{"id":5521,"date":"2026-07-05T11:22:21","date_gmt":"2026-07-05T02:22:21","guid":{"rendered":"https:\/\/www.ktech.biz\/jp\/?p=5521"},"modified":"2026-07-05T12:06:38","modified_gmt":"2026-07-05T03:06:38","slug":"10-1_ode","status":"publish","type":"post","link":"https:\/\/www.ktech.biz\/jp\/num\/10-1_ode\/","title":{"rendered":"10.1 \u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u57fa\u672c\u6570\u5024\u89e3\u6cd5"},"content":{"rendered":"\n<h3>10.1.1 \u6570\u5024\u89e3\u6cd5\u306e\u5206\u985e<\/h3>\n<p>\u5909\u6570 \\(t\\) \u3092\u96e2\u6563\u5316\u3057\u3066, \\(t_0, t_1, \\dots, t_n, \\dots\\) \u3068\u8868\u3059. \u3053\u308c\u3089\u306e\u70b9\u306e\u9593\u9694\u3092\u30b9\u30c6\u30c3\u30d7\u5e45\u3042\u308b\u3044\u306f\u30b9\u30c6\u30c3\u30d7\u5e45 (step size) \u3068\u3044\u3044, \\(h_n\\) \u3067\u8868\u3059.<br \/>\n\\[<br \/>\n  h_n = t_{n+1} &#8211; t_n<br \/>\n\\]\n\u30b9\u30c6\u30c3\u30d7\u5e45\u306f\u5fc5\u305a\u3057\u3082\u4e00\u5b9a\u3067\u306a\u304f\u3066\u3088\u3044\u304c, \\(n\\) \u306b\u3088\u3089\u305a\u4e00\u5b9a\u306e\u5834\u5408\u306b\u306f\u5358\u306b \\(h\\) \u3068\u8868\u3059\u3053\u3068\u304c\u3042\u308b. \u672a\u77e5\u95a2\u6570 \\(y(t)\\) \u306e \\(t_n\\) \u306b\u304a\u3051\u308b\u5024 \\(y(t_n)\\) \u306e\u8fd1\u4f3c\u5024\u3092 \\(y_n\\) \u3068\u8868\u3059\u3053\u3068\u306b\u3059\u308b.<\/p>\n<p>\u5dee\u5206\u6cd5 (\u5fae\u5206\u3092\u5dee\u5206\u3067\u8fd1\u4f3c\u3059\u308b\u65b9\u6cd5) \u306b\u3088\u308a\u6570\u5024\u89e3\u6cd5\u3092\u5c0e\u304f\u3053\u3068\u3092\u8003\u3048\u308b. \u4ee3\u8868\u7684\u306a\u5dee\u5206\u306e\u6c42\u3081\u65b9\u306b\u306f, \u6b21\u306e\u3088\u3046\u306a\u3082\u306e\u304c\u3042\u308b.<\/p>\n<ul>\n<li>\u524d\u9032\u5dee\u5206: \\((y_{n+1} &#8211; y_n)\/h\\)<\/li>\n<li>\u5f8c\u9000\u5dee\u5206: \\((y_n &#8211; y_{n-1})\/h\\)<\/li>\n<li>\u4e2d\u5fc3\u5dee\u5206: \\((y_{n+1} &#8211; y_{n-1})\/2h\\)<\/li>\n<\/ul>\n<p>\u307e\u305a, \u524d\u9032\u5dee\u5206\u3092\u4f7f\u3063\u3066\u5fae\u5206\u3092\u8fd1\u4f3c\u3059\u308b.<br \/>\n\\[<br \/>\n  y_n&#8217; = f(t_n, y_n) \\simeq (y_{n+1} &#8211; {y_n})\/h<br \/>\n\\]\n\u3053\u308c\u3088\u308a\u6b21\u306e\u8fd1\u4f3c\u5f0f\u304c\u5f97\u3089\u308c\u308b.<br \/>\n\\[<br \/>\n  y_{n+1} = y_n + hf(t_n, y_n)<br \/>\n\\]\n\u3053\u306e\u5f0f\u3092\u4f7f\u3044, \u521d\u671f\u5024 \\(y_0\\) \u304b\u3089\u59cb\u3081\u3066, \\(y_1, y_2, \\sim\\) \u3068\u6b21\u3005\u306b\u5404\u70b9\u306b\u304a\u3051\u308b\u8fd1\u4f3c\u89e3\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u3053\u308c\u3092<strong>\u30aa\u30a4\u30e9\u30fc\u6cd5<\/strong> (Euler method) \u3068\u3044\u3046.<\/p>\n<p>\u5f8c\u9000\u5dee\u5206\u3092\u4f7f\u3063\u305f\u5834\u5408\u306b\u306f, \u540c\u69d8\u306b\u3057\u3066\u6b21\u5f0f\u306b\u3088\u308a\u8fd1\u4f3c\u89e3\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<br \/>\n\\[<br \/>\n  y_{n+1} = y_n + hf(t_{n+1}, y_{n+1})<br \/>\n\\]\n\u3053\u308c\u3092<strong>\u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5<\/strong> (backward Euler method) \u3068\u3044\u3046.<\/p>\n<p>\u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5\u3067\u306f\u53f3\u8fba\u306e \\(f()\\) \u306e\u4e2d\u306b \\(y_{n+1}\\) \u304c\u542b\u307e\u308c\u3066\u304a\u308a, \\(y_{n+1}\\) \u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u65b9\u7a0b\u5f0f\u3092\u89e3\u304b\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044 (\u4e00\u822c\u7684\u306b\u306f\u975e\u7dda\u5f62\u65b9\u7a0b\u5f0f\u306b\u306a\u308b). \u3053\u306e\u3088\u3046\u306a\u89e3\u6cd5\u3092<strong>\u9670\u89e3\u6cd5<\/strong> (\u9670\u7684\u516c\u5f0f, implicit formula) \u3068\u3044\u3046. \u3053\u308c\u306b\u5bfe\u3057\u3066, \u30aa\u30a4\u30e9\u30fc\u6cd5\u306e\u5834\u5408\u306b\u306f\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3053\u3068\u306a\u304f\u76f4\u63a5 \\(y_{n+1}\\) \u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u305f. \u305d\u306e\u3088\u3046\u306a\u89e3\u6cd5\u3092<strong>\u967d\u89e3\u6cd5<\/strong> (\u967d\u7684\u516c\u5f0f, explicit formula) \u3068\u3044\u3046.<\/p>\n<p>\u6b21\u306b, \u4e2d\u5fc3\u5dee\u5206\u3092\u4f7f\u3063\u305f\u5834\u5408\u306b\u306f, \u6b21\u5f0f\u306b\u3088\u308a\u8fd1\u4f3c\u89e3\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<br \/>\n\\[<br \/>\n  y_{n+1} = y_{n-1} + 2hf(t_n, y_n)<br \/>\n\\]\n\u3053\u308c\u306f\u4e2d\u70b9\u5247\u3068\u3088\u3070\u308c\u308b.<\/p>\n<p>\u4e2d\u70b9\u5247\u3067\u306f, \\(y_{n+1}\\) \u306e\u8a08\u7b97\u306b\u306f \\(y_n\\) \u306e\u4ed6\u306b\u904e\u53bb\u306e\u5024 \\(y_{n-1}\\) \u3082\u5fc5\u8981\u3067, 2 \u3064\u306e\u5024\u3092\u4f7f\u3046\u306e\u3067 2 \u6bb5\u6cd5\u3068\u3088\u3070\u308c, <strong>\u591a\u6bb5\u6cd5<\/strong> (\u591a\u6bb5\u968e\u6cd5, multistep method) \u3068\u3088\u3070\u308c\u308b\u89e3\u6cd5\u306e\u3072\u3068\u3064\u306b\u306a\u308b. \u3053\u308c\u306b\u5bfe\u3057\u3066, \u30aa\u30a4\u30e9\u30fc\u6cd5\u3068\u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5\u306f \\(y_n\\) \u3060\u3051\u3057\u304b\u4f7f\u308f\u306a\u3044\u306e\u3067, <strong>1 \u6bb5\u6cd5<\/strong> (1 \u6bb5\u968e\u6cd5, one-step method) \u3068\u3088\u3070\u308c\u308b\u89e3\u6cd5\u306b\u5206\u985e\u3055\u308c\u308b.<\/p>\n<p>\u307e\u3068\u3081\u308b\u3068, \u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u89e3\u6cd5\u3092\u5927\u304d\u304f\u5206\u985e\u3059\u308b\u3068 1 \u6bb5\u6cd5\u3068\u591a\u6bb5\u6cd5\u304c\u3042\u308a, \u305d\u308c\u305e\u308c\u306b\u967d\u89e3\u6cd5\u3068\u9670\u89e3\u6cd5\u304c\u3042\u308b.<\/p>\n<h3>10.1.2 \u6570\u5024\u89e3\u6cd5\u306e\u6b21\u6570<\/h3>\n<p>\u95a2\u6570 \\(y(t)\\) (\u771f\u306e\u89e3) \u3092 \\(t_n\\) \u306b\u304a\u3044\u3066\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\ny(t_{n+1}) &#038; = y(t_n) + hy'(t_n) + (1\/2)h^2y&#8221;(t_n) + (1\/6)h^3y&#8221;'(t_n) + \\dots + (1\/p)h^py^(p)(t_n) + \\dots \\\\<br \/>\n&#038; = y(t_n) + hf(t_n, y(t_n)) + (1\/2)h^2f'(t_n, y(t_n))+ (1\/6)h^3f&#8221;(t_n, y(t_n)) + \\dots + (1\/p)h^pf^{p-1}(t_n, y(t_n)) + \\dots \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\u6570\u5024\u89e3\u6cd5\u306b\u3088\u308b\u8fd1\u4f3c\u89e3 \\(y_{n+1}\\) \u306e\u53f3\u8fba\u304c\u4e0a\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b \\(y(t_{n+1})\\) \u306e p \u6b21\u306e\u9805\u307e\u3067\u4e00\u81f4\u3059\u308b\u3068\u304d, \u3059\u306a\u308f\u3061\u6b21\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3068\u304d\u306b\u305d\u306e\u6570\u5024\u89e3\u6cd5\u306f p \u6b21\u306e\u65b9\u6cd5\u3067\u3042\u308b\u3068\u3044\u3046.<br \/>\n\\[<br \/>\n  y_{n+1} &#8211; y(t_{n+1}) = O(h^{p+1})<br \/>\n\\]\n\u30aa\u30a4\u30e9\u30fc\u6cd5\u306f\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306e 1 \u6b21 \\((h)\\) \u306e\u9805\u307e\u3067\u3067\u6253\u3061\u5207\u3063\u305f\u3082\u306e\u306b\u7b49\u3057\u3044\u304b\u3089\u660e\u3089\u304b\u306b 1 \u6b21\u306e\u65b9\u6cd5\u3067\u3042\u308b. \u307e\u305f, \u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5\u306f 1 \u6b21, \u4e2d\u70b9\u5247\u306f 2 \u6b21\u306e\u65b9\u6cd5\u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u304b\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n<h4>\u6570\u5024\u5b9f\u9a13 (1)<\/h4>\n<p>\u6700\u3082\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3092\u8003\u3048\u308b.<br \/>\n\\[<br \/>\n  y&#8217; = y, \\space y(0) = 1<br \/>\n\\]\n\u3053\u306e\u554f\u984c\u306f\u89e3\u6790\u89e3\u304c\u308f\u304b\u3063\u3066\u3044\u3066 \\(y = e^t\\) \u3067\u3042\u308b. \u4e0a\u306e 3 \u3064\u306e\u516c\u5f0f\u3092\u7528\u3044\u3066\u30b9\u30c6\u30c3\u30d7\u5e45\u3092 \\(0.1, 0.05, 0.01, \\dots\\) \u3068\u3044\u3046\u3088\u3046\u306b\u5c0f\u3055\u304f\u3057\u3066\u3044\u304d\u306a\u304c\u3089\u9806\u306b\u8a08\u7b97\u3059\u308b. \\(t = 1\\) \u306b\u304a\u3044\u3066\u5f97\u3089\u308c\u305f\u8fd1\u4f3c\u89e3\u306e\u771f\u5024 \\(y = e\\) \u304b\u3089\u306e\u8aa4\u5dee\u3068\u5fc5\u8981\u3068\u3057\u305f\u30b9\u30c6\u30c3\u30d7\u6570\u306e\u5024\u3092\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3059\u308b\u3068\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u3063\u305f. \u6a2a\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee (\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044), \u7e26\u8ef8\u306f\u30b9\u30c6\u30c3\u30d7\u6570 (= \u95a2\u6570\u8a55\u4fa1 (\\(f()\\) \u306e\u8a08\u7b97) \u56de\u6570) \u3067\u3042\u308b.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_1_exp1.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5619\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_1_exp1.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_1_exp1-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\u30aa\u30a4\u30e9\u30fc\u6cd5\u3068\u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5\u3067\u306f, 1,000 \u30b9\u30c6\u30c3\u30d7 (\\(h = 10^{-3}\\)) \u3067\u304a\u3088\u305d 3 \u6841\u534a\u306e\u7cbe\u5ea6, 1,000,000 \u30b9\u30c6\u30c3\u30d7 (\\(h = 10^{-6}\\)) \u3067\u304a\u3088\u305d 6 \u6841\u534a\u306e\u7cbe\u5ea6\u3068\u306a\u308a, 1 \u6b21\u306e\u65b9\u6cd5\u3067\u3042\u308b\u3053\u3068\u304c\u78ba\u304b\u3081\u3089\u308c\u305f. \u4e00\u65b9, \u4e2d\u70b9\u5247\u3067\u306f, 1,000 \u30b9\u30c6\u30c3\u30d7 (\\(h = 10^{-3}\\)) \u3067\u304a\u3088\u305d 6 \u6841\u534a\u306e\u7cbe\u5ea6, 1,000,000 \u30b9\u30c6\u30c3\u30d7 (\\(h = 10^{-6}\\)) \u3067\u304a\u3088\u305d 12 \u6841\u534a\u306e\u7cbe\u5ea6\u304c\u3042\u308a, 2 \u6b21\u306e\u65b9\u6cd5\u3068\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u540c\u3058\u8a08\u7b97\u91cf\u3067\u3042\u308c\u3070\u4e2d\u70b9\u5247\u306f\u30aa\u30a4\u30e9\u30fc\u6cd5\u306e\u500d\u306e\u6841\u6570\u3092\u6b63\u3057\u304f\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<\/p>\n<h3>10.1.3 1 \u6bb5\u6cd5<\/h3>\n<h4>10.1.3.1 \u30c6\u30a4\u30e9\u30fc\u6cd5<\/h4>\n<p>\u95a2\u6570 y(t) \u306e t_n \u306b\u304a\u3051\u308b\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3092\u518d\u5ea6\u6b21\u306b\u793a\u3059.<br \/>\n\\[<br \/>\n  y(t_{n+1}) = y(t_n) + hf(t_n, y(t_n)) + (1\/2)h^2f'(t_n, y(t_n)) + (1\/6)h^3f&#8221;(t_n, y(t_n)) + \\dots + (1\/p)h^pf^{p-1}(t_n, y(t_n)) + \\dots<br \/>\n\\]\n\u30aa\u30a4\u30e9\u30fc\u6cd5\u306f\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3092 1 \u6b21\u306e\u9805\u307e\u3067\u3067\u6253\u3061\u5207\u3063\u3066\u8fd1\u4f3c\u3057\u305f\u3082\u306e\u306b\u76f8\u5f53\u3057\u305f. \u305d\u3053\u3067, \u3088\u308a\u9ad8\u6b21\u306e\u9805\u307e\u3067\u4f7f\u3063\u305f\u8fd1\u4f3c\u5f0f\u3092\u4f7f\u3048\u3070\u30aa\u30a4\u30e9\u30fc\u6cd5\u3088\u308a\u7cbe\u5ea6\u306e\u3088\u3044, \u3059\u306a\u308f\u3061, \u3088\u308a\u6b21\u6570\u306e\u9ad8\u3044\u89e3\u6cd5\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3068\u8003\u3048\u3089\u308c\u308b. \u3053\u308c\u3092\u30c6\u30a4\u30e9\u30fc\u6cd5\u3068\u3044\u3046.<\/p>\n<p>\u4e0a\u5f0f\u3067 \\(f'(t, y(t))\\) \u3068 \\(f^{\\prime\\prime}(t, y(t))\\) \u306f\u5177\u4f53\u7684\u306b\u306f\u6b21\u306e\u3088\u3046\u306b\u8a08\u7b97\u3059\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; f'(t, y(t)) = f_t(t, y(t)) + f_y(t, y(t))f(t, y(t)) \\\\<br \/>\n&#038; f^{\\prime\\prime}(t, y(t)) = f_{tt}(t, y(t)) + 2f_{ty}(t, y(t))f(t, y(t)) + f_{yy}(t, y(t))f^2(t, y(t)) + f_t(t, y(t))f_y(t, y(t)) + f_y^2(t, y(t))f(t, y(t)) \\\\<br \/>\n\\end{align}<br \/>\n\\]\n2 \u6b21\u306e\u9805\u3092\u8a08\u7b97\u3059\u308b\u306e\u3067\u3055\u3048\u7d50\u69cb\u8907\u96d1\u306a\u3053\u3068\u304c\u308f\u304b\u308b. \u305d\u306e\u305f\u3081, \\(f()\\) \u306e\u9ad8\u968e\u306e\u5fae\u5206\u304c\u7c21\u5358\u306b\u8a08\u7b97\u3067\u304d\u308b\u95a2\u6570\u3067\u306a\u3044\u9650\u308a\u3053\u306e\u65b9\u6cd5\u3092\u5b9f\u969b\u306b\u9069\u7528\u3059\u308b\u306e\u306f\u96e3\u3057\u3044.<\/p>\n<h4>\u6570\u5024\u5b9f\u9a13 (2)<\/h4>\n<p>\u6570\u5024\u89e3\u6cd5\u306e\u6b21\u6570\u306e\u5b9f\u9a13\u3068\u540c\u3058\u6700\u3082\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3092\u4f7f\u3063\u3066 1 \u6b21 (\u30aa\u30a4\u30e9\u30fc\u6cd5), 2 \u6b21, 3 \u6b21\u306e\u30c6\u30a4\u30e9\u30fc\u6cd5\u306b\u3088\u308b\u8a08\u7b97\u3092\u5b9f\u969b\u306b\u884c\u3063\u3066\u6bd4\u8f03\u3059\u308b.<br \/>\n\\[<br \/>\n  y&#8217; = y, \\space y(0) = 1<br \/>\n\\]\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_2_exp2.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5621\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_2_exp2.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_2_exp2-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\u6a2a\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee (\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044), \u7e26\u8ef8\u306f\u30b9\u30c6\u30c3\u30d7\u6570\u306e\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3067\u3042\u308b. \u305d\u308c\u305e\u308c 1 \u6b21 \uff5e 3 \u6b21\u306e\u50be\u304d\u3092\u793a\u3057\u3066\u3044\u308b. \u305f\u3060\u3057, 1 \u30b9\u30c6\u30c3\u30d7\u306e\u8a08\u7b97\u306b\u306f\u4e0a\u306e \\(f'(t, y(t))\\) \u3084 \\(f^{\\prime\\prime}(t, y(t))\\) \u306e\u8a08\u7b97\u304c\u542b\u307e\u308c\u308b\u306e\u3067, 1 \u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a\u306e\u8a08\u7b97\u91cf\u306f, 2 \u6b21\u306e\u30c6\u30a4\u30e9\u30fc\u6cd5\u3067\u306f \\(f()\\) \u306e\u8a08\u7b97\u304c 3 \u56de\u76f8\u5f53, 3 \u6b21\u306e\u30c6\u30a4\u30e9\u30fc\u6cd5\u3067\u306f 6 \u56de\u76f8\u5f53\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308b. \u305d\u306e\u305f\u3081\u8a08\u7b97\u91cf\u3092\u6307\u6a19\u3068\u3059\u308b\u3068\u4e00\u822c\u7684\u306b\u306f\u30aa\u30a4\u30e9\u30fc\u6cd5\u3068\u306e\u5dee\u306f\u3084\u3084\u7e2e\u307e\u308b. \u305f\u3060\u3057, \u4e0a\u306e\u4f8b\u306e\u65b9\u7a0b\u5f0f\u3067\u306f\u5fae\u5206\u4fc2\u6570\u304c 0 \u304b 1 \u3068\u7c21\u5358\u306a\u305f\u3081 1 \u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a\u306e\u8a08\u7b97\u91cf\u306e\u5dee\u306f\u5c11\u306a\u3044.<\/p>\n<h4>10.1.3.2 2 \u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/h4>\n<p><strong>\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/strong> (Runge-Kutta method) \u306f, \\(f()\\) \u306e\u5fae\u5206\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u306a\u304f\u9ad8\u6b21\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306b\u76f8\u5f53\u3059\u308b\u7cbe\u5ea6\u3092\u5f97\u308b 1 \u6bb5\u6cd5\u3067\u3042\u308b. \u3044\u304f\u3064\u304b\u306e \\(t\\) \u3068 \\(y\\) \u306b\u3064\u3044\u3066 \\(f(t, y)\\) \u3092\u8a08\u7b97\u3057\u305d\u308c\u3089\u306e\u91cd\u307f\u4ed8\u304d\u5e73\u5747\u3092\u4f5c\u3063\u3066\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306e p \u6b21\u306e\u9805\u307e\u3067\u4e00\u81f4\u3055\u305b\u308b\u3053\u3068\u306b\u3088\u308a p \u6b21\u306e\u516c\u5f0f\u3092\u4f5c\u308b.<\/p>\n<p>2\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u4e00\u822c\u5f62\u306f\u6b21\u5f0f\u306e\u3068\u304a\u308a\u3067\u3042\u308b. \\(a, b_1, b_2, c\\) \u306f\u30d1\u30e9\u30e1\u30fc\u30bf\u3067\u3042\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; k_1 = f(t_n, y_n) \\\\<br \/>\n&#038; k_2 = f(t_n + ch, y_n + h ak_1) \\\\<br \/>\n&#038; y_n+1 = y_n + h(b_1k_1 + b_2k_2) \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\\(k_2\\) \u306e \\((t_n, y_n)\\) \u306b\u304a\u3051\u308b\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<br \/>\n\\[<br \/>\n  k_2 = f_n + c h f_{t,n} + a k_1 f_{y,n} + \\dots<br \/>\n\\]\n\u3053\u308c\u3092\u4e0a\u306e\u5f0f\u306b\u4ee3\u5165\u3057\u3066, \u6b21\u5f0f\u304c\u5f97\u3089\u308c\u308b.<br \/>\n\\[<br \/>\n  y_{n+1} = y_n + (b_1 + b_2) h f_n + b_2 h^2(c f_{t,n} + a f_{y,n}f_n + \\dots )<br \/>\n\\]\n\u3053\u306e\u5f0f\u3068\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306e\u5f0f\u3092\u6bd4\u8f03\u3057\u3066, \\(h\\) \u3068 \\(h^2\\) \u306e\u4fc2\u6570\u304c\u7b49\u3057\u3044\u3068\u3059\u308b\u3068\u6b21\u5f0f\u304c\u5f97\u3089\u308c\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; b_1 + b_2 = 1 \\\\<br \/>\n&#038; b_2c = 1\/2 \\\\<br \/>\n&#038; b_2a = 1\/2 \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\u672a\u77e5\u6570\u304c 4 \u500b\u3067\u5f0f\u304c 3 \u672c\u306a\u306e\u3067\u81ea\u7531\u5ea6\u304c 1 \u3042\u308b. \u3088\u304f\u4f7f\u308f\u308c\u308b\u3082\u306e\u3068\u3057\u3066\u306f, \\(c = 1\\) \u3068\u3057\u305f\u3082\u306e\u3092<strong>\u30db\u30a4\u30f3\u6cd5<\/strong> (Heun method) \u3068\u3088\u3076. \u307e\u305f, \\(c = 1\/2\\) \u3068\u3057\u305f\u3082\u306e\u3092<strong>\u4fee\u6b63\u30aa\u30a4\u30e9\u30fc\u6cd5<\/strong>\u3068\u3088\u3076\u3053\u3068\u304c\u3042\u308b (2 \u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3068\u306f\u3088\u3070\u308c\u306a\u3044\u304c, \u3053\u308c\u3089\u306e\u7dcf\u79f0\u3068\u3068\u3089\u3048\u3066\u3088\u3044).<\/p>\n<p>\u30db\u30a4\u30f3\u6cd5\u306e\u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; k_1 = f(t_n, y_n) \\\\<br \/>\n&#038; k_2 = f(t_n + h, y_n + h k_1) \\\\<br \/>\n&#038; y_{n+1} = y_n + h(k_1 + k_2)\/2 \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\u4fee\u6b63\u30aa\u30a4\u30e9\u30fc\u6cd5\u306e\u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; k_1 = f(t_n, y_n) \\\\<br \/>\n&#038; k_2 = f(t_n + h\/2, y_n + h k_1\/2) \\\\<br \/>\n&#038; y_{n+1} = y_n + h k_2 \\\\<br \/>\n\\end{align}<br \/>\n\\]\n<h4>\u6570\u5024\u5b9f\u9a13 (3)<\/h4>\n<p>\u518d\u3073\u6700\u3082\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3067\u30db\u30a4\u30f3\u6cd5\u3068\u4fee\u6b63\u30aa\u30a4\u30e9\u30fc\u6cd5\u306e\u7cbe\u5ea6\u3092\u78ba\u8a8d\u3059\u308b.<br \/>\n\\[<br \/>\n  y&#8217; = y, \\space y(0) = 1<br \/>\n\\]\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_3_exp3.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5623\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_3_exp3.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_3_exp3-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\u6a2a\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee (\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044), \u7e26\u8ef8\u306f\u30b9\u30c6\u30c3\u30d7\u6570\u306e\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3067\u3042\u308b. \u30db\u30a4\u30f3\u6cd5\u3068\u4fee\u6b63\u30aa\u30a4\u30e9\u30fc\u6cd5\u306e\u8a08\u7b97\u7d50\u679c\u306f\u307b\u307c\u540c\u3058\u3067\u3042\u308b. 2 \u6b21\u306e\u50be\u304d\u3092\u78ba\u8a8d\u3067\u304d, \u6700\u521d\u306e\u65b9\u3067\u793a\u3057\u305f\u4e2d\u70b9\u5247\u3068\u4f3c\u305f\u30b0\u30e9\u30d5\u306b\u306a\u3063\u305f. \u305f\u3060\u3057, \u8a08\u7b97\u5f0f\u304b\u3089\u308f\u304b\u308b\u3088\u3046\u306b\u30db\u30a4\u30f3\u6cd5\u3068\u4fee\u6b63\u30aa\u30a4\u30e9\u30fc\u6cd5\u3067\u306f 1 \u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a 2 \u56de\u306e\u95a2\u6570\u8a55\u4fa1 (\\(f()\\) \u306e\u8a08\u7b97) \u3092\u884c\u3063\u3066\u3044\u308b\u306e\u306b\u5bfe\u3057\u3066, \u4e2d\u70b9\u5247\u3067\u306f 1 \u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a 1 \u56de\u306e\u95a2\u6570\u8a55\u4fa1\u3057\u304b\u884c\u308f\u306a\u304f\u3066\u3088\u3044.<\/p>\n<h4>10.1.3.3 \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u4e00\u822c\u5f62<\/h4>\n<p>\u4e0a\u3068\u540c\u69d8\u306b 3 \u6b21, 4 \u6b21 \u30fb\u30fb\u30fb \u3068\u9ad8\u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u516c\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d, \u591a\u304f\u306e\u516c\u5f0f\u304c\u63d0\u6848\u3055\u308c\u3066\u3044\u308b.<\/p>\n<p>\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306f\u4e00\u822c\u7684\u306b\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; y_{n+1} = y_n + h \\sum_{i=1}^s b_ik_i \\\\<br \/>\n&#038; k_i = f(t_n + c_ih, y_n + h \\sum_{j=1}^s a_{ij}k_j) \\space (i = 1, 2, \\dots, s) \\\\<br \/>\n\\end{align}<br \/>\n\\]\ns \u306f<strong>\u6bb5\u6570<\/strong>(stage)\u3068\u3088\u3070\u308c, 1 \u30b9\u30c6\u30c3\u30d7\u9032\u3080\u306e\u306b\u5fc5\u8981\u306a \\(f(t, y)\\) \u306e\u8a08\u7b97\u56de\u6570\u3092\u8868\u3059 (1 \u6bb5\u6cd5\u3084\u591a\u6bb5\u6cd5\u306e&#8221;\u6bb5&#8221;\u3068\u306f\u7570\u306a\u308b). \\(a_{ij}\\), \\(b_i\\) \u304a\u3088\u3073 \\(c_i\\) \u306f\u30d1\u30e9\u30e1\u30fc\u30bf\u3067, \\(b_i\\) \u304a\u3088\u3073 \\(c_i\\) \u306f\u6b21\u5f0f\u3092\u6e80\u305f\u3059.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; c_i = \\sum_{j=1}^s a_{ij} \\space (i = 1, 2, \\dots, s) \\\\<br \/>\n&#038; \\sum_{i=1}^s b_i = 1 \\\\<br \/>\n\\end{align}<br \/>\n\\]\n<p>\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u8868\u3059\u306b\u306f<strong>\u30d6\u30c3\u30c1\u30e3\u30fc\u914d\u5217<\/strong>\u3068\u547c\u3070\u308c\u308b\u6b21\u306e\u3088\u3046\u306a\u8868\u8a18\u6cd5\u304c\u3042\u308b.<br \/>\n\\[<br \/>\n\\begin{array}{c|cccc}<br \/>\nc_1 &#038; a_11 &#038; a_12 &#038; \\dots &#038; a_1s \\\\<br \/>\nc_2 &#038; a_21 &#038; a_22 &#038; \\dots &#038; a_2s \\\\<br \/>\nc_3 &#038; a_31 &#038; a_32 &#038; \\dots &#038; a_3s \\\\<br \/>\n\\vdots &#038; \\vdots &#038; \\vdots &#038; \\vdots &#038; \\vdots \\\\<br \/>\nc_s &#038; a_s1 &#038; a_s2 &#038; \\dots &#038; a_ss \\\\<br \/>\n\\hline<br \/>\n    &#038; b_1 &#038; b_2 &#038; \\dots &#038; b_s \\\\<br \/>\n\\end{array}<br \/>\n\\]\n<p>\\(a_{ij}\\) \u306e\u5024\u306b\u3088\u3063\u3066 3 \u3064\u306e\u5834\u5408\u306b\u5206\u3051\u3089\u308c\u308b.<\/p>\n<p>(i) \\(a_{ij} = 0 \\space (i \\le j)\\) \u306e\u5834\u5408 (\u53f3\u4e0a\u4e09\u89d2\u90e8\u5206\u3068\u5bfe\u89d2\u90e8\u5206\u304c 0):<br \/>\n\u967d\u89e3\u6cd5. \u5358\u306b\u300c\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u300d\u3042\u308b\u3044\u306f\u300c<strong>\u967d\u7684<\/strong> (explicit) <strong>\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/strong>\u300d\u3068\u3044\u3046. \\(c_1 = 0\\) \u3067\u3042\u308b. \u30d6\u30c3\u30c1\u30e3\u30fc\u914d\u5217\u3067\u306f \\(a_{ij}\\) \u306e\u53f3\u4e0a\u4e09\u89d2\u90e8\u5206\u3068\u5bfe\u89d2\u90e8\u5206\u3092\u66f8\u304b\u306a\u3044 (\u7a7a\u767d\u306b\u3059\u308b).<\/p>\n<p>(ii) \\(a_{ij} = 0 \\space (i < j)\\) \u306e\u5834\u5408 (\u53f3\u4e0a\u4e09\u89d2\u90e8\u5206\u304c 0):<br \/>\n\u534a\u9670\u89e3\u6cd5. \u300c<strong>\u534a\u9670\u7684<\/strong> (semi-implicit) <strong>\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/strong>\u300d\u3068\u3044\u3046.<\/p>\n<p>(iii) \\(a_{ij} \\ne 0 \\space (i < j)\\) \u306e\u5834\u5408 (\u53f3\u4e0a\u4e09\u89d2\u90e8\u5206\u306b 0 \u3067\u306a\u3044\u3082\u306e\u304c\u3042\u308b):<br \/>\n\u9670\u89e3\u6cd5. \u300c<strong>\u9670\u7684<\/strong> (implicit) <strong>\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/strong>\u300d\u3068\u3044\u3046.<\/p>\n<p>\u4ee5\u4e0b (i) \u306b\u3064\u3044\u3066\u8aac\u660e\u3059\u308b. (ii), (iii) \u306b\u3064\u3044\u3066\u306f\u5225\u9014\u300c10.3 \u30b9\u30c6\u30a3\u30d5\u306a\u65b9\u7a0b\u5f0f\u300d\u3067\u89e6\u308c\u308b.<\/p>\n<p>\u4e0a\u3067\u793a\u3057\u305f\u30db\u30a4\u30f3\u6cd5\u3068\u4fee\u6b63\u30aa\u30a4\u30e9\u30fc\u6cd5\u3092\u30d6\u30c3\u30c1\u30e3\u30fc\u914d\u5217\u3067\u8868\u3059\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n<h4>\u30db\u30a4\u30f3\u6cd5<\/h4>\n<p>\\[<br \/>\n\\begin{array}{c|cc}<br \/>\n0 \\\\<br \/>\n1 &#038; 1 \\\\<br \/>\n\\hline<br \/>\n  &#038; 1\/2 &#038; 1\/2 \\\\<br \/>\n\\end{array}<br \/>\n\\]\n<h4>\u4fee\u6b63\u30aa\u30a4\u30e9\u30fc\u6cd5<\/h4>\n<p>\\[<br \/>\n\\begin{array}{c|cc}<br \/>\n0 \\\\<br \/>\n1\/2 &#038; 1\/2 \\\\<br \/>\n\\hline<br \/>\n    &#038; 0 &#038; 1 \\\\<br \/>\n\\end{array}<br \/>\n\\]\n(\u967d\u7684) \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u306f\u3067\u304d\u308b\u3060\u3051\u9ad8\u6b21\u306e\u516c\u5f0f\u306b\u306a\u308b\u3088\u3046\u306b\u30d1\u30e9\u30e1\u30fc\u30bf\u3092\u6c7a\u3081\u308b\u304c, s \u6bb5\u306a\u3089\u3070 s \u6b21\u306b\u306a\u308b\u308f\u3051\u3067\u306f\u306a\u304f, s \u6bb5\u306e\u516c\u5f0f\u304c\u9054\u6210\u3057\u3046\u308b\u6700\u5927\u306e\u6b21\u6570 p \u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u3066\u3044\u308b.<\/p>\n<table style=\"height: 54px;\" width=\"400\">\n<tbody>\n<tr>\n<td>\u6bb5\u6570 s<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>8<\/td>\n<td>9<\/td>\n<td>10<\/td>\n<td>11<\/td>\n<\/tr>\n<tr>\n<td>\u6700\u5927\u6b21\u6570 p<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>7<\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>10.1.3.4 4 \u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/h4>\n<p>4 \u6bb5 4 \u6b21\u306e\u516c\u5f0f\u3092\u4f5c\u308d\u3046\u3068\u3059\u308b\u3068, \u30d1\u30e9\u30e1\u30fc\u30bf\u304c 12 \u500b\u3067\u81ea\u7531\u5ea6\u304c 4 \u306a\u306e\u3067\u69d8\u3005\u306a\u516c\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u304c\u53ef\u80fd\u3067\u3042\u308b. \u305d\u306e\u4e2d\u3067\u6700\u826f\u306e\u3082\u306e\u3092\u9078\u3076\u305f\u3081\u306e\u591a\u304f\u306e\u7814\u7a76\u304c\u884c\u308f\u308c\u305f. \u4e2d\u3067\u3082\u30af\u30c3\u30bf\u304c\u9078\u3093\u3060\u4ee5\u4e0b\u306e 2 \u7d44\u306e\u4fc2\u6570\u304c\u6709\u540d\u3067\u3042\u308b.<\/p>\n<h4>(1) \u53e4\u5178\u7684\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5<\/h4>\n<p>\u5358\u306b\u300c\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u300d\u3068\u3044\u3046\u3068\u3053\u306e\u516c\u5f0f\u3092\u6307\u3059.<br \/>\n\\[<br \/>\n\\begin{array}{c|cccc}<br \/>\n0    \\\\<br \/>\n1\/2  &#038; 1\/2 \\\\<br \/>\n1\/2  &#038; 0   &#038; 1\/2 \\\\<br \/>\n1    &#038; 0   &#038; 0   &#038; 1 \\\\<br \/>\n\\hline<br \/>\n     &#038; 1\/6 &#038; 1\/3 &#038; 1\/3 &#038; 1\/6 \\\\<br \/>\n\\end{array}<br \/>\n\\]\n\u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; k_1 = f(t_n, y_n) \\\\<br \/>\n&#038; k_2 = f(t_n + h\/2, y_n + h k_1\/2) \\\\<br \/>\n&#038; k_3 = f(t_n + h\/2, y_n + h k_2\/2) \\\\<br \/>\n&#038; k_4 = f(t_n + h, y_n + h k_3) \\\\<br \/>\n&#038; y_{n+1} = y_n + h(k_1 + 2 k_2 + 2 k_3 + k_4)\/6 \\\\<br \/>\n\\end{align}<br \/>\n\\]\n<h4>(2) \u30af\u30c3\u30bf\u306e 3\/8 \u516c\u5f0f<\/h4>\n<p>\\[<br \/>\n\\begin{array}{c|cccc}<br \/>\n0   \\\\<br \/>\n1\/3 &#038; 1\/3 \\\\<br \/>\n2\/3 &#038; -1\/3 &#038; 1\/2 \\\\<br \/>\n1   &#038; 1    &#038; -1 &#038; 1 \\\\<br \/>\n\\hline<br \/>\n    &#038; 1\/8 &#038; 3\/8 &#038; 3\/8 &#038; 1\/8 \\\\<br \/>\n\\end{array}<br \/>\n\\]\n\u8a08\u7b97\u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; k_1 = f(t_n, y_n) \\\\<br \/>\n&#038; k_2 = f(t_n + h\/2, y_n + hk_1\/3) \\\\<br \/>\n&#038; k_3 = f(t_n + h\/2, y_n + h(-k_1\/3 + k_2)) \\\\<br \/>\n&#038; k_4 = f(, t_n + h, y_n + h(k_1 &#8211; k_2 + k_3)) \\\\<br \/>\n&#038; y_{n+1} = y_n + h(k_1 + 3k_2 + 3k_3 + k_4)\/8 \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u4e00\u65cf\u306b\u304a\u3044\u3066\u6700\u3082\u3088\u304f\u4f7f\u308f\u308c\u3066\u3044\u308b\u306e\u306f 4 \u6b21\u306e\u53e4\u5178\u7684\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u3042\u308b. \u3044\u304f\u3064\u304b\u3042\u308b 4 \u6bb5 4 \u6b21\u306e\u516c\u5f0f\u9593\u3067\u306e\u7cbe\u5ea6\u306e\u5dee\u306f\u307b\u3068\u3093\u3069\u306a\u3044\u304c, \u3053\u306e\u516c\u5f0f\u306f\u4fc2\u6570 \\(a_{ij}\\) \u306b 0 \u304c\u591a\u304f, \u3059\u3079\u3066\u306e\u4fc2\u6570\u304c\u6b63\u3067\u3042\u308b\u306a\u3069\u306e\u7406\u7531\u306b\u3088\u308a\u4ed6\u306e\u3082\u306e\u3088\u308a\u4f7f\u3044\u3084\u3059\u3044. \u307e\u305f, 5 \u6b21\u306e\u516c\u5f0f\u306f 6 \u6bb5\u4ee5\u4e0a\u3067\u306a\u3044\u3068\u4f5c\u308c\u306a\u3044\u305f\u3081, \u3053\u308c\u4ee5\u4e0a\u9ad8\u6b21\u306e\u3082\u306e\u3092\u6c42\u3081\u308b\u3068\u8907\u96d1\u306b\u306a\u308b\u305f\u3081\u3053\u308c\u304c\u666e\u53ca\u3057\u305f\u306e\u304b\u3082\u3057\u308c\u306a\u3044.<\/p>\n<h4>\u6570\u5024\u5b9f\u9a13 (4)<\/h4>\n<p>\u6700\u3082\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3067 4 \u6b21\u306e\u53e4\u5178\u7684\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u304a\u3088\u30733\/8\u516c\u5f0f\u306e\u7cbe\u5ea6\u3092\u78ba\u8a8d\u3059\u308b.<br \/>\n\\[<br \/>\n  y&#8217; = y, \\space y(0) = 1<br \/>\n\\]\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_4_exp4.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5624\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_4_exp4.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_4_exp4-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\u6a2a\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee (\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044), \u7e26\u8ef8\u306f\u30b9\u30c6\u30c3\u30d7\u6570\u306e\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3067\u3042\u308b. \u53e4\u5178\u7684\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u304a\u3088\u3073 3\/8 \u516c\u5f0f\u306e\u8a08\u7b97\u7d50\u679c\u306f\u307b\u307c\u540c\u3058\u3067\u3042\u308b. \u4e21\u65b9\u3068\u3082 4 \u6b21\u306e\u50be\u304d\u3092\u3057\u3066\u3044\u308b\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u308b. \u305f\u3060\u3057, 1 \u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a 4 \u56de\u306e\u95a2\u6570\u8a55\u4fa1 (\\(f()\\) \u306e\u8a08\u7b97) \u304c\u5fc5\u8981\u3067\u3042\u308b.<\/p>\n<h3>10.1.4 \u591a\u6bb5\u6cd5<\/h3>\n<p>1 \u6bb5\u6cd5\u306f \\(y_{n+1}\\) \u306e\u8a08\u7b97\u306b\u73fe\u5728\u306e\u5024 \\(y_n\\) \u306e\u307f\u3092\u4f7f\u3046\u89e3\u6cd5\u3067\u3042\u3063\u305f\u304c, \u904e\u53bb\u306e\u5024 (\u305d\u308c\u307e\u3067\u306b\u8a08\u7b97\u6e08\u306e\u5024) \\(y_{n-1}, y_{n-2}, \\dots\\) \u3082\u4f7f\u3063\u305f\u65b9\u304c\u3088\u3044\u7d50\u679c\u304c\u5f97\u3089\u308c\u305d\u3046\u306b\u601d\u308f\u308c\u308b. \u3053\u306e\u30a2\u30a4\u30c7\u30a2\u3092\u5b9f\u73fe\u3057\u305f\u306e\u304c\u591a\u6bb5\u6cd5\u3067\u3042\u308b.<\/p>\n<p>1 \u968e\u306e\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u7a4d\u5206\u5f62\u3092\u8003\u3048\u308b.<br \/>\n\\[<br \/>\ny(t_{n+1}) = y(t_n) + \\int_{t_n}^{t_{n+1}} f(t, y(t)) dt<br \/>\n\\]\n\u3053\u3053\u3067, \u88ab\u7a4d\u5206\u95a2\u6570 \\(f()\\) \u3092\u88dc\u9593\u591a\u9805\u5f0f\u3067\u8fd1\u4f3c\u3057\u3066\u53f3\u8fba\u306e\u7a4d\u5206\u3092\u6c42\u3081\u308b\u65b9\u6cd5\u3092\u4e00\u822c\u306b<strong>\u30a2\u30c0\u30e0\u30b9\u6cd5<\/strong> (Adams method) \u3068\u3044\u3046.<\/p>\n<h4>10.1.4.1 \u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u6cd5<\/h4>\n<p>\\(y(t_{n+1})\\) \u3068 \\(y(t_n)\\) \u306f\u8fd1\u4f3c\u5024 \\(y_{n+1}\\) \u3068 \\(y_n\\) \u3092\u4f7f\u3044, \u53f3\u8fba\u306e\u7a4d\u5206\u306f \\((t_i, f_i) \\space (i = n &#8211; k +1, \\dots, n)\\) \u3092\u901a\u308b\u88dc\u9593\u591a\u9805\u5f0f \\(p(t)\\) \u3067 \\(f(t, y(t))\\) \u3092\u7f6e\u304d\u63db\u3048\u308b\u3068, \u6b21\u306e\u3088\u3046\u306b k \u6bb5\u306e\u967d\u89e3\u6cd5\u306e\u516c\u5f0f\u304c\u5f97\u3089\u308c, <strong>\u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u6cd5<\/strong> (Adams-Bashforth method) \u3068\u3088\u3070\u308c\u308b. \u305f\u3060\u3057, \\(f_i = f(t_i, y_i)\\) \u3092\u8868\u3057, \\(y_i \\space (i = n &#8211; k + 1, \\dots, n)\\) \u304c\u8a08\u7b97\u6e08\u306a\u306e\u3067 \\(f_i\\) \u306f\u3059\u3050\u306b\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b. \u3053\u306e\u516c\u5f0f\u306f, k \u6b21\u306e\u516c\u5f0f\u306b\u306a\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\ny_{n+1} &#038; = y_n + \\int_{t_n}^{t_{n+1}} p(t) dt \\\\<br \/>\n        &#038; = y_n + h \\sum_{j=0}^{k-1} \\gamma_j \\nabla^j f_n \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\u305f\u3060\u3057,<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; \\nabla^0 f_n = f_n, \\nabla^{j+1} f_n = \\nabla^j f_n &#8211; \\nabla^j f_{n-1} \\space (\u5f8c\u9000\u5dee\u5206) \\\\<br \/>\n&#038; \\gamma_j = (-1)^j \\int_0^1 C(-s, j) ds \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\u3067\u3042\u308b. \u3053\u3053\u3067, \\(C(n, k)\\) \u306f\u4e8c\u9805\u4fc2\u6570\u3092\u8868\u3059. \u306a\u304a, \\(\\gamma_j\\) \u306f\u76f4\u63a5\u8a08\u7b97\u305b\u305a\u306b\u6b21\u306e\u6f38\u5316\u5f0f\u3092\u7528\u3044\u3066\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; \\gamma_0 = 1 \\\\<br \/>\n&#038; \\gamma_j + (1\/2)\\gamma_{j-1} + (1\/3)\\gamma_{j-2} + \\dots + (1\/(j + 1))\\gamma_0 = 1 \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\\(j = 0 \\sim 7\\) \u306e\u3068\u304d\u306e\u5024\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<\/p>\n<table>\n<tbody>\n<tr>\n<td>\\(j\\)<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>\\(\\gamma_j\\)<\/td>\n<td>1<\/td>\n<td>1\/2<\/td>\n<td>5\/12<\/td>\n<td>3\/8<\/td>\n<td>251\/720<\/td>\n<td>95\/288<\/td>\n<td>19087\/60480<\/td>\n<td>5257\/17280<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u3053\u306e\u65b9\u6cd5\u3092\u56f3\u306b\u793a\u3059\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b. \u533a\u9593 \\([t_n, t_{n+1}]\\) \u306e\u7a4d\u5206\u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306b, \u533a\u9593 \\([t_{n-k+1}, t_n]\\) \u306e\u88dc\u9593\u30c7\u30fc\u30bf\u3092\u4f7f\u7528\u3057\u305f\u88dc\u9593\u95a2\u6570 \\(p(t)\\) \u3092\u4f7f\u3046. \u56f3\u3067, \u9ed2\u4e38\u306f\u88dc\u9593\u70b9, \u8272\u3064\u304d\u306e\u90e8\u5206\u306f\u7a4d\u5206\u7bc4\u56f2\u3092\u8868\u3059.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-710 size-medium\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-1-300x150.png\" alt=\"ode_fig-1\" width=\"300\" height=\"150\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-1-300x150.png 300w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-1.png 501w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>\\(k = 1 \\sim 5\\) \u306b\u3064\u3044\u3066\u6574\u7406\u3059\u308b\u3068\u6b21\u5f0f\u306e\u3088\u3046\u306b\u306a\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; k = 1 : y_{n+1} = y_n + hf_n \\\\<br \/>\n&#038; k = 2 : y_{n+1} = y_n + h\/2(3f_n &#8211; f_{n-1}) \\\\<br \/>\n&#038; k = 3 : y_{n+1} = y_n + h\/12(23f_n &#8211; 16f_{n-1} + 5f_{n-2}) \\\\<br \/>\n&#038; k = 4 : y_{n+1} = y_n + h\/24(55f_n &#8211; 59f_{n-1} + 37f_{n-2} &#8211; 9f_{n-3}) \\\\<br \/>\n&#038; k = 5 : y_{n+1} = y_n + h\/720(1901f_n &#8211; 2774f_{n-1} + 2616f_{n-2} &#8211; 1274f_{n-3} + 251f_{n-4}) \\\\<br \/>\n\\end{align}<br \/>\n\\]\nk = 1 \u306e\u5834\u5408\u306f 1 \u6bb5\u6cd5\u3068\u307f\u308b\u3053\u3068\u304c\u3067\u304d\u30aa\u30a4\u30e9\u30fc\u6cd5\u306b\u4e00\u81f4\u3059\u308b.<\/p>\n<h4>10.1.4.2 \u30a2\u30c0\u30e0\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5<\/h4>\n<p>\u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u6cd5\u3067\u306f\u88dc\u9593\u95a2\u6570\u3092\u4f7f\u3063\u3066\u5916\u633f\u306b\u3088\u308a\u7a4d\u5206\u3092\u6c42\u3081\u305f\u304c, \u7cbe\u5ea6\u3092\u4e0a\u3052\u308b\u305f\u3081\u306b \\(t_{n+1}\\) \u3092\u52a0\u3048\u3066\u533a\u9593 \\([t_{n-k+1}, t_{n+1}]\\) \u306e\u88dc\u9593\u30c7\u30fc\u30bf\u3092\u4f7f\u3046, \u3059\u306a\u308f\u3061, \\((t_i, f_i) \\space (i = n &#8211; k + 1, \\dots, n, n + 1)\\) \u3092\u901a\u308b\u88dc\u9593\u591a\u9805\u5f0f \\(p^*(t)\\) \u3092\u4f7f\u3044, \u5916\u633f\u3059\u308b\u3053\u3068\u3092\u907f\u3051\u305f\u306e\u304c<strong>\u30a2\u30c0\u30e0\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5<\/strong> (Adams-Moulton method) \u3067\u3042\u308b. \u3053\u3061\u3089\u306e\u65b9\u304c\u7cbe\u5ea6\u304c\u3088\u3055\u305d\u3046\u3067\u3042\u308b\u304c, \u305d\u306e\u4ee3\u308f\u308a\u53f3\u8fba\u306b \\(y_{n+1}\\) \u304c\u73fe\u308c, \u9670\u89e3\u6cd5\u306b\u306a\u308b. \u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b. \u3053\u306e\u516c\u5f0f\u306f, k + 1 \u6b21\u306e\u516c\u5f0f\u306b\u306a\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\ny_{n+1} &#038; = y_n + \\int_{t_n}^{t_{n+1}} p^*(x) dx \\\\<br \/>\n        &#038; = y_n + h \\sum_{j=0}^k \\gamma^*_j \\nabla^j f_{n+1}<br \/>\n\\end{align}<br \/>\n\\]\n\u305f\u3060\u3057,<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; \\nabla^0 f_n = f_n, \\space \\nabla^{j+1} f_n = \\nabla^j f_n &#8211; \\nabla^j f_{n-1} \\space (\u5f8c\u9000\u5dee\u5206) \\\\<br \/>\n&#038; \\gamma^*_j = (-1)^j \\int_0^1 C(-s+1, j) ds \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\u3067\u3042\u308b. \u3053\u3053\u3067, \\(C(n, k)\\) \u306f\u4e8c\u9805\u4fc2\u6570\u3067\u3042\u308b. \u306a\u304a, \\(\\gamma_j^{*}\\) \u306f\u76f4\u63a5\u8a08\u7b97\u305b\u305a\u306b\u6b21\u306e\u6f38\u5316\u5f0f\u3092\u7528\u3044\u3066\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; \\gamma^*_0 = 1 \\\\<br \/>\n&#038; \\gamma^*_j + (1\/2)\\gamma^*_{j-1} + (1\/3)\\gamma^*_{j-2} + \\dots + (1\/(j + 1))\\gamma^*_0 = 0 \\\\<br \/>\n\\end{align}<br \/>\n\\]\n\\(j = 0 \\sim 7\\) \u306e\u3068\u304d\u306e\u5024\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3042\u308b.<\/p>\n<table>\n<tbody>\n<tr>\n<td>\\(j\\)<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>\\(\\gamma^*_j\\)<\/td>\n<td>1<\/td>\n<td>-1\/2<\/td>\n<td>-1\/12<\/td>\n<td>-1\/24<\/td>\n<td>-19\/720<\/td>\n<td>-3\/160<\/td>\n<td>-863\/60480<\/td>\n<td>-275\/24192<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u3053\u306e\u65b9\u6cd5\u3092\u56f3\u306b\u793a\u3059\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308b. \u4e0b\u56f3\u3067, \u9ed2\u4e38\u306f\u88dc\u9593\u70b9, \u8272\u3064\u304d\u306e\u90e8\u5206\u306f\u7a4d\u5206\u7bc4\u56f2\u3092\u8868\u3059. \u7a4d\u5206\u7bc4\u56f2\u306f\u88dc\u9593\u533a\u9593\u306b\u542b\u307e\u308c\u3066\u304a\u308a, \u5916\u633f\u3055\u308c\u3066\u3044\u306a\u3044.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-711 size-medium\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-2-300x150.png\" alt=\"ode_fig-2\" width=\"300\" height=\"150\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-2-300x150.png 300w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-2.png 501w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>\\(k = 0 \\sim 5\\) \u306b\u3064\u3044\u3066\u6574\u7406\u3059\u308b\u3068\u6b21\u5f0f\u306e\u3088\u3046\u306b\u306a\u308b.<br \/>\n\\[<br \/>\n\\begin{align}<br \/>\n&#038; k = 0 : y_{n+1} = y_n + hf_{n+1} \\\\<br \/>\n&#038; k = 1 : y_{n+1} = y_n + h\/2(f_{n+1} + f_n) \\\\<br \/>\n&#038; k = 2 : y_{n+1} = y_n + h\/12(5f_{n+1} + 8f_n &#8211; f_{n-1}) \\\\<br \/>\n&#038; k = 3 : y_{n+1} = y_n + h\/24(9f_{n+1} + 19f_n &#8211; 5f_{n-1} + f_{n-2}) \\\\<br \/>\n&#038; k = 4 : y_{n+1} = y_n + h\/720(251f_{n+1} + 646f_n &#8211; 264f_{n-1} + 106f_{n-2} &#8211; 19f_{n-3}) \\\\<br \/>\n&#038; k = 5 : y_{n+1} = y_n + h\/1440 (475f_{n+1} + 1427f_n &#8211; 798f_{n-1} + 482f_{n-2} &#8211; 173f_{n-3} + 27f_{n-4}) \\\\<br \/>\n\\end{align}<br \/>\n\\]\nk = 0 \u306e\u5834\u5408\u306f 1 \u6bb5\u6cd5\u3068\u307f\u308b\u3053\u3068\u304c\u3067\u304d\u5f8c\u9000\u30aa\u30a4\u30e9\u30fc\u6cd5\u306b\u4e00\u81f4\u3059\u308b. k = 1 \u306e\u5834\u5408\u306f\u53f0\u5f62\u5247\u3068\u3088\u3070\u308c\u308b\u516c\u5f0f\u306b\u4e00\u81f4\u3059\u308b.<\/p>\n<h4>\u6570\u5024\u5b9f\u9a13 (5)<\/h4>\n<p>\u6700\u3082\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3067 k = 4 \u306e\u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u6cd5 (\u967d\u89e3\u6cd5) \u3068\u30a2\u30c0\u30e0\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5 (\u9670\u89e3\u6cd5) \u5358\u4f53(\u6ce8)\u3067\u8a08\u7b97\u3092\u884c\u3044\u7cbe\u5ea6\u3092\u78ba\u8a8d\u3059\u308b. (\u6ce8: \u5b9f\u969b\u306b\u306f\u5f8c\u3067\u8aac\u660e\u3059\u308b\u3088\u3046\u306b\u7d44\u307f\u5408\u308f\u305b\u3066\u4f7f\u7528\u3057, \u5358\u4f53\u3067\u306f\u7528\u3044\u306a\u3044)<br \/>\n\\[<br \/>\n  y&#8217; = y, \\space y(0) = 1<br \/>\n\\]\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_5_exp5.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5625\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_5_exp5.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_5_exp5-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\u6a2a\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee (\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044), \u7e26\u8ef8\u306f\u30b9\u30c6\u30c3\u30d7\u6570\u306e\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3067\u3042\u308b. 4 \u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306e\u8a08\u7b97\u7d50\u679c\u3082\u793a\u3057\u305f. \u591a\u6bb5\u6cd5\u3067\u306f\u9670\u89e3\u6cd5\u306e\u65b9\u304c\u7cbe\u5ea6\u304c\u3088\u3044\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u308b. \u591a\u6bb5\u6cd5\u306e\u307b\u3046\u304c\u540c\u3058\u6b21\u6570\u3067\u306f\u5c11\u3057\u7cbe\u5ea6\u304c\u60aa\u3044\u3088\u3046\u306b\u611f\u3058\u3089\u308c\u308b\u304c, \u30b9\u30c6\u30c3\u30d7\u6570\u3067\u8868\u793a\u3057\u3066\u3044\u308b\u7e26\u8ef8\u3092\u95a2\u6570\u8a55\u4fa1\u56de\u6570 (\u8a08\u7b97\u91cf) \u3067\u8868\u793a\u3057\u76f4\u3059\u3068\u307b\u3068\u3093\u3069\u5dee\u304c\u306a\u304f\u306a\u308b.<\/p>\n<p>\u591a\u6bb5\u6cd5\u3067\u306f, \u306f\u3058\u3081\u306b\u51fa\u767a\u5024 \\(y_1, y_2, \\dots, y_{k-1}\\) \u306e\u5024\u3092\u6c42\u3081\u3066\u304a\u304f\u5fc5\u8981\u304c\u3042\u308b\u304c, \u305d\u306e\u305f\u3081\u306b 1 \u6bb5\u89e3\u6cd5\u306a\u3069\u3092\u88dc\u52a9\u7684\u306b\u4f7f\u308f\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044. \u3053\u306e\u7cbe\u5ea6\u304c\u60aa\u3044\u3068\u4ee5\u964d\u306e\u8a08\u7b97\u5024\u306b\u5f71\u97ff\u3059\u308b\u305f\u3081, \u4f7f\u7528\u3059\u308b\u591a\u6bb5\u6cd5\u306e\u516c\u5f0f\u306e\u7cbe\u5ea6\u306b\u5341\u5206\u898b\u5408\u3046\u3060\u3051\u306e\u7cbe\u5ea6\u3092\u6301\u3063\u305f\u65b9\u6cd5\u3092\u4f7f\u7528\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b. \u3053\u3053\u3067\u306f\u6700\u521d\u306e 4 \u30b9\u30c6\u30c3\u30d7\u76ee\u307e\u3067\u306f\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5 (4 \u6b21) \u306b\u3088\u308a\u8a08\u7b97\u3092\u884c\u3063\u305f.<\/p>\n<h4>10.1.4.3 \u4e88\u6e2c\u5b50\u30fb\u4fee\u6b63\u5b50\u6cd5<\/h4>\n<p>\u591a\u6bb5\u6cd5\u306b\u304a\u3044\u3066\u306f, \u967d\u89e3\u6cd5\u306f\u8a08\u7b97\u304c\u5bb9\u6613\u3067\u3042\u308b\u304c\u6e80\u8db3\u306e\u3044\u304f\u7cbe\u5ea6\u304c\u5f97\u3089\u308c\u306a\u3044. \u4e00\u65b9, \u9670\u89e3\u6cd5\u306f\u7cbe\u5ea6\u306f\u3088\u3044\u304c\u4e00\u822c\u7684\u306b\u306f\u975e\u7dda\u5f62\u65b9\u7a0b\u5f0f\u3092\u6bce\u56de\u89e3\u304b\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u305f\u3081\u8a08\u7b97\u304c\u96e3\u3057\u3044.<\/p>\n<p>\u305d\u3053\u3067\u5e83\u304f\u4f7f\u308f\u308c\u3066\u3044\u308b\u65b9\u6cd5\u304c<strong>\u4e88\u6e2c\u5b50\u30fb\u4fee\u6b63\u5b50\u6cd5<\/strong> (predictor-corrector method) \u3067\u3042\u308b. \u307e\u305a\u4e88\u6e2c\u5b50 (predictor) \u3068\u3088\u3070\u308c\u308b\u967d\u89e3\u6cd5\u306b\u3088\u308a \\(y_{n+1}\\) \u306e\u7b2c 1 \u8fd1\u4f3c\u3092\u6c42\u3081\u308b. \u6b21\u306b\u305d\u306e\u5024\u3092\u4f7f\u3063\u3066\u4fee\u6b63\u5b50 (corrector) \u3068\u3088\u3070\u308c\u308b\u9670\u89e3\u6cd5\u3092 1 \u56de\uff5e\u6570\u56de\u9069\u7528\u3057\u3053\u308c\u3092\u6539\u826f\u3059\u308b\u65b9\u6cd5\u3067\u3042\u308b.<\/p>\n<p>\u4e88\u6e2c\u5b50\u3068\u4fee\u6b63\u5b50\u306e\u7d44\u307f\u5408\u308f\u305b\u306b\u306f\u30d0\u30ea\u30a8\u30fc\u30b7\u30e7\u30f3\u304c\u3042\u308b\u304c, \u4e88\u6e2c\u5b50\u306b\u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u6cd5, \u4fee\u6b63\u5b50\u306b\u30a2\u30c0\u30e0\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5\u3092\u4f7f\u3063\u305f\u4e88\u6e2c\u5b50\u30fb\u4fee\u6b63\u5b50\u6cd5\u306f\u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5 (\u3042\u308b\u3044\u306f\u5358\u306b\u30a2\u30c0\u30e0\u30b9\u6cd5) \u3068\u547c\u3070\u308c, \u5e83\u304f\u4f7f\u308f\u308c\u3066\u3044\u308b.<\/p>\n<p>\u4e88\u6e2c\u5b50\u30fb\u4fee\u6b63\u5b50\u6cd5\u306e\u8a08\u7b97\u624b\u9806\u306f, (1) \u4e88\u6e2c\u5b50\u3092\u9069\u7528\u3057\u3066 \\(y_{n+1}\\) \u3092\u6c42\u3081\u308b (P), (2) \u305d\u308c\u3092\u4f7f\u3063\u3066\u95a2\u6570\u5024 \\(f_{n+1}\\) \u3092\u8a08\u7b97\u3059\u308b (E), (3) \u4fee\u6b63\u5b50\u3092\u9069\u7528\u3057\u3066 \\(y_{n+1}\\) \u3092\u66f4\u65b0\u3059\u308b (C), (4) \u66f4\u65b0\u3057\u305f \\(y_{n+1}\\) \u3092\u4f7f\u3063\u3066 \\(f_{n+1}\\) \u3092\u66f4\u65b0\u3059\u308b (E) \u3068\u306a\u308b. \u3053\u308c\u3092 PECE \u3068\u8868\u3059\u3053\u3068\u306b\u3059\u308b\u3068, \u6700\u5f8c\u306e E \u3092\u7701\u7565\u3057\u3066 PEC \u3068\u3059\u308b\u65b9\u6cd5, \u4fee\u6b63\u5b50\u306e\u9069\u7528\u3092 2 \u56de\u7e70\u308a\u8fd4\u3057\u3066 PECECE \u3068\u3059\u308b\u65b9\u6cd5\u306a\u3069, \u30d0\u30ea\u30a8\u30fc\u30b7\u30e7\u30f3\u304c\u3042\u308b.<\/p>\n<h4>\u6570\u5024\u5b9f\u9a13 (6)<\/h4>\n<p>\u6700\u3082\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3067 k = 4 \u306e\u4e88\u6e2c\u5b50\u30fb\u4fee\u6b63\u5b50\u6cd5 (\u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5) \u306e\u7cbe\u5ea6\u3092\u78ba\u8a8d\u3059\u308b.<br \/>\n\\[<br \/>\n  y&#8217; = y, \\space y(0) = 1<br \/>\n\\]\n\u3053\u3053\u3067\u306f PEC \u3068 PECE \u306e 2 \u7a2e\u985e\u306b\u3064\u3044\u3066\u8a66\u3057\u3066\u307f\u305f. \u51fa\u767a\u5024\u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u306f 4 \u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3092\u4f7f\u7528\u3057\u305f.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_6_exp6.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5626\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_6_exp6.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_6_exp6-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\u6a2a\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee (\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044), \u7e26\u8ef8\u306f\u4eca\u56de\u306f\u95a2\u6570\u8a55\u4fa1\u56de\u6570 (\u30b9\u30c6\u30c3\u30d7\u6570\u3067\u306f\u306a\u3044) \u306e\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3068\u3057\u305f. \u30b9\u30c6\u30c3\u30d7\u3042\u305f\u308a\u306e\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u304c PEC \u3068 PECE \u3067\u7570\u306a\u308b\u305f\u3081, \u7dcf\u5408\u7684\u306b\u8a55\u4fa1\u3059\u308b\u3088\u3046\u306b\u7e26\u8ef8\u306e\u30d7\u30ed\u30c3\u30c8\u3092\u5909\u3048\u305f.<\/p>\n<p>\u305d\u306e PEC \u3068 PECE \u3067\u3042\u308b\u304c, \u3053\u306e\u4f8b\u306f\u7c21\u5358\u306a\u95a2\u6570\u3067\u3042\u308b\u305f\u3081\u304b\u307b\u3068\u3093\u3069\u8a08\u7b97\u7d50\u679c\u306b\u5dee\u304c\u51fa\u306a\u304b\u3063\u305f. \u305d\u306e\u305f\u3081, 2 \u56de\u76ee\u306e E \u306e\u8a08\u7b97\u306e\u3076\u3093\u3060\u3051 PECE \u306e\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u304c\u5897\u3048\u305f\u3060\u3051\u3068\u3044\u3046\u7d50\u679c\u306b\u306a\u3063\u305f. \u306a\u304a, \u5225\u9014\u96e3\u3057\u3044\u95a2\u6570\u3067\u8a66\u3057\u3066\u307f\u308b\u3068 PECE \u306e\u307b\u3046\u304c\u3088\u304f\u306a\u308b\u3053\u3068\u304c\u3042\u3063\u305f.<\/p>\n<h3>10.1.5 \u4f7f\u308f\u308c\u306a\u304f\u306a\u3063\u305f\u89e3\u6cd5<\/h3>\n<p>\u3088\u304f\u77e5\u3089\u308c\u3066\u3044\u308b\u304c\u73fe\u5728\u3067\u306f\u4f7f\u308f\u308c\u306a\u304f\u306a\u3063\u305f\u89e3\u6cd5\u306b\u3064\u3044\u3066\u3053\u3053\u3067\u89e6\u308c\u3066\u304a\u304f.<\/p>\n<h4>10.1.5.1 \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30b8\u30eb\u6cd5<\/h4>\n<p>\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30b8\u30eb\u6cd5 (Runge-Kutta-Gill method) \u306f\u4fc2\u6570\u304c\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u308b 4 \u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u3042\u308b.<br \/>\n\\[<br \/>\n\\begin{array}{c|cccc}<br \/>\n0 \\\\<br \/>\n1\/2 &#038; 1\/2 \\\\<br \/>\n1\/2 &#038; (\u221a2-1)\/2 &#038; (2-\u221a2)\/2 \\\\<br \/>\n1 &#038; 0 &#038; (-\u221a2)\/2 &#038; 1+(\u221a2)\/2 \\\\<br \/>\n\\hline<br \/>\n&#038; 1\/6 &#038; (2-\u221a2)\/6 &#038; (2+\u221a2)\/6 &#038; 1\/6 \\\\<br \/>\n\\end{array}<br \/>\n\\]\n\u901a\u5e38\u306e 4 \u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3068\u6027\u80fd (\u53ce\u675f\u901f\u5ea6) \u306f\u307b\u307c\u540c\u3058\u3067\u3042\u308b\u304c Gill \u306e\u63d0\u6848\u306b\u306f\u6b21\u306e\u7279\u9577\u304c\u3042\u308b.<\/p>\n<p>(i) \u901a\u5e38\u306e 4 \u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u306f\u8a08\u7b97\u306e\u305f\u3081\u306b 4 \u500b\u306e\u5909\u6570 (\u9023\u7acb\u65b9\u7a0b\u5f0f\u306e\u5834\u5408\u306b\u306f\u914d\u5217) \u304c\u5fc5\u8981\u306a\u306e\u306b\u5bfe\u3057, \u3053\u306e\u65b9\u6cd5\u3067\u306f\u4fc2\u6570\u3092\u5de5\u592b\u3057\u3066 3 \u500b\u306e\u5909\u6570 (\u914d\u5217) \u3060\u3051\u3067\u8a08\u7b97\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u3063\u3066\u3044\u308b. [\u30e1\u30e2\u30ea\u306e\u7bc0\u7d04]\n<p>(ii) \u4e38\u3081\u8aa4\u5dee\u3092\u6e1b\u3089\u3059\u305f\u3081\u306e\u624b\u9806 (\u4e0b\u8a18\u53c2\u7167) \u304c\u542b\u307e\u308c\u3066\u3044\u308b (\u3053\u308c\u306f\u4ed6\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u306b\u3082\u9069\u7528\u53ef\u80fd\u3067\u3042\u308b). [\u7cbe\u5ea6\u306e\u5411\u4e0a]\n<p>\u30e1\u30e2\u30ea\u304c\u5c11\u306a\u304f, \u304a\u305d\u3089\u304f\u5358\u7cbe\u5ea6\u8a08\u7b97\u304c\u4e3b\u6d41\u3067\u3042\u3063\u305f, \u521d\u671f\u306e\u8a08\u7b97\u6a5f\u306b\u304a\u3051\u308b\u30d7\u30ed\u30b0\u30e9\u30e0\u306b\u9069\u3057\u305f\u65b9\u6cd5\u3060\u3063\u305f\u3068\u601d\u308f\u308c\u308b. \u304b\u3064\u3066\u5e83\u304f\u4f7f\u308f\u308c\u3066\u3044\u305f\u304c, \u30e1\u30e2\u30ea\u304c\u8c4a\u5bcc\u3067\u500d\u7cbe\u5ea6\u8a08\u7b97\u304c\u666e\u901a\u306e\u73fe\u4ee3\u3067\u306f\u5fc5\u8981\u6027\u304c\u8584\u308c\u3066\u304d\u305f. \u3055\u3089\u306b, \u3088\u308a\u9ad8\u6b21\u306e\u516c\u5f0f\u306e\u5b9f\u7528\u5316\u3084\u30b9\u30c6\u30c3\u30d7\u5e45\u306e\u81ea\u52d5\u8abf\u7bc0\u6a5f\u80fd\u306e\u666e\u53ca\u3082\u3042\u308a\u51fa\u756a\u304c\u5c11\u306a\u304f\u306a\u3063\u305f\u3068\u601d\u308f\u308c\u308b. <\/p>\n<p>\u3053\u3053\u3067, (ii)\u306b\u3064\u3044\u3066\u5c11\u3057\u8aac\u660e\u3059\u308b.<\/p>\n<p>\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u57fa\u672c\u7684\u306a\u6570\u5024\u89e3\u6cd5\u3067\u306f\u5404\u30b9\u30c6\u30c3\u30d7\u306b\u304a\u3044\u3066\u95a2\u6570\u5024\u306e\u66f4\u65b0\u5dee\u5206 (\u3053\u3053\u3067\u306f k \u3068\u8868\u3059) \u3092\u8a08\u7b97\u3057, \u6b21\u306e\u3088\u3046\u306b\u3057\u3066\u95a2\u6570\u5024\u3092\u66f4\u65b0\u3059\u308b\u3082\u306e\u3068\u3059\u308b.<br \/>\n\\[<br \/>\n  y_{n+1} = y_n + k<br \/>\n\\]\n\u3053\u306e\u8a08\u7b97\u3092\u8a08\u7b97\u6a5f\u306e\u6d6e\u52d5\u5c0f\u6570\u30c7\u30fc\u30bf (\u30ef\u30fc\u30c9\u9577\u306f\u6709\u9650) \u3067\u884c\u3046\u3068\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-704 size-full\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-5.png\" alt=\"ode_fig-5\" width=\"453\" height=\"162\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-5.png 453w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2015\/01\/ode_fig-5-300x107.png 300w\" sizes=\"auto, (max-width: 453px) 100vw, 453px\" \/><\/p>\n<p>\u3053\u3053\u3067, k \u3092\u30ef\u30fc\u30c9\u9577\u3044\u3063\u3071\u3044\u6b63\u3057\u304f\u6c42\u3081\u305f\u3068\u3057\u3066\u3082, \u3053\u306e\u66f4\u65b0\u306b\u4f7f\u7528\u3055\u308c\u308b\u306e\u306f\u8272\u306e\u3064\u3044\u305f\u4e00\u90e8\u3060\u3051\u3067\u6b8b\u308a\u306e\u90e8\u5206\u306f\u6368\u3066\u3089\u308c\u3066\u3057\u307e\u3046 (\u4e38\u3081\u8aa4\u5dee\u3068\u3044\u3046). \u305d\u3053\u3067, \u3053\u306e\u6368\u3066\u3089\u308c\u308b\u90e8\u5206\u3092\u5225\u306e\u5834\u6240\u306b\u7d2f\u7a4d\u3057\u3066\u304a\u304d, \u8272\u306e\u3064\u3044\u305f\u90e8\u5206\u306b\u982d\u304c\u51fa\u308b\u307b\u3069\u306b\u8caf\u307e\u308b\u305f\u3073\u306b k \u306b\u52a0\u3048\u3066\u95a2\u6570\u5024\u306e\u66f4\u65b0\u306b\u4f7f\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u4e38\u3081\u8aa4\u5dee\u3092\u6e1b\u3089\u3059\u65b9\u6cd5\u304c\u8003\u6848\u3055\u308c\u305f. \u3053\u306e\u65b9\u6cd5\u304c\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30b8\u30eb\u6cd5\u306b\u4f7f\u308f\u308c\u3066\u3044\u308b.<\/p>\n<h4>\u6570\u5024\u5b9f\u9a13 (7)<\/h4>\n<p>\u6700\u3082\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3067\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30b8\u30eb\u6cd5\u306e\u7cbe\u5ea6\u3092\u78ba\u8a8d\u3059\u308b.<br \/>\n\\[<br \/>\n  y&#8217; = y, \\space y(0) = 1<br \/>\n\\]\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_7_exp7.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5627\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_7_exp7.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_7_exp7-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\u6a2a\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee (\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044), \u7e26\u8ef8\u306f\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u3092\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3068\u3057\u305f. \u6bd4\u8f03\u306e\u305f\u3081\u666e\u901a\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3082\u30d7\u30ed\u30c3\u30c8\u3057\u3066\u3044\u308b. \u8a08\u7b97\u306f 64 \u30d3\u30c3\u30c8\u500d\u7cbe\u5ea6\u6f14\u7b97\u3067\u884c\u308f\u308c\u3066\u3044\u308b.<\/p>\n<p>\u76f8\u5bfe\u8aa4\u5dee\u304c \\(10^{-14}\\) \u4ed8\u8fd1\u307e\u3067\u306f\u5dee\u304c\u306a\u3044\u304c, \u666e\u901a\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3067\u306f \\(10^{-15}\\) \u3042\u305f\u308a\u3067\u982d\u6253\u3061\u306b\u306a\u308a, \u305d\u306e\u5148\u306f\u523b\u307f\u5e45\u3092\u5c0f\u3055\u304f\u3059\u308b\u3068\u3080\u3057\u308d\u8aa4\u5dee\u304c\u5927\u304d\u304f\u306a\u308b\u3053\u3068\u3082\u3042\u308b. \u3053\u308c\u306f, 64 \u30d3\u30c3\u30c8\u500d\u7cbe\u5ea6\u3067\u306f 1 \u56de\u306e\u6f14\u7b97\u306e\u7cbe\u5ea6\u306e\u9650\u754c\u306f \\(2^{-16}\\) \u306a\u306e\u3067, \u3053\u306e\u5834\u5408\u306f\u7dcf\u5408\u7684\u306b\u306f \\(10^{-15}\\) \u304f\u3089\u3044\u304c\u6700\u9ad8\u7cbe\u5ea6\u3060\u3068\u8003\u3048\u3089\u308c, \u305d\u308c\u4ee5\u4e0a\u8a08\u7b97\u3092\u7e70\u308a\u8fd4\u3059\u3068\u4e38\u3081\u8aa4\u5dee\u304c\u7a4d\u307f\u91cd\u306a\u3063\u3066\u3044\u304d\u304b\u3048\u3063\u3066\u7cbe\u5ea6\u304c\u843d\u3061\u3066\u3057\u307e\u3046\u305f\u3081\u3068\u8003\u3048\u3089\u308c\u308b.<\/p>\n<p>\u4e00\u65b9\u3067, \u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u30fb\u30b8\u30eb\u6cd5\u3067\u306f\u6700\u5f8c\u306e\u307b\u3046\u306e\u8aa4\u5dee\u306f \\(1.6 \\times 10^{-16}\\) \u307e\u305f\u306f 0 \u3068\u306a\u3063\u305f (\u56f3\u3067\u7dda\u304c\u3068\u3093\u3067\u3044\u308b\u3068\u3053\u308d\u306f\u5b8c\u5168\u306b 0 \u306e\u305f\u3081\u30d7\u30ed\u30c3\u30c8\u3055\u308c\u3066\u3044\u306a\u3044). \u3064\u307e\u308a, \u6700\u7d42 1 \u30d3\u30c3\u30c8\u304c\u9055\u3046\u304b\u3069\u3046\u304b\u306e\u3081\u3044\u3063\u3071\u3044\u306e\u7cbe\u5ea6\u3067\u8a08\u7b97\u3055\u308c\u305f\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b.<\/p>\n<h4>10.1.5.2 \u30df\u30eb\u30f3\u6cd5<\/h4>\n<p>\u30a2\u30c0\u30e0\u30b9\u6cd5\u3068\u4f3c\u3066\u3044\u308b\u304c, \u7a4d\u5206\u7bc4\u56f2\u304c\u7570\u306a\u308b\u6b21\u306e\u7a4d\u5206\u5f62\u3092\u8003\u3048\u308b.<br \/>\n\\[<br \/>\n  y(t_{n+1}) = y(t_{n-1}) + \\int_{t_{n-1}}^{t_{n+1}} f(t, y(t)) dt<br \/>\n\\]\n\u30a2\u30c0\u30e0\u30b9\u6cd5\u3068\u540c\u69d8\u306b \\((t_i, f_i) \\space (i = n &#8211; k + 1, \\dots, n)\\) \u3092\u901a\u308b\u88dc\u9593\u591a\u9805\u5f0f \\(p(t)\\) \u3067\u7a4d\u5206\u3092\u8fd1\u4f3c\u3059\u308b\u3068, \u6b21\u306e\u967d\u89e3\u6cd5\u306e\u516c\u5f0f\u304c\u5f97\u3089\u308c\u308b.<br \/>\n\\[<br \/>\n  y_{n+1} = y_{n-1} + h \\sum_{j=0}^{k-1} \\kappa_j \\nabla^j f_n<br \/>\n\\]\n\u305f\u3060\u3057,<br \/>\n\\[<br \/>\n  \\kappa_j = (-1)^j \\int_{-1}^1 C(-s, j) ds<br \/>\n\\]\n\u307e\u305f, \\((t_i, f_i) \\space (i = n &#8211; k +1, \\dots, n+1)\\) \u3092\u901a\u308b\u88dc\u9593\u591a\u9805\u5f0f \\(p^*(x)\\) \u3067\u7a4d\u5206\u3092\u8fd1\u4f3c\u3059\u308b\u3068, \u6b21\u306e\u9670\u89e3\u6cd5\u306e\u516c\u5f0f\u304c\u5f97\u3089\u308c\u308b.<br \/>\n\\[<br \/>\n  y_{n+1} = y_{n-1} + h \\sum_{j=0}^k \\kappa^*_j \\nabla^j f_{n+1}<br \/>\n\\]\n\u305f\u3060\u3057,<br \/>\n\\[<br \/>\n  \\kappa^*_j = (-1)^j \\int_{-1}^1 C(-s+1, j) ds<br \/>\n\\]\n\u4e88\u6e2c\u5b50\u30fb\u4fee\u6b63\u5b50\u6cd5\u306e\u4e00\u7a2e\u306b\u30df\u30eb\u30f3\u6cd5 (Milne method) \u3068\u3088\u3070\u308c\u308b\u65b9\u6cd5\u304c\u3042\u308a, \u4e0a\u306e\u9670\u89e3\u6cd5\u306e\u5f0f\u306b\u304a\u3044\u3066 k = 2 \u3068\u3057\u305f\u6b21\u5f0f\u304c\u305d\u306e\u4fee\u6b63\u5b50\u3068\u3057\u3066\u4f7f\u308f\u308c\u3066\u3044\u308b.<br \/>\n\\[<br \/>\n  y_{n+1} = y_{n-1} + h\/3 (f_{n+1} + 4f_n + f_{n-1})<br \/>\n\\]\n\u30df\u30eb\u30f3\u6cd5\u306e\u4e88\u6e2c\u5b50\u306b\u306f, \u7a4d\u5206\u7bc4\u56f2\u3092 \\([t_{n-3}, t_{n+1}]\\) \u306b\u5e83\u3052\u305f\u7a4d\u5206\u5f62\u3088\u308a\u5c0e\u304b\u308c\u308b\u6b21\u5f0f\u304c\u4f7f\u308f\u308c\u3066\u3044\u308b.<br \/>\n\\[<br \/>\n  y_{n+1} = y_{n-3} + (4h\/3) (2f_n &#8211; f_{n-1} + 2f_{n-2})<br \/>\n\\]\n\u30df\u30eb\u30f3\u6cd5\u3067\u306f\u4e88\u6e2c\u5b50\u3067\u6c42\u3081\u305f\u8fd1\u4f3c\u306b\u5bfe\u3057\u3066\u88dc\u6b63\u5b50 (modifier) \u3092\u9069\u7528\u3057\u3066\u52a0\u901f\u3059\u308b\u3053\u3068\u304c\u63d0\u6848\u3055\u308c\u3066\u3044\u308b. \u3059\u306a\u308f\u3061, \u4e88\u6e2c\u5b50\u3067\u6c42\u3081\u305f\u8fd1\u4f3c\u3092 \\(y_{n+1}^p\\) \u3067\u8868\u3059\u3068, \u6b21\u306e\u3088\u3046\u306a\u88dc\u6b63\u3092\u884c\u3046.<br \/>\n\\[<br \/>\n  y_{n+1} = y_{n+1}^p + (28\/29) (y_n &#8211; y_n^p)<br \/>\n\\]\n\u3053\u308c\u3092 M \u3068\u8868\u3059\u3068\u30df\u30eb\u30f3\u6cd5\u306e\u4e88\u6e2c\u5b50\u30fb\u4fee\u6b63\u5b50\u6cd5\u306f PMECE&#8230; \u3068\u306a\u308b.<\/p>\n<p>\u30df\u30eb\u30f3\u6cd5\u306f 4 \u6b21\u306e\u516c\u5f0f\u3067, \u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u304c\u666e\u53ca\u3059\u308b\u4ee5\u524d\u306e\u624b\u8a08\u7b97\u306e\u6642\u4ee3\u306b\u3088\u304f\u4f7f\u308f\u308c\u3066\u3044\u305f. \u3057\u304b\u3057, \u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u3067\u4f7f\u308f\u308c\u308b\u3088\u3046\u306b\u306a\u308b\u3068\u5b89\u5b9a\u6027\u306e\u9762\u3067\u554f\u984c\u304c\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a, \u73fe\u5728\u3067\u306f\u4f7f\u308f\u308c\u306a\u304f\u306a\u3063\u305f.<\/p>\n<p>\u306a\u304a, \u5b89\u5b9a\u6027\u306e\u554f\u984c\u3092\u89e3\u6c7a\u3059\u308b\u305f\u3081\u306b, \u30df\u30eb\u30f3\u6cd5\u306e\u4e88\u6e2c\u5b50\u3068\u7d44\u307f\u5408\u308f\u305b\u3066\u4f7f\u3046\u5b89\u5b9a\u306a\u4fee\u6b63\u5b50\u3068\u3057\u3066\u6b21\u306e\u30cf\u30df\u30f3\u30b0 (Hamming) \u306e\u4fee\u6b63\u5b50\u304c\u63d0\u6848\u3055\u308c\u3066\u3044\u308b.<br \/>\n\\[<br \/>\n  y_{n+1} = 9\/8 y_n &#8211; 1\/8 y_{n-2} + h\/8 (3f_{n+1} + 6f_n &#8211; 3f_{n-1})<br \/>\n\\]\n\u306a\u304a, \u88dc\u6b63\u5b50\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308b.<br \/>\n\\[<br \/>\n  y_{n+1} = y_{n+1}^p + (112\/121) (y_n &#8211; y_n^p)<br \/>\n\\]\n<h4>\u6570\u5024\u5b9f\u9a13 (8)<\/h4>\n<p>\u6700\u3082\u7c21\u5358\u306a\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u521d\u671f\u5024\u554f\u984c\u306e\u4f8b\u3067\u30df\u30eb\u30f3\u6cd5\u306e\u7cbe\u5ea6\u3092\u78ba\u8a8d\u3059\u308b.<br \/>\n\\[<br \/>\n  y&#8217; = y, \\space y(0) = 1<br \/>\n\\]\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_8_exp8.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5628\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_8_exp8.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_8_exp8-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\u6a2a\u8ef8\u306f\u76f8\u5bfe\u8aa4\u5dee (\u53f3\u306b\u884c\u304f\u307b\u3069\u7cbe\u5ea6\u304c\u3088\u3044), \u7e26\u8ef8\u306f\u95a2\u6570\u8a55\u4fa1\u56de\u6570\u3092\u5bfe\u6570\u30d7\u30ed\u30c3\u30c8\u3068\u3057\u305f. \u6bd4\u8f03\u306e\u305f\u3081\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3068\u30a2\u30c0\u30e0\u30b9\u30fb\u30d0\u30b7\u30e5\u30d5\u30a9\u30fc\u30b9\u30fb\u30e2\u30eb\u30c8\u30f3\u6cd5\u3082\u30d7\u30ed\u30c3\u30c8\u3057\u3066\u3044\u308b. \u306a\u304a, \u51fa\u767a\u5024\u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u306f 4 \u6b21\u306e\u30eb\u30f3\u30b2\u30fb\u30af\u30c3\u30bf\u6cd5\u3092\u4f7f\u7528\u3057\u305f.<\/p>\n<p>\u30df\u30eb\u30f3\u6cd5\u306f\u516c\u5f0f\u304c\u6bd4\u8f03\u7684\u7c21\u5358\u306a\u5272\u306b, \u56f3\u306e\u3088\u3046\u306b\u6027\u80fd\u306e\u3088\u3044\u89e3\u6cd5\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n<h4>\u6570\u5024\u5b9f\u9a13 (9)<\/h4>\n<p>\u6b21\u306e\u521d\u671f\u5024\u554f\u984c\u3092\u30df\u30eb\u30f3\u6cd5\u306b\u3088\u308a\u89e3\u304f.<br \/>\n\\[<br \/>\n  y&#8217; = -y + 1, \\space y(0) = 2<br \/>\n\\]\n\u3053\u306e\u65b9\u7a0b\u5f0f\u306e\u89e3\u306f \\(y(t) = e^{-t} + 1\\) \u3067\u3042\u308b. \u30b9\u30c6\u30c3\u30d7\u5e45\u3092 0.01 \u304a\u3088\u3073 0.001 \u3068\u3057\u3066\u8a08\u7b97\u3057\u305f\u7d50\u679c, \u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f (\u898b\u3084\u3059\u3055\u306e\u305f\u3081\u30c7\u30fc\u30bf\u70b9\u3092\u9593\u5f15\u304d\u3057\u3066\u8868\u793a\u3057\u305f. \u5b9f\u969b\u306f\u3082\u3063\u3068\u7d30\u304b\u304f\u632f\u52d5\u3059\u308b).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_9_exp9.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5629\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_9_exp9.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_9_exp9-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\u30b9\u30c6\u30c3\u30d7\u5e45\u304c\u5341\u5206\u5c0f\u3055\u3044\u306b\u3082\u304b\u304b\u308f\u3089\u305a\u8a08\u7b97\u304c\u9032\u3093\u3067\u30b9\u30c6\u30c3\u30d7\u6570\u304c\u591a\u304f\u306a\u308b\u3068\u632f\u52d5\u3092\u59cb\u3081\u767a\u6563\u3057\u305f. \u3053\u308c\u306f, \u89e3\u6cd5\u306e\u5dee\u5206\u65b9\u7a0b\u5f0f\u306e\u4e00\u822c\u89e3\u304c\u771f\u306e\u89e3\u306e\u8fd1\u4f3c (\u6709\u52b9\u6210\u5206) \u306e\u4ed6\u306b\u5fc5\u8981\u306e\u306a\u3044\u6210\u5206\u3092\u542b\u307f, \u6761\u4ef6\u306b\u3088\u3063\u3066\u305d\u308c\u304c\u6709\u52b9\u6210\u5206\u3088\u308a\u3082\u5927\u304d\u304f\u306a\u3063\u305f\u3068\u304d\u306b\u8d77\u304d\u308b\u73fe\u8c61\u3067\u3042\u308b.<\/p>\n<p>\u3053\u306e\u4e0d\u5b89\u5b9a\u6027\u306e\u5bfe\u7b56\u3068\u3057\u3066\u30cf\u30df\u30f3\u30b0\u306e\u4fee\u6b63\u5b50\u304c\u3042\u308b\u306e\u3067, \u305d\u308c\u3092\u4f7f\u3063\u3066\u540c\u3058\u8a08\u7b97\u3092\u884c\u3046\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u3063\u305f.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_10_exp9.png\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5630\" srcset=\"https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_10_exp9.png 640w, https:\/\/www.ktech.biz\/jp\/wp-content\/uploads\/sites\/2\/2026\/06\/10-1_10_exp9-300x225.png 300w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<p>\u30cf\u30df\u30f3\u30b0\u306e\u4fee\u6b63\u5b50\u3092\u4f7f\u3046\u3053\u3068\u306b\u3088\u308a\u4e0d\u5b89\u5b9a\u6027\u304c\u53ce\u307e\u308b\u3053\u3068\u304c\u78ba\u8a8d\u3055\u308c\u305f.<\/p>\n<p>\u6570\u5024\u5b9f\u9a13 (1) \u3067\u306f\u4e2d\u70b9\u5247\u306e\u6027\u80fd\u304c\u3088\u304b\u3063\u305f\u304c, \u4e2d\u70b9\u5247\u306b\u3082\u3053\u306e\u7a2e\u306e\u4e0d\u5b89\u5b9a\u6027\u304c\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u3066\u3044\u308b. \u306a\u304a, 1 \u6bb5\u6cd5\u3067\u306f\u3053\u306e\u7a2e\u306e\u4e0d\u5b89\u5b9a\u6027\u306f\u751f\u3058\u306a\u3044.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>10.1.1 \u6570\u5024\u89e3\u6cd5\u306e\u5206\u985e \u5909\u6570 \\(t\\) \u3092\u96e2\u6563\u5316\u3057\u3066, \\(t_0, t_1, \\dots, t_n, \\dots\\) \u3068\u8868\u3059. \u3053\u308c\u3089\u306e\u70b9\u306e\u9593\u9694\u3092\u30b9\u30c6\u30c3\u30d7\u5e45\u3042\u308b\u3044\u306f\u30b9\u30c6\u30c3\u30d7\u5e45 (step size) \u3068\u3044\u3044,  [&hellip;]<\/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,"footnotes":""},"categories":[13],"tags":[],"class_list":["post-5521","post","type-post","status-publish","format-standard","hentry","category-num"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.1.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"10.1.1 \u6570\u5024\u89e3\u6cd5\u306e\u5206\u985e \u5909\u6570 \\(t\\) \u3092\u96e2\u6563\u5316\u3057\u3066, \\(t_0, t_1, \\dots, t_n,\" \/>\n\t<meta name=\"robots\" content=\"max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n\t<meta name=\"author\" content=\"Administrator\"\/>\n\t<meta name=\"google-site-verification\" content=\"5PEQMQVhF0wv5i-xMYW9a3HzuBwixzVduoR2arr5UDM\" \/>\n\t<meta name=\"msvalidate.01\" content=\"EEB9704E04960338720AEA89E77E749F\" \/>\n\t<link rel=\"canonical\" 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