u56fa\u6709\u5024\u03bbmin<\/sub>\u306e\u5b9f\u6570\u90e8\u306e\u5024\u306e\u6bd4\u3068\u5b9a\u7fa9\u3059\u308b.<\/p>\n
  \n  |\u03bbmax<\/sub>\u306e\u5b9f\u6570\u90e8|\/|\u03bbmin<\/sub>\u306e\u5b9f\u6570\u90e8|\n\n<\/pre>\n
 
\n\u3053\u306e\u786c\u5ea6\u6bd4\u304c\u5927\u304d\u3044(> 104<\/sup> \u304f\u3089\u3044)\u3068\u304d\u306b\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f\u3068\u3044\u3046\u3053\u3068\u304c\u591a\u3044\u3088\u3046\u3067\u3042\u308b. \u4e0a\u306e\u4f8b\u306f\u307b\u307c\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f\u3068\u3044\u3048\u308b.<\/p>\n
\u3057\u304b\u3057, \u786c\u5ea6\u6bd4\u304c\u5c0f\u3055\u304f\u3066\u3082\u30b9\u30c6\u30a3\u30d5\u306a\u30b1\u30fc\u30b9\u3084, \u786c\u5ea6\u6bd4\u304c\u5927\u304d\u304f\u3066\u3082\u521d\u671f\u5024\u306b\u3088\u3063\u3066\u306f\u30b9\u30c6\u30a3\u30d5\u3067\u306f\u306a\u304f\u89e3\u3051\u308b\u30b1\u30fc\u30b9\u304c\u3042\u308a, \u6587\u732e[1]\u3067\u306f\u6b21\u306e\u3088\u3046\u306a\u5b9a\u7fa9\u3082\u4e0e\u3048\u3066\u3044\u308b.<\/p>\n
\u975e\u6589\u6b21\u9805\u306e\u5909\u52d5\u91cfM(t)\u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b.<\/p>\n
  \n  M(t) = ||\u03c6'(t)||\/||\u03c6(t)||,  \u305f\u3060\u3057 \u03c6(t) = 0 \u306e\u3068\u304d\u306f M(t) = 0 \u3068\u3059\u308b\n\n<\/pre>\n
 
\n\u6b21\u306b\u6e1b\u8870\u6bd4(damping ratio)\u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b.<\/p>\n
  \n  |\u03bbmax<\/sub>\u306e\u5b9f\u6570\u90e8|\/M(t)\n\n<\/pre>\n
 
\n\u3053\u306e\u3068\u304d\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b.<\/p>\n
– \u6e1b\u8870\u6bd4\u304c\u5927\u304d\u306a\u554f\u984c\u3092, \u305d\u306et\u306e\u8fd1\u508d\u3067\u786c\u3044\u7cfb\u3068\u3044\u3046.
\n– \u786c\u5ea6\u6bd4\u304c\u5927\u304d\u3044\u304c, \u6e1b\u8870\u6bd4\u304c\u305d\u308c\u3088\u308a\u305a\u3063\u3068\u5c0f\u3055\u3044\u554f\u984c\u306f\u7591\u4f3c\u7684\u306b\u786c\u3044\u7cfb\u3068\u3044\u3046.<\/p>\n
\u4e0a\u306e\u4f8b\u306f\u786c\u5ea6\u6bd4\u306f2000\u3067, \u6e1b\u8870\u6bd4\u306ft\u306b\u3088\u308a\u5909\u52d5\u30571\uff5e\u7d0419992<\/sup>\u3068\u306a\u3063\u3066\u3044\u308b.<\/p>\n
\u6570\u5024\u5b9f\u9a13 (21)<\/h3>\n
<\/span>\u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5\u306b\u3088\u308b\u8a08\u7b97\u4f8b<\/div>
\n\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f\u3067\u3042\u3063\u3066\u3082 A\u5b89\u5b9a\u306a\u89e3\u6cd5\u306a\u3089\u3070\u89e3\u3051\u308b\u306f\u305a\u3067\u3042\u308b. \u305d\u3053\u3067, \u4e0a\u306e\u4f8b\u984c\u3092\u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5\u3067\u89e3\u3044\u3066\u307f\u308b.<\/p>\n
\"\"<\/p>\n
\u4e0a\u56f3\u306f h = 0.2 \u3068\u3057\u3066\u8a08\u7b97\u3057\u305f y1<\/sub>, y2<\/sub> \u3092\u30d7\u30ed\u30c3\u30c8\u3057\u305f\u3082\u306e\u3067\u3042\u308b. \u3053\u306e\u3088\u3046\u306b\u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5\u306a\u3089\u3070\u5927\u304d\u306a\u30b9\u30c6\u30c3\u30d7\u5e45\u3067\u3082\u554f\u984c\u306a\u304f\u89e3\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u305f.
\n<\/div><\/div>\n
\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5<\/h3>\n
\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u305f\u3081\u306b\u306f\u30b9\u30c6\u30c3\u30d7\u5e45\u306b\u5236\u7d04\u304c\u306a\u3044\u89e3\u6cd5, \u3059\u306a\u308f\u3061, A\u5b89\u5b9a\u306a\u89e3\u6cd5\u3092\u4f7f\u3046\u5fc5\u8981\u304c\u3042\u308b.<\/p>\n
A\u5b89\u5b9a\u306a\u89e3\u6cd5\u3068\u3057\u3066\u306f\u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5\u3084\u9670\u7684\u53f0\u5f62\u5247\u304c\u3042\u308b\u304c, \u5b9f\u7528\u7684\u306b\u306f\u3082\u3063\u3068\u9ad8\u6b21\u306e\u516c\u5f0f\u304c\u5fc5\u8981\u3067\u3042\u308a\u7a2e\u3005\u306e\u89e3\u6cd5\u304c\u7814\u7a76\u3055\u308c\u3066\u304d\u305f. \u4ee5\u4e0b, \u4ee3\u8868\u7684\u306a\u89e3\u6cd5\u3068\u3057\u3066, \u5f8c\u9000\u5fae\u5206\u516c\u5f0f(BDF)\u304a\u3088\u3073\u9670\u7684\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf(IRK)\u6cd5\u306b\u3064\u3044\u3066\u8aac\u660e\u3059\u308b. \u307e\u305f, \u30ed\u30fc\u30bc\u30f3\u30d6\u30ed\u30c3\u30af\u6cd5\u3068\u88dc\u5916\u6cd5\u306b\u3064\u3044\u3066\u3082\u5c11\u3057\u89e6\u308c\u308b.<\/p>\n
\u5f8c\u9000\u5fae\u5206\u516c\u5f0f(BDF)<\/h3>\n
\u30a2\u30c0\u30e0\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5\u3067\u306f (ti<\/sub>, fi<\/sub>) (i = n – k +1, …, n+1) \u3092\u88dc\u9593\u30c7\u30fc\u30bf\u3068\u3057\u3066 f \u3092\u88dc\u9593\u3057\u305f\u304c, y \u305d\u306e\u3082\u306e\u3092\u88dc\u9593\u3059\u308b\u65b9\u6cd5\u304c\u3042\u308b. \u3059\u306a\u308f\u3061, (ti<\/sub>, yi<\/sub>) (i = n – k +1, …, n+1) \u3092\u88dc\u9593\u30c7\u30fc\u30bf\u3068\u3057\u3066\u88dc\u9593\u591a\u9805\u5f0f q(t) \u3092\u6c42\u3081, \u305d\u308c\u304c\u5c11\u306a\u304f\u3066\u3082\u683c\u5b50\u70b9\u306e1\u304b\u6240(i=n+1-r \u306e\u70b9)\u3067\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u3059\u308b.<\/p>\n
  \n  q'(tn+1-r<\/sub>) = f(yn+1-r<\/sub>, tn+1-r<\/sub>)\n\n<\/pre>\n
 
\n\u3053\u3053\u3067, r = 1 \u3068\u3059\u308b\u3068\u967d\u89e3\u6cd5\u306b\u306a\u308a, k = 1 \u3068 k = 2 \u306e\u5834\u5408\u306f\u305d\u308c\u305e\u308c\u30aa\u30a4\u30e9\u30fc\u6cd5\u3068\u4e2d\u70b9\u5247\u306b\u306a\u308b. \u3057\u304b\u3057, k \u2267 3 \u306e\u5834\u5408\u306b\u306f\u4e0d\u5b89\u5b9a\u306b\u306a\u308a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5\u3068\u3057\u3066\u4f7f\u3046\u3053\u3068\u306f\u3067\u304d\u306a\u3044.<\/p>\n
r = 0 \u3068\u3059\u308b\u3068\u9670\u89e3\u6cd5\u306b\u306a\u308b\u304c k \u2266 6 \u3067\u5b89\u5b9a\u306a\u89e3\u6cd5\u304c\u5f97\u3089\u308c\u308b. \u3053\u308c\u306f\u5f8c\u9000\u5fae\u5206\u516c\u5f0f (Backward differentiation formula (BDF)<\/strong> \u307e\u305f\u306f \u30ae\u30e4\u6cd5(Gear method)) \u3068\u3088\u3070\u308c, k \u6b21\u306e\u516c\u5f0f\u306b\u306a\u308b.<\/p>\n
  \n  \u03a3 (1\/j)\u25bdj<\/sup>yn+1<\/sub> = hfn+1<\/sub>  (\u03a3\u306f j = 1, ... , k)\n\n<\/pre>\n
 
\n\u305f\u3060\u3057,<\/p>\n
  \n  \u25bd0<\/sup>fn<\/sub> = fn<\/sub>,  \u25bdj+1<\/sup>fn<\/sub> = \u25bdj<\/sup>fn<\/sub> - \u25bdj<\/sup>fn-1<\/sub>  (\u5f8c\u9000\u5dee\u5206)\n\n<\/pre>\n
 
\nk = 1\uff5e6 \u306b\u3064\u3044\u3066, yn+1<\/sub>, yn<\/sub>, yn-1<\/sub>, … \u306b\u3064\u3044\u3066\u6574\u7406\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
  \n  k = 1 : yn+1<\/sub> - yn<\/sub> = hfn+1<\/sub>  (\u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5\u306b\u4e00\u81f4)\n  k = 2 : (3\/2)yn+1<\/sub> - 2yn<\/sub> + (1\/2)yn-1<\/sub> = hfn+1<\/sub>\n  k = 3 : (11\/6)yn+1<\/sub> - 3yn<\/sub> + (3\/2)yn-1<\/sub> - (1\/3)yn-2<\/sub> = hfn+1<\/sub>\n  k = 4 : (25\/12)yn+1<\/sub> - 4yn<\/sub> + 3yn-1<\/sub> - (4\/3)yn-2<\/sub> + (1\/4)yn-3<\/sub> = hfn+1<\/sub>\n  k = 5 : (137\/60)yn+1<\/sub> - 5yn<\/sub> + 5yn-1<\/sub> - (10\/3)yn-2<\/sub> + (5\/4)yn-3<\/sub> - (1\/5)yn-4<\/sub> = hfn+1<\/sub>\n  k = 6 : (147\/60)yn+1<\/sub> - 6yn<\/sub> + (15\/2)yn-1<\/sub> - (20\/3)yn-2<\/sub> + (15\/4)yn-3<\/sub> - (6\/5)yn-4<\/sub> + (1\/6)yn-5<\/sub> = hfn+1<\/sub>\n\n<\/pre>\n
 
\nBDF\u306e\u7d76\u5bfe\u5b89\u5b9a\u9818\u57df\u3092 k = 1\uff5e4 \u306b\u3064\u3044\u3066\u4e0b\u56f3\u306b\u793a\u3059.<\/p>\n
\"ode_bdf\"<\/p>\n
\u56f3\u306e\u66f2\u7dda\u306e\u5916\u5074\u304c\u5b89\u5b9a\u9818\u57df\u3067\u3042\u308b. k = 1 (\u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5) \u3068 k = 2 \u306f A\u5b89\u5b9a\u3067\u3042\u308b\u304c, \u305d\u308c\u4ee5\u5916\u3067\u306f\u5c11\u3057\u5de6\u534a\u5e73\u9762\u306b\u306f\u307f\u3060\u3057\u3066\u3044\u308b\u90e8\u5206\u304c\u3042\u308a A\u5b89\u5b9a\u3067\u306f\u306a\u3044. k \u304c\u5927\u304d\u304f\u306a\u308b\u307b\u3069\u5b89\u5b9a\u9818\u57df\u306f\u72ed\u304f\u306a\u308a, k \u2267 7 \u3067\u306f\u539f\u70b9\u3067\u3082\u4e0d\u5b89\u5b9a\u306b\u306a\u308b.<\/p>\n
\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u969b\u306b, \u5b9f\u969b\u306b\u306f\u591a\u304f\u306e\u5834\u5408\u306b\u5b8c\u5168\u306bA\u5b89\u5b9a\u3067\u3042\u308b\u5fc5\u8981\u306f\u306a\u3044. \u305d\u3053\u3067, \u6761\u4ef6\u3092\u7de9\u3081\u305f\u4ee3\u308f\u308a\u306b\u9ad8\u6b21\u306a\u89e3\u6cd5\u3068\u3057\u3066BDF\u304c\u3088\u304f\u4f7f\u308f\u308c\u3066\u3044\u308b. 4\uff5e6\u6b21\u306eBDF\u306fA\u5b89\u5b9a\u306e\u6761\u4ef6\u3092\u7de9\u3081\u305fA(\u03b1)\u5b89\u5b9a\u3067\u3042\u308b.<\/p>\n
\u539f\u70b9\u3092\u9802\u70b9\u3068\u3057\u3066, \u5de6\u5074\u306b\u975e\u6709\u754c\u3067\u3042\u308a, \u5b9f\u8ef8\u306b\u5bfe\u3057\u3066\u4e0a\u306b\u03b1\u00b0\u4e0b\u306b\u03b1\u00b0\u306e\u6247\u578b\u306e\u9818\u57df\u304c\u7d76\u5bfe\u5b89\u5b9a\u9818\u57df\u306b\u542b\u307e\u308c\u308b\u5834\u5408, A(\u03b1)\u5b89\u5b9a\u3067\u3042\u308b\u3068\u3044\u3046. A(90\u00b0)\u5b89\u5b9a\u306fA\u5b89\u5b9a\u306b\u7b49\u3057\u3044. \u65b9\u7a0b\u5f0f\u306e\u4fc2\u6570\u884c\u5217\u306e\u56fa\u6709\u5024\u304c\u3059\u3079\u3066\u3053\u306e\u4e2d\u306b\u5165\u3063\u3066\u3044\u308b\u306a\u3089\u3070, \u5b9f\u8cea\u7684\u306b A\u5b89\u5b9a\u3068\u540c\u3058\u3067\u3042\u308b. 3\u6b21\u306eBDF\u306fA(86\u00b0)\u5b89\u5b9a, 4\u6b21\u306eBDF\u306fA(73\u00b0)\u5b89\u5b9a, 5\u6b21\u306eBDF\u306fA(52\u00b0)\u5b89\u5b9a, 6\u6b21\u306eBDF\u306fA(18\u00b0)\u5b89\u5b9a\u3067\u3042\u308b.<\/p>\n
\u5de6\u534a\u5e73\u9762\u304c\u307b\u3068\u3093\u3069\u7d76\u5bfe\u5b89\u5b9a\u9818\u57df\u306b\u542b\u307e\u308c\u308b\u304c, \u539f\u70b9\u4ed8\u8fd1\u3092\u542b\u307e\u306a\u3044\u865a\u8ef8\u8fd1\u508d\u3060\u3051\u304c\u7d76\u5bfe\u5b89\u5b9a\u9818\u57df\u306b\u542b\u307e\u308c\u306a\u3044\u5834\u5408\u3092\u786c\u5b89\u5b9a\u3068\u3044\u3046. \u865a\u8ef8\u306b\u8fd1\u3044\u3068\u3053\u308d\u306b\u4fc2\u6570\u884c\u5217\u306e\u56fa\u6709\u5024\u306a\u3044\u9650\u308a, \u3053\u308c\u3082\u5b9f\u8cea\u7684\u306b A\u5b89\u5b9a\u3068\u540c\u3058\u3067\u3042\u308b. 4\uff5e6\u6b21\u306eBDF\u306f\u786c\u5b89\u5b9a\u3067\u3042\u308b.<\/p>\n
\u6570\u5024\u5b9f\u9a13 (22)<\/h3>\n
<\/span>BDF\u306b\u3088\u308b\u8a08\u7b97\u4f8b<\/div>
\n\u4e0a\u306e\u4f8b\u984c\u3092BDF\u3092\u4f7f\u3063\u3066\u89e3\u3044\u3066\u307f\u308b.<\/p>\n
\"\"<\/p>\n
\u4e0a\u56f3\u306f, 2\u6b21\uff5e6\u6b21\u306eBDF\u3092\u7528\u3044\u3066 h = 1\uff5e0.001 \u3068\u5909\u5316\u3055\u305b\u3066\u8a08\u7b97\u3057\u305f\u3068\u304d\u306e y2<\/sub> \u306e\u76f8\u5bfe\u8aa4\u5dee\u306b\u5bfe\u3059\u308b\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u3092\u30d7\u30ed\u30c3\u30c8\u3057\u305f\u3082\u306e\u3067\u3042\u308b. \u51fa\u767a\u5024\u306f3\u6b21\u306eIRK\u6cd5\u3067\u8a08\u7b97\u3057\u305f.<\/p>\n
2\u6b21\uff5e6\u6b21\u307e\u3067\u554f\u984c\u306a\u304f\u89e3\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u305f.
\n<\/div><\/div>\n
\u9670\u7684\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/h3>\n
s\u6bb5\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u4e00\u822c\u5f62\u306f\u6b21\u306e\u3088\u3046\u306a\u3082\u306e\u3067\u3042\u3063\u305f.<\/p>\n
  \n  ki<\/sub> = f(tn<\/sub> + ci<\/sub>h, yn<\/sub> + h\u03a3aij<\/sub>kj<\/sub>) (\u03a3 \u306f j = 1, 2, ... , s) (i = 1, 2, ... , s)\n  yn+1<\/sub> = yn<\/sub> + h\u03a3bi<\/sub>ki<\/sub> (\u03a3 \u306f i = 1, 2, ... , s)\n\n<\/pre>\n
 
\naij<\/sub>, bi<\/sub> \u304a\u3088\u3073 ci<\/sub> \u306f\u30d1\u30e9\u30e1\u30fc\u30bf\u3067, bi<\/sub> \u304a\u3088\u3073 ci<\/sub> \u306f\u6b21\u5f0f\u3092\u6e80\u305f\u3059.<\/p>\n
  \n  ci<\/sub> = \u03a3aij<\/sub> (\u03a3 \u306f j = 1, 2, ... , s) (i = 1, 2, ... , s)\n  \u03a3bi<\/sub> = 1 (\u03a3 \u306f i = 1, 2, ... , s)\n\n<\/pre>\n
 
\naij<\/sub> = 0 (i \u2266 j) \u306e\u5834\u5408, \u967d\u89e3\u6cd5\u306b\u306a\u308b. \u305d\u308c\u4ee5\u5916\u306e\u5834\u5408, \u9670\u7684(implicit)\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf(IRK)\u6cd5<\/strong>\u3068\u3044\u3046. IRK\u6cd5\u306f A\u5b89\u5b9a\u306a\u3082\u306e\u3092\u4f5c\u308c\u308b\u304c, 1 \u30b9\u30c6\u30c3\u30d7\u3054\u3068\u306b\u9023\u7acb\u975e\u7dda\u5f62\u65b9\u7a0b\u5f0f\u3092\u89e3\u304b\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u305f\u3081\u8a08\u7b97\u306e\u624b\u9593\u304c\u304b\u304b\u308b\u3068\u3044\u3046\u554f\u984c\u304c\u3042\u308b.<\/p>\n
\u30ac\u30a6\u30b9\u6cd5<\/h4>\n
\u30ac\u30a6\u30b9\u7a4d\u5206\u5247\u306b\u57fa\u3065\u304f\u6700\u5927\u6b21\u6570\u306eIRK\u6cd5\u3067\u3042\u308b. s\u6bb5\u30ac\u30a6\u30b9\u6cd5\u306e\u6b21\u6570\u306f2s\u6b21\u3067\u3042\u308b. c1<\/sub>, c2<\/sub>, …, cs<\/sub> \u306f\u533a\u9593[0, 1]\u306b\u5909\u6570\u5909\u63db\u3055\u308c\u305f\u30eb\u30b8\u30e3\u30f3\u30c9\u30eb\u591a\u9805\u5f0f<\/p>\n
  \n  ds<\/sup>\/dxs<\/sup> (xs<\/sup>(x - 1)s<\/sup>)\n\n<\/pre>\n
 
\n\u306e\u30bc\u30ed\u70b9\u3067\u4e0e\u3048\u3089\u308c\u308b.<\/p>\n
\u30af\u30f3\u30c4\u30de\u30f3-\u30d6\u30c3\u30c1\u30e3\u30fc\u6cd5\u3068\u3088\u3070\u308c\u308b4\u6b21\u30686\u6b21\u306e\u516c\u5f0f\u306e\u4fc2\u6570\u3092\u4ee5\u4e0b\u306b\u793a\u3059. \u3053\u306e\u65b9\u6cd5\u306fA\u5b89\u5b9a\u3067\u3042\u308b.<\/p>\n
2\u6bb54\u6b21\u30af\u30f3\u30c4\u30de\u30f3-\u30d6\u30c3\u30c1\u30e3\u30fc\u6cd5<\/p>\n
  \n  1\/2-\u221a3\/6 |    1\/4     1\/4-\u221a3\/6\n  1\/2+\u221a3\/6 | 1\/4+\u221a3\/6      1\/4\n  ----------------------------------\n            |    1\/2        1\/2\n    \n<\/pre>\n
 
\n3\u6bb56\u6b21\u30af\u30f3\u30c4\u30de\u30f3-\u30d6\u30c3\u30c1\u30e3\u30fc\u6cd5<\/p>\n
  \n  1\/2-\u221a15\/10 |    5\/36      2\/9-\u221a15\/15  5\/36-\u221a15\/30\n       1\/2    | 5\/36+\u221a15\/24      2\/9     5\/36-\u221a15\/24\n  1\/2+\u221a15\/10 | 5\/36+\u221a15\/30  2\/9+\u221a15\/15     5\/36\n  ------------------------------------------------------\n              |    5\/18          4\/9        5\/18\n    \n<\/pre>\n
 <\/p>\n
\u30e9\u30c0\u30a6\u6cd5\u3068\u30ed\u30d0\u30c3\u30c8\u6cd5<\/h4>\n
\u30e9\u30c0\u30a6\u3068\u30ed\u30d0\u30c3\u30c8\u306e\u7a4d\u5206\u5247\u306b\u57fa\u3065\u304fIRK\u6cd5\u3067\u3042\u308b. c1<\/sub>, c2<\/sub>, …, cs<\/sub> \u306f\u6b21\u306e\u591a\u9805\u5f0f\u306e\u30bc\u30ed\u70b9\u3067\u4e0e\u3048\u3089\u308c\u308b.<\/p>\n
  \n  ds-1<\/sup>\/dxs-1<\/sup> (xs<\/sup>(x - 1)s-1<\/sup>)     (\u2160\u578b: \u30e9\u30c0\u30a6\u53f3)\n  ds-1<\/sup>\/dxs-1<\/sup> (xs-1<\/sup>(x - 1)s<\/sup>)     (\u2161\u578b: \u30e9\u30c0\u30a6\u5de6)\n  ds-2<\/sup>\/dxs-2<\/sup> (xs-1<\/sup>(x - 1)s-1<\/sup>)   (\u2162\u578b: \u30ed\u30d0\u30c3\u30c8)\n\n<\/pre>\n
 
\n\u2160\u578b\u306e\u30e9\u30c0\u30a6\u2160A\u6cd5(c1<\/sub> = 0 \u306b\u306a\u308b), \u2161\u578b\u306e\u30e9\u30c0\u30a6\u2161A\u6cd5(cs<\/sub> = 1 \u306b\u306a\u308b)\u306e\u6b21\u6570\u306fs\u6bb5\u306e\u3068\u304d2s-1\u6b21\u3067\u3042\u308b. \u4e21\u65b9\u3068\u3082A\u5b89\u5b9a\u3067\u3042\u308b. 1\u6bb5\u30e9\u30c0\u30a6\u2161A\u6cd5\u306f\u9670\u7684\u30aa\u30a4\u30e9\u30fc\u6cd5\u306b\u306a\u308b.<\/p>\n
2\u6bb53\u6b21\u30e9\u30c0\u30a6\u2161A\u6cd5<\/p>\n
  \n  1\/3  |  5\/12  -1\/12\n   1   |  3\/4    1\/4\n  ---------------------\n       |  3\/4    1\/4\n\n<\/pre>\n
 
\n3\u6bb55\u6b21\u30e9\u30c0\u30a6\u2161A\u6cd5<\/p>\n
  \n  (4-\u221a6)\/10 |   (88-7\u221a6)\/360    (296-169\u221a6)\/1800  (-2+3\u221a6)\/225\n  (4+\u221a6)\/10 | (296+169\u221a6)\/1800    (88+7\u221a6)\/360    (-2-3\u221a6)\/225\n       1     |    (16-\u221a6)\/36        (16+\u221a6)\/36           1\/9\n  ------------------------------------------------------------------\n             |    (16-\u221a6)\/36        (16+\u221a6)\/36           1\/9\n    \n<\/pre>\n
 
\n\u2162\u578b\u306e\u65b9\u6cd5\u306f\u30ed\u30d0\u30c3\u30c8\u2162A, \u2162B, \u2162C \u306e\uff13\u3064\u304c\u3042\u308b. s\u6bb5\u306e\u3068\u304d2s-2\u6b21\u3068\u306a\u308b. \u3044\u305a\u308c\u3082A\u5b89\u5b9a\u3067\u3042\u308b.
\n\u4ee5\u4e0b\u306b\u30ed\u30d0\u30c3\u30c8\u2162A\u6cd5\u306e\u4f8b\u3092\u793a\u3059\u3002<\/p>\n
3\u6bb54\u6b21\u30ed\u30d0\u30c3\u30c8\u2162A\u6cd5<\/p>\n
  \n     0   |    0     0      0\n    1\/2  |  5\/24   1\/3   -1\/24\n     1   |   1\/6   2\/3    1\/6\n  ------------------------------\n         |   1\/6   2\/3    1\/6\n\n<\/pre>\n
 
\n4\u6bb56\u6b21\u30ed\u30d0\u30c3\u30c8\u2162A\u6cd5<\/p>\n
  \n       0     |       0             0              0             0\n  (5-\u221a5)\/10 | (11+\u221a5)\/120   (25-\u221a5)\/120   (25-13\u221a5)\/120  (-1+\u221a5)\/120\n  (5+\u221a5)\/10 | (11-\u221a5)\/120  (25+13\u221a5)\/120   (25+\u221a5)\/120   (-1-\u221a5)\/120\n       1     |     1\/12          5\/12           5\/12          1\/12\n  -------------------------------------------------------------------------\n             |     1\/12          5\/12           5\/12          1\/12\n\n<\/pre>\n
 <\/p>\n
\u6570\u5024\u5b9f\u9a13 (23)<\/h3>\n
<\/span>IRK\u6cd5\u306b\u3088\u308b\u8a08\u7b97\u4f8b<\/div>
\n\u4e0a\u306e\u4f8b\u984c\u3092\u3044\u304f\u3064\u304b\u306eIRK\u6cd5\u3092\u4f7f\u3063\u3066\u89e3\u3044\u3066\u307f\u308b.<\/p>\n
\"\"<\/p>\n
\u4e0a\u56f3\u306f, 2\u6bb54\u6b21\u30af\u30f3\u30c4\u30de\u30f3-\u30d6\u30c3\u30c1\u30e3\u30fc\u6cd5(\u30ac\u30a6\u30b9\u6cd5), 3\u6bb56\u6b21\u30af\u30f3\u30c4\u30de\u30f3-\u30d6\u30c3\u30c1\u30e3\u30fc\u6cd5(\u30ac\u30a6\u30b9\u6cd5), 2\u6bb53\u6b21\u30e9\u30c0\u30a6\u2161A\u6cd5, 3\u6bb55\u6b21\u30e9\u30c0\u30a6\u2161A\u6cd5, 3\u6bb54\u6b21\u30ed\u30d0\u30c3\u30c8\u2162A\u6cd5, 4\u6bb56\u6b21\u30ed\u30d0\u30c3\u30c8\u2162A\u6cd5\u3092\u7528\u3044\u3066 h = 1\uff5e0.001 \u3068\u5909\u5316\u3055\u305b\u3066\u8a08\u7b97\u3057\u305f\u3068\u304d\u306e y2<\/sub> \u306e\u76f8\u5bfe\u8aa4\u5dee\u306b\u5bfe\u3059\u308b\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u3092\u30d7\u30ed\u30c3\u30c8\u3057\u305f\u3082\u306e\u3067\u3042\u308b.<\/p>\n
IRK\u6cd5\u3067\u306f\u30b9\u30c6\u30c3\u30d7\u3054\u3068\u306bn\u00d7s\u5143\u975e\u7dda\u5f62\u9023\u7acb\u65b9\u7a0b\u5f0f(\u3053\u306e\u4f8b\u3067\u306f2\u00d7s\u5143)\u3092\u89e3\u304b\u306a\u3051\u308c\u3070\u306a\u3089\u305a, BDF\u306b\u6bd4\u3079\u308b\u3068s\u500d\u5927\u304d\u3044\u65b9\u7a0b\u5f0f\u306b\u306a\u308b. \u95a2\u6570f()\u304c\u7c21\u5358\u306a\u5834\u5408\u306b\u306f\u3053\u306e\u8a08\u7b97\u91cf\u306f\u7121\u8996\u3067\u304d\u306a\u3044\u306e\u3067, \u56f3\u306e\u3088\u3046\u306b\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u3060\u3051\u3092\u8a08\u7b97\u91cf\u306e\u6307\u6a19\u306b\u306f\u3067\u304d\u306a\u3044\u304b\u3082\u3057\u308c\u306a\u3044.
\n<\/div><\/div>\n
\u5bfe\u89d2\u9670\u7684\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3068\u30ed\u30fc\u30bc\u30f3\u30d6\u30ed\u30c3\u30af\u6cd5<\/h3>\n
IRK\u6cd5\u3067\u306f\u30b9\u30c6\u30c3\u30d7\u3054\u3068\u306bn*s\u5143\u975e\u7dda\u5f62\u9023\u7acb\u65b9\u7a0b\u5f0f\u3092\u89e3\u304b\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u304c, \u5834\u5408\u306b\u3088\u3063\u3066\u306f\u5927\u304d\u306a\u65b9\u7a0b\u5f0f\u306b\u306a\u3063\u3066\u3057\u307e\u3046\u306e\u3067\u3053\u306e\u8ca0\u62c5\u3092\u8efd\u6e1b\u3059\u308b\u5de5\u592b\u304c\u691c\u8a0e\u3055\u308c\u3066\u3044\u308b.<\/p>\n
aij<\/sub> = 0 (i < j) \u3067 aii<\/sub> \u2260 0 (\u5c11\u306a\u304f\u3066\u3082 1\u3064\u306e i \u306b\u3064\u3044\u3066) \u3068\u3057\u305f\u3082\u306e\u3092\u5bfe\u89d2\u9670\u7684(diagonally implicit)\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf(DIRK)\u6cd5<\/strong>\u3068\u3044\u3046. \u534a\u9670\u7684(semi-implicit)\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3068\u3082\u3044\u3046. \u3055\u3089\u306b, \u3059\u3079\u3066\u306e\u5bfe\u89d2\u6210\u5206\u304c\u540c\u4e00\u306e\u5024\u3067\u3042\u308b\u5834\u5408, \u5358\u7d14\u5bfe\u89d2\u9670\u7684(singly diagonally implicit)\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf(SDIRK)\u6cd5<\/strong>\u3068\u3044\u3046.<\/p>\n
DIRK\u6cd5\u306e\u5834\u5408, s\u500b\u306en\u5143\u975e\u7dda\u5f62\u65b9\u7a0b\u5f0f\u3092\u9010\u6b21\u89e3\u304f\u3053\u3068\u306b\u3088\u308a\u89e3\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308b. n\u5143\u975e\u7dda\u5f62\u65b9\u7a0b\u5f0f\u3092\u30cb\u30e5\u30fc\u30c8\u30f3\u6cd5\u3067\u89e3\u304f\u3068\u304daii<\/sub>\u3092\u542b\u3080\u4fc2\u6570\u306e\u7dda\u5f62\u9023\u7acb\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u5fc5\u8981\u304c\u3042\u308b\u304c, SDIRK\u6cd5\u3067\u3042\u308c\u3070LU\u5206\u89e3\u30921\u56de\u3067\u6e08\u307e\u305b\u3066\u7d50\u679c\u3092\u7e70\u308a\u8fd4\u3057\u4f7f\u3046\u3053\u3068\u304c\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308b.<\/p>\n
DIRK\u6cd5\u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b.<\/p>\n
  \n  ki<\/sub> = f(yn<\/sub> + h\u03a3aij<\/sub>kj<\/sub> + aii<\/sub>ki<\/sub>) (\u03a3 \u306f j = 1, 2, ... , i-1) (i = 1, 2, ... , s)\n  yn+1<\/sub> = yn<\/sub> + h\u03a3bi<\/sub>ki<\/sub> (\u03a3 \u306f i = 1, 2, ... , s)\n\n<\/pre>\n
 
\n\u5de5\u592b\u3092\u9032\u3081\u3066\u6b21\u306e\u3088\u3046\u306b\u5909\u3048\u305f\u306e\u304c\u30ed\u30fc\u30bc\u30f3\u30d6\u30ed\u30c3\u30af\u6cd5(Rosenbrock method)\u3067\u3042\u308b.<\/p>\n
  \n  (I - h\u03b3ii<\/sub>J)ki<\/sub> = f(yn<\/sub> + h\u03a3\u03b1ij<\/sub>kj<\/sub>) + hJ\u03a3\u03b3ij<\/sub>kj<\/sub>  (\u03a3\u306f j = 1, 2, ... , i-1) (i = 1, 2, ... , s)\n  yn+1<\/sub> = yn<\/sub> + h\u03a3bi<\/sub>ki<\/sub>  (\u03a3 \u306f i = 1, 2, ... , s)\n\n<\/pre>\n
 
\n\u03b1ij<\/sub>, \u03b3ij<\/sub>, bi<\/sub>\u306f\u4fc2\u6570, J = f'(yn<\/sub>) \u3067\u3042\u308b. DIRK\u6cd5\u3092\u7dda\u5f62\u5316\u3057\u305f\u3082\u306e\u3068\u307f\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\u7dda\u5f62\u9670\u7684(linearly implicit)\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3068\u3088\u3070\u308c\u308b. \u5404\u30b9\u30c6\u30c3\u30d7\u3067\u306fki<\/sub>\u3092\u672a\u77e5\u6570\u3068\u3057(I – h\u03b3ii<\/sub>J)\u3092\u4fc2\u6570\u3068\u3059\u308b\u7dda\u5f62\u9023\u7acb\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3053\u3068\u3067\u8a08\u7b97\u3092\u9032\u3081\u3089\u308c\u308b\u3088\u3046\u306b\u306a\u308b.<\/p>\n
\u3053\u308c\u306b\u5c5e\u3059\u308b\u7a2e\u3005\u306e\u516c\u5f0f\u304c\u63d0\u6848\u3055\u308c\u3066\u304a\u308a, \u516c\u5f0f\u306b\u3088\u308aA\u5b89\u5b9a\u3042\u308b\u3044\u306fA(\u03b1)\u5b89\u5b9a\u3067\u3042\u308b.<\/p>\n
\u88dc\u5916\u6cd5<\/h3>\n
\u88dc\u5916\u6cd5\u3092\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f\u306b\u9069\u7528\u3059\u308b\u305f\u3081\u306bGBS\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u7dda\u5f62\u9670\u7684\u306a(\u30ed\u30fc\u30bc\u30f3\u30d6\u30ed\u30c3\u30af\u30bf\u30a4\u30d7\u306e)\u62e1\u5f35\u304c\u7814\u7a76\u3055\u308c\u3066\u3044\u308b.<\/p>\n
  \n  (I - hj<\/sub>J)(y1<\/sub> - y0<\/sub>) = hj<\/sub>f(t0<\/sub>, y0<\/sub>)  [\u7dda\u5f62\u9670\u7684\u30aa\u30a4\u30e9\u30fc\u6cd5]\n  (I - hj<\/sub>J)(yi+1<\/sub> - yi<\/sub>) = -(I + hj<\/sub>J)(yi<\/sub> - yi-1<\/sub>) + 2hj<\/sub>f(ti<\/sub>, yi<\/sub>)  [\u7dda\u5f62\u9670\u7684\u4e2d\u70b9\u5247]\n\n  Tj1<\/sub> = (1\/2)(ynj<\/sub>-1<\/sub> + ynj<\/sub>+1<\/sub>)\n\n<\/pre>\n
 
\nJ\u306f\u30e4\u30b3\u30d3\u884c\u5217\u306e\u8fd1\u4f3c\u3067, J = 0 \u3068\u304a\u304f\u3068GBS\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306b\u4e00\u81f4\u3059\u308b.<\/p>\n
\u7dda\u5f62\u9670\u7684\u4e2d\u70b9\u5247\u3084\u7dda\u5f62\u9670\u7684\u30aa\u30a4\u30e9\u30fc\u6cd5\u3092\u4f7f\u3063\u305f\u88dc\u5916\u6cd5\u3082\u3044\u304f\u3064\u304b\u5b9f\u88c5\u3055\u308c\u3066\u304a\u308a, \u03b1\u304c90\u00b0\u306b\u8fd1\u3044A(\u03b1)\u5b89\u5b9a\u3067\u3042\u308b.
\n<\/div>\n
\n
\u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f<\/h2>\n
\u6b21\u306e\u3088\u3046\u306b\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3068\u4ee3\u6570\u65b9\u7a0b\u5f0f\u304c\u9023\u7acb\u3057\u305f\u65b9\u7a0b\u5f0f\u3092\u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f<\/strong>(Differential algebraic equations (DAE))\u3068\u3044\u3046.<\/p>\n
  \n  x' = f(t, x, z)\n  0 = g(t, x, z)\n\n<\/pre>\n
 
\n\u4ee3\u6570\u65b9\u7a0b\u5f0f\u306f\u5236\u7d04\u6761\u4ef6\u3092\u8868\u3057, \u3053\u306e\u5236\u7d04\u306e\u3082\u3068\u3067\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3068\u307f\u308b\u3053\u3068\u304c\u3067\u304d\u308b. \u4ee3\u6570\u5236\u7d04\u5f0fg()\u3092\u89e3\u6790\u7684\u306b\u5fae\u5206\u3068\u6d88\u53bb\u3092\u7e70\u308a\u8fd4\u3057 z’ = h(t, x, z) \u306e\u3088\u3046\u306a\u5f62\u306b\u5909\u63db\u3059\u308c\u3070, \u3059\u3079\u3066\u306e\u672a\u77e5\u6570\u306b\u3064\u3044\u3066\u967d\u7684\u306a\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u5f97\u3089\u308c\u308b(\u65b9\u7a0b\u5f0f\u304c\u7279\u7570\u3067\u306a\u3051\u308c\u3070\u53ef\u80fd\u3067\u3042\u308b). \u3053\u306e\u5909\u63db\u306b\u5fc5\u8981\u306a\u5fae\u5206\u56de\u6570\u3092\u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f\u306e\u6307\u6570<\/strong>\u3068\u3044\u3046.<\/p>\n
\u5358\u632f\u308a\u5b50\u306e\u4f8b<\/h3>\n
\u4e0b\u56f3\u306e\u3088\u3046\u306b, \u9577\u3055l[m]\u306e\u7cf8\u306b\u91cd\u3055m[kg]\u306e\u304a\u3082\u308a\u304c\u3064\u3044\u305f\u632f\u308a\u5b50\u3092\u8003\u3048\u308b. \u7cf8\u3068\u925b\u76f4\u7dda\u306e\u89d2\u5ea6\u304c\u03b8[rad]\u3067\u91cd\u529b\u52a0\u901f\u5ea6\u3092g[m\/s2<\/sup>]\u3068\u3059\u308b.<\/p>\n
\"\"<\/p>\n
\u3053\u306e\u632f\u308a\u5b50\u306e\u904b\u52d5\u306f\u03b8\u3092\u4f7f\u3063\u3066\u6b21\u306e2\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3067\u8868\u3055\u308c\u308b.<\/p>\n
  \n  \u03b8'' = -(g\/l)sin(\u03b8), \u03b8(0) = \u03b80<\/sub>\n\n<\/pre>\n
 
\n\u03b80<\/sub>\u3067\u624b\u3092\u96e2\u3059\u3068\u632f\u308a\u5b50\u306f\u5468\u671f\u904b\u52d5\u3092\u958b\u59cb\u3057, \u305d\u306e\u5468\u671fT\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u3066\u3044\u308b.<\/p>\n
  \n  T = 4sqrt(l\/g)K(sin(\u03b80<\/sub>\/2))\n\n<\/pre>\n
 
\n\u305f\u3060\u3057, K\u306f\u7b2c1\u7a2e\u5b8c\u5168\u6955\u5186\u7a4d\u5206\u3067\u3042\u308b.<\/p>\n
\u3053\u306e\u554f\u984c\u306f1\u968e\u9023\u7acb\u5f62\u306b\u5909\u63db\u3057\u3066\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u3066, \u521d\u671f\u5024\u3092 \u03b8(0) = \u03c0\/4 \u3068\u3057, Derkf\u3092\u4f7f\u3063\u3066 t = 0.1 \u3054\u3068\u306b\u30d7\u30ed\u30c3\u30c8\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f. \u306a\u304a, \u5468\u671fT\u3054\u3068\u306e\u5024\u3092\u8d64\u3067\u793a\u3057\u305f.<\/p>\n
\"\"<\/p>\n
\u03b80<\/sub> = \u03c0\/4, l = 1 \u3068\u3057\u30662\u5468\u671f\u5206\u306e\u632f\u308a\u5b50\u306e\u52d5\u304d\u3092\u793a\u3059\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
\"\"<\/p>\n
\u3055\u3066, \u540c\u3058\u554f\u984c\u3092 x1<\/sub> = l\u30fbsin(\u03b8), x2<\/sub> = -l\u30fbcos(\u03b8) \u3068\u3057\u3066\u76f4\u4ea4\u5ea7\u6a19\u7cfb\u3067\u8868\u3059\u3068\u6b21\u306e\u65b9\u7a0b\u5f0f\u306b\u306a\u308b. z\u306f\u672a\u5b9a\u4e57\u6570\u3067\u3042\u308b.<\/p>\n
  \n  mx1<\/sub>'' = -zx1<\/sub>\n  mx2<\/sub>'' = -zx2<\/sub> - mg\n  x1<\/sub>2<\/sup> + x2<\/sub>2<\/sup> = l2<\/sup>\n\n<\/pre>\n
 
\n\u3053\u306e\u65b9\u7a0b\u5f0f\u306f\u03b8\u3092\u4f7f\u3063\u3066\u5909\u6570\u5909\u63db\u3057z\u3092\u6d88\u53bb\u3059\u308c\u3070\u4e0a\u306e\u03b8\u3092\u4f7f\u3063\u305f\u65b9\u7a0b\u5f0f\u306b\u623b\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u304c, \u3053\u3053\u3067\u306f\u3053\u306e\u307e\u307e\u8a08\u7b97\u3092\u884c\u3046\u3053\u3068\u306b\u3059\u308b.<\/p>\n
q1<\/sub> = x1<\/sub>, q2<\/sub> = x2<\/sub>, v1<\/sub> = x1<\/sub>‘, v2<\/sub> = x2<\/sub>‘, \u03bb = z\/m \u3068\u5909\u6570\u5909\u63db\u3057\u30661\u968e\u9023\u7acb\u5f62\u306b\u66f8\u304d\u76f4\u3059\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
  \n  q1<\/sub>' = v1<\/sub>\n  q2<\/sub>' = v2<\/sub>\n  v1<\/sub>' = -\u03bbq1<\/sub>\n  v2<\/sub>' = -\u03bbq2<\/sub> - g\n  0 = q1<\/sub>2<\/sup> + q2<\/sub>2<\/sup> - l2<\/sup>\n\n<\/pre>\n
 
\n\u3053\u308c\u306f\u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f\u306e\u5f62\u306b\u306a\u3063\u3066\u3044\u308b. \u6700\u5f8c\u306e\u5f0f\u304c\u4ee3\u6570\u5236\u7d04\u5f0f\u3067\u3042\u308a, \u4f4d\u7f6e\u306e\u5236\u7d04\u6761\u4ef6\u3092\u8868\u3059(\u7269\u7406\u7684\u306b\u306f\u304a\u3082\u308a\u306e\u8ecc\u9053\u304c\u534a\u5f84l\u306e\u5186\u5468\u4e0a\u306b\u3042\u308b\u3053\u3068\u3092\u793a\u3059). \u3053\u308c\u30921\u56de\u5fae\u5206\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
  \n  q1<\/sub>q1<\/sub>' + q2<\/sub>q2<\/sub>' =\n  q1<\/sub>v1<\/sub> + q2<\/sub>v2<\/sub> = 0\n\n<\/pre>\n
 
\n\u3053\u308c\u306f\u901f\u5ea6\u306e\u5236\u7d04\u6761\u4ef6\u3092\u8868\u3059. \u3082\u30461\u56de\u5fae\u5206\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n
  \n  q1<\/sub>v1<\/sub>' + q2<\/sub>v2<\/sub>' + v1<\/sub>2<\/sup> + v2<\/sub>2<\/sup> =\n  -\u03bb - gq2<\/sub> + v1<\/sub>2<\/sup> + v2<\/sub>2<\/sup> = 0\n\n<\/pre>\n
 
\n\u3053\u308c\u306f\u52a0\u901f\u5ea6\u306e\u5236\u7d04\u6761\u4ef6\u3092\u8868\u3059. \u3053\u308c\u3092\u3082\u30461\u56de\u5fae\u5206\u3059\u308b\u3068\u03bb’\u306e\u5f0f\u304c\u5f97\u3089\u308c\u308b\u306e\u3067\u3053\u306e\u65b9\u7a0b\u5f0f\u306e\u6307\u6570\u306f3\u3067\u3042\u308b.<\/p>\n
\u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5 (1)<\/h3>\n
\u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f\u306f\u6307\u6570\u304c\u9ad8\u3044\u307b\u3069\u89e3\u304d\u306b\u304f\u3044\u306e\u3067, \u5236\u7d04\u5f0f\u306e\u89e3\u6790\u7684\u5fae\u5206\u3092\u7e70\u308a\u8fd4\u3057\u6307\u6570\u3092\u4e0b\u3052\u3066\u89e3\u304f\u306e\u304c1\u3064\u306e\u65b9\u6cd5\u3067\u3042\u308b.<\/p>\n
\u5358\u632f\u308a\u5b50\u306e\u5236\u7d04\u5f0f\u30922\u56de\u5fae\u5206\u3057\u3066\u5f97\u3089\u308c\u305f\u03bb\u3092\u5143\u306e\u9023\u7acb\u65b9\u7a0b\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068\u6b21\u306e4\u5143\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f(\u6307\u65700)\u304c\u5f97\u3089\u308c\u308b.<\/p>\n
  \n  q1<\/sub>' = v1<\/sub>\n  q2<\/sub>' = v2<\/sub>\n  v1<\/sub>' = (gq2<\/sub> - v1<\/sub>2<\/sup> - v2<\/sub>2<\/sup>)q1<\/sub>\n  v2<\/sub>' = (gq2<\/sub> - v1<\/sub>2<\/sup> - v2<\/sub>2<\/sup>)q2<\/sub> - g\n\n<\/pre>\n
 
\n\u3053\u306e\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f\u666e\u901a\u306b\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u3066, \u521d\u671f\u5024\u3092 q1<\/sub>(0) = l\u30fbsin(\u03c0\/4), q2<\/sub>(0) = -l\u30fbcos(\u03c0\/4) \u3068\u3057, Derkf\u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3057\u3066\u307f\u308b\u3068\u03b8\u3092\u4f7f\u3063\u305f\u65b9\u7a0b\u5f0f\u3068\u540c\u3058\u89e3\u304c\u5f97\u3089\u308c\u305f.<\/p>\n
\u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f\u306e\u5236\u7d04\u5f0f\u3092\u5fae\u5206\u3057\u3066\u5f97\u3089\u308c\u305f\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306f\u4e0d\u5909\u6027\u3092\u6301\u3064\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3068\u3044\u308f\u308c, \u5236\u7d04\u6761\u4ef6\u304c\u967d\u306b\u306f\u793a\u3055\u308c\u3066\u3044\u306a\u3044\u304c\u9023\u7acb\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u4e2d\u306b\u542b\u307e\u308c\u3066\u3044\u308b. \u3053\u306e\u3088\u3046\u306a\u65b9\u7a0b\u5f0f\u306b\u306f\u5b89\u5b9a\u6027\u306e\u554f\u984c\u304c\u3042\u308a, \u8a08\u7b97\u3092\u9032\u3081\u3066\u3044\u3063\u305f\u3068\u304d\u306b\u8aa4\u5dee\u306e\u7d2f\u7a4d\u306b\u3088\u308a\u5236\u7d04\u6761\u4ef6\u304c\u6e80\u305f\u3055\u308c\u306a\u304f\u306a\u3063\u3066\u3044\u304f\u3053\u3068\u304c\u3042\u308b.<\/p>\n
\u9014\u4e2d\u3067\u8aa4\u5dee\u304c\u84c4\u7a4d\u3057\u3066\u304d\u305f\u3068\u4eee\u5b9a\u3057\u3066q2<\/sub>\u306e\u521d\u671f\u5024\u3092 -l*cos(\u03c0\/4) + 0.05 \u3068\u3057\u3066Derkf\u3067\u8a08\u7b97\u3057\u3066\u307f\u308b(\u672c\u6765\u3053\u306e\u3088\u3046\u306a\u521d\u671f\u5024\u306f\u8a31\u3055\u308c\u306a\u3044\u304c). \u4e0b\u56f3\u306e\u3088\u3046\u306b\u5236\u7d04\u6761\u4ef6\u304b\u3089\u5916\u308c\u305f\u8ecc\u8de1\u3092\u63cf\u304f\u3088\u3046\u306b\u306a\u308b\u306e\u304c\u308f\u304b\u308b. x-y\u5e73\u9762\u3067\u306e\u632f\u308a\u5b50\u306e\u8ecc\u8de1\u3092\u8868\u3059. \u8d64\u7dda\u306f\u5236\u7d04\u6761\u4ef6\u3092\u6e80\u305f\u3059\u672c\u6765\u306e\u8ecc\u8de1\u3067\u3042\u308b.<\/p>\n
\"\"<\/p>\n
\u3053\u308c\u3092\u5b89\u5b9a\u5316\u3059\u308b\u65b9\u6cd5\u306e\u4e00\u4f8b\u3068\u3057\u3066, X’ = f(t, x) \u306e\u4ee3\u308f\u308a\u306b\u6b21\u306e\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u65b9\u6cd5\u304c\u3042\u308b. h(x)\u306f\u4e0d\u5909\u65b9\u7a0b\u5f0f\u3067, h(x) = 0 \u306e\u3068\u304d\u306b\u5236\u7d04\u6761\u4ef6\u3092\u6e80\u305f\u3059. \u5f93\u3063\u3066, \u5236\u7d04\u6761\u4ef6\u3092\u6e80\u305f\u3057\u3066\u3044\u308c\u3070\u3053\u306e\u65b9\u7a0b\u5f0f\u306f\u5143\u306e\u65b9\u7a0b\u5f0f\u3068\u540c\u3058\u89e3\u3092\u6301\u3064.<\/p>\n
  \n  x' = f(t, x) - \u03b3F(x)h(x)\n\n<\/pre>\n
 
\n\u8a73\u7d30\u306f\u7701\u7565\u3059\u308b\u304c, \u4eca\u8003\u3048\u3066\u3044\u308b\u5358\u632f\u308a\u5b50\u306e\u554f\u984c\u306e\u5834\u5408, \u5177\u4f53\u7684\u306b\u306f\u6b21\u306e\u3088\u3046\u306b\u3059\u308b(\u6587\u732e[4]\u53c2\u7167).<\/p>\n
  \n  h(x) = ( q1<\/sub>2<\/sup> + q2<\/sub>2<\/sup> - l2<\/sup> )\n         ( q1<\/sub>v1<\/sub> + q2<\/sub>v2<\/sub>    )\n\n  F(x) = D(HD)-1<\/sup>\n  DT<\/sup> = H = ( 2q1<\/sub>  2q2<\/sub>  0   0  )\n           ( v1<\/sub>   v2<\/sub>   q1<\/sub>  q2<\/sub> )\n\n<\/pre>\n
 
\n\u03b3 = 10 \u3068\u3057\u3066\u4e0a\u56f3\u3068\u540c\u69d8\u306b\u8a08\u7b97\u3057\u3066\u307f\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f.<\/p>\n
\"\"<\/p>\n
\u5236\u7d04\u6761\u4ef6\u3092\u6e80\u305f\u3059\u65b9\u5411\u306b\u5b89\u5b9a\u5316\u3057\u3066\u3044\u308b\u306e\u304c\u308f\u304b\u308b.<\/p>\n
\u3053\u306e\u3088\u3046\u306b, \u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f\u306f, \u554f\u984c\u306e\u5b9a\u5f0f\u5316\u306b\u3044\u304f\u3064\u304b\u306e\u3084\u308a\u65b9\u304c\u3042\u308b\u5834\u5408\u306b\u306f\u6307\u6570\u306e\u5c0f\u3055\u306a\u3082\u306e\u3092\u9078\u3073, \u6307\u6570\u304c\u5927\u304d\u3051\u308c\u3070\u89e3\u6790\u7684\u306b\u5fae\u5206\u3092\u7e70\u308a\u8fd4\u3059\u3053\u3068\u306b\u3088\u308a\u6307\u6570\u3092\u4e0b\u3052\u308b\u3053\u3068\u306b\u3088\u308a\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306b\u5909\u63db\u3057\u3066\u89e3\u304f\u306e\u304c\u3088\u3044. \u306a\u304a, \u554f\u984c\u306b\u3088\u3063\u3066\u306f\u5236\u7d04\u6761\u4ef6\u3092\u6e80\u305f\u3059\u3088\u3046\u306b\u5b89\u5b9a\u5316\u3092\u56f3\u308b\u65b9\u304c\u3088\u3044.<\/p>\n
\u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f\u306e\u89e3\u6cd5 (2)<\/h3>\n
\u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f\u3092\u76f4\u63a5\u96e2\u6563\u5316\u3059\u308b\u3068\u975e\u5e38\u306b\u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f\u306b\u306a\u308b. \u3057\u304b\u3057, RADAU5(Radau IIA\u6cd5\u3092\u4f7f\u7528)\u3084DASSL(BDF\u3092\u4f7f\u7528)\u306e\u3088\u3046\u306b\u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f\u3092\u76f4\u63a5\u96e2\u6563\u5316\u3057\u3066\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u30d7\u30ed\u30b0\u30e9\u30e0\u3082\u3067\u304d\u3066\u3044\u308b. \u6307\u6570\u3092\u4e0b\u3052\u308b\u306a\u3069\u306e\u518d\u5b9a\u5f0f\u5316\u306b\u306f\u624b\u9593\u304c\u304b\u304b\u308b\u306e\u3067, \u3053\u308c\u3092\u4f7f\u3063\u3066\u89e3\u3051\u308b\u306e\u3067\u3042\u308c\u3070\u305d\u306e\u65b9\u304c\u3088\u3044. \u3057\u304b\u3057, \u591a\u304f\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u306f\u6307\u65701\u306b\u3057\u304b\u4f7f\u3048\u306a\u3044\u306a\u3069, \u3053\u306e\u5834\u5408\u3067\u3082\u6307\u6570\u304c\u5c0f\u3055\u304f\u306a\u308b\u3088\u3046\u306b\u5b9a\u5f0f\u5316\u3057\u3066\u304a\u304f\u306e\u304c\u3088\u3044\u3053\u3068\u306f\u5909\u308f\u3089\u306a\u3044.<\/p>\n
XLPack\u306b\u306f\u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f\u306b\u4f7f\u3048\u308b\u3044\u304f\u3064\u304b\u306e\u30d7\u30ed\u30b0\u30e9\u30e0\u304c\u53ce\u9332\u3055\u308c\u3066\u3044\u308b. \u8a08\u7b97\u4f8b\u306f\u5b9f\u7528\u30eb\u30fc\u30c1\u30f3\u306e\u30d9\u30f3\u30c1\u30de\u30fc\u30af\u7de8\u3092\u53c2\u7167\u305b\u3088.
\n<\/div><\/div><\/div>\n
\n
\u53c2\u8003\u6587\u732e<\/h4>\n[1] \u4e09\u4e95\u658c\u53cb\u300c\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u6cd5\u300d(2003) \u5ca9\u6ce2\u66f8\u5e97
\n[2] E. Hairer, \u4ed6\u300c\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u6cd5 I\u300d(2007) \u30b7\u30e5\u30d7\u30ea\u30f3\u30ac\u30fc\u30fb\u30b8\u30e3\u30d1\u30f3
\n[3] E. Hairer, \u4ed6\u300c\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u6cd5 II\u300d(2008) \u30b7\u30e5\u30d7\u30ea\u30f3\u30ac\u30fc\u30fb\u30b8\u30e3\u30d1\u30f3
\n[4] U.M.\u30a2\u30c3\u30b7\u30e3\u30fc, \u4ed6\u300c\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u3068\u5fae\u5206\u4ee3\u6570\u65b9\u7a0b\u5f0f\u306e\u6570\u5024\u89e3\u6cd5\u300d(2006) \u57f9\u98a8\u9928<\/p>\n","protected":false},"excerpt":{"rendered":"
\u76ee\u6b21 \u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f (1): \u6982\u8981, \uff11\u6bb5\u6cd5, \u591a\u6bb5\u6cd5, \u4f7f\u308f\u308c\u306a\u304f\u306a\u3063\u305f\u89e3\u6cd5 \u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f (2): \u8aa4\u5dee\u306e\u63a8\u5b9a, \u88dc\u5916\u6cd5, \u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u81ea\u52d5\u8abf\u7bc0, \u5bc6\u51fa\u529b, 2\u968e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f, \u9045\u5ef6\u5fae\u5206\u65b9\u7a0b\u5f0f \u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f (3): […]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[13],"tags":[],"class_list":["post-3019","post","type-post","status-publish","format-standard","hentry","category-num"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/3019","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/comments?post=3019"}],"version-history":[{"count":3,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/3019\/revisions"}],"predecessor-version":[{"id":3146,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/posts\/3019\/revisions\/3146"}],"wp:attachment":[{"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/media?parent=3019"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/categories?post=3019"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ktech.biz\/jp\/wp-json\/wp\/v2\/tags?post=3019"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}