<\/span><\/h3>\r\n\r\n\r\n\r\n\u3042\u304f\u307e\u3067\u3001\u53c2\u8003\u7528\u3067\u3059\u3002\u59cb\u3081\u306b\u81ea\u5206\u3067\u89e3\u304d\u65b9\u3092\u8003\u3048\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\r\n\r\n\r\n\r\n
\uff11\u6b21\u5143\u6ce2\u52d5\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u70ba\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002
\u5b89\u5b9a\u6027\u6761\u4ef6\u3092\u6e80\u305f\u3059\u305f\u3081![\"\lambda \le 1\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-04d952c17f440a45deec89032ffab1a3_l3.png\" "\\"Rendered")
\u3068\u306a\u308b\u3088\u3046\u306b![\"h, \tau\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-ca07b5b7b125ed877692a4358b37f17c_l3.png\" "\\"Rendered")
\u3092\u4e0e\u3048\u307e\u3059\u3002<\/p>\r\n\r\n\r\n\r\n
\u62e1\u6563\u65b9\u7a0b\u5f0f\u306e\u3068\u304d\u306b\u306f\u300cuj<\/sub> \u300d\u3001\u300c new_uj<\/sub> \u300d\u306e\u307f\u5b9a\u7fa9\u3057\u3066\u3044\u307e\u3057\u305f\u304c\u3001\u4eca\u56de\u306f\u300cold_uj<\/sub> \u300d\u3082\u5fc5\u8981\u306b\u306a\u308a\u307e\u3059\u3002\u307e\u305f\u3001\u305d\u308c\u305e\u308c\u306e\u30ea\u30b9\u30c8\u306e\u500b\u6570\u306f\u5883\u754c\u6761\u4ef6\u306e\u8a08\u7b97\u7528\u306b2\u3064\u52a0\u3048\u305fN+2\u500b\u3068\u306a\u308a\u307e\u3059\u3002\uff08new_uj <\/sub>\u306b\u3064\u3044\u3066\u306f\u3001\u5b9f\u969b\u306b\u306f\u5883\u754c\u51e6\u7406\u7528\u306e\u8981\u7d20\u306f\u7528\u3044\u306a\u3044\u3067\u3059\u304c…\uff09<\/p>\r\n\r\n\r\n\r\n\r\n\r\n(\uff11)\u521d\u671f\u30d1\u30e9\u30e1\u30fc\u30bf\u8a2d\u5b9a\r\nT\uff08\u6642\u9593\u4e0a\u9650\uff09, M\uff08\u6642\u9593\u523b\u307f\u6570\uff09, L\uff08\u7a7a\u9593\u5e45\uff09, N\uff08\u7a7a\u9593\u523b\u307f\u6570\uff09, c \u3092\u8a2d\u5b9a\u3059\u308b\r\n\u03bb = \u03c4*c\/h (\u03bb \u2266 1 \u3068\u306a\u308b\u3088\u3046\u306b. h, \u03c4 \u6c7a\u3081\u3001N = L\/h, M = T\/\u03c4\uff08\u3044\u305a\u308c\u3082\u81ea\u7136\u6570\uff09\u3068\u3059\u308b\u3002 \u30d7\u30ed\u30b0\u30e9\u30e0\u4e2d\u3067\u306f, \u03bb\u306f\u300clamb\u300d, \u03c4\u306f\u300ctau\u300d\u306a\u3069\u3068\u3059\u308b\u3068\u3088\u3044.)\r\n \r\n\uff08\uff12\uff09 \u521d\u671f\u5024\u8a2d\u5b9a\r\nj := 1,.....,N \u306e\u9806\u306b\r\n x_j = (j - 0.5)*h\r\n old_u[j] = f(x_j) # t=0\u306e\u3068\u304d\u306eu\r\n\u3092\u7e70\u308a\u8fd4\u3059 ( f(x_j) \u306f\u8ab2\u984c\uff11,\uff12\u3092\u53c2\u7167 )\r\nold_u[0] = -old_u[1], old_u[N+1] = -old_u[N] \uff08\u5883\u754c\u6761\u4ef6\uff09\r\nj := 1,.....,N \u306e\u9806\u306b\r\n u[j] = old_u[j] + tau*g(x_j) + (\u03bb*\u03bb\/2) * (old_u[j-1] - 2 * old_u[j] + old_u[j+1]) # t=h\u306e\u3068\u304d\u306eu\r\n\u3092\u7e70\u308a\u8fd4\u3059 ( g(x) \u306f\u8ab2\u984c\uff11,\uff12\u3092\u53c2\u7167 )\r\nu[0] = -u[1], u[N+1] = -u[N] \uff08\u5883\u754c\u6761\u4ef6\uff09\r\n\r\n\uff08\uff13\uff09\u8a08\u7b97, \u52d5\u753b\u63cf\u753b\r\nk := 0,....,M \u306e\u9806\u306b\r\n t = k*\u03c4\r\n \uff08i\uff09\u63cf\u753b\r\n \u4e00\u5b9a\u9593\u9694\u6bce\u306b(\u4f8b\u3048\u3070 k%100==0 \u306e\u6642\u306b, \u7b49)\r\n (x[1], old_u[1]) \uff5e (x[N], old_u[N]) \u3092\u753b\u9762\u306b\u8868\u793a\r\n (x[0], old_u[0], x[N+1], old_u[N+1] \u306f\u5883\u754c\u51e6\u7406\u7528\u306a\u306e\u3067\u8868\u793a\u3057\u306a\u3044)\r\n\r\n \uff08ii\uff09\u8a08\u7b97\r\n j := 1,.....,N \u306e\u9806\u306b\r\n new_u[j] = 2*u[j] - old_u[j] +\u03bb*\u03bb(u[j-1] - 2 * u[j] + u[j+1]) # t=(k+2)*\u03c4\u306e\u3068\u304d\u306eu\r\n \u3092\u7e70\u308a\u8fd4\u3059\r\n new_u[0] = -new_u[1]\r\n new_u[N+1] = -new_u[N]\r\n \r\n \uff08iii\uff09\u66f4\u65b0\uff08\u9806\u756a\u306b\u6ce8\u610f\uff09\r\n old_u[:] = u[:]\r\n u[:] = new_u[:]\r\n \r\n\r\n\u3092\u7e70\u308a\u8fd4\u3059<\/pre>\r\n\r\n\r\n
\r\n<\/span>\u8ab2\u984c1
<\/u><\/strong><\/span><\/h4>\r\n\r\n\r\n\r\n![\"L=1, c=1, f(x) = 2x(1-x), g(x) = 0, u(0, t)=u(1, t) = 0\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-7d2ef36e61317e9c918a0731352881f7_l3.png\" "\\"Rendered")
\u3068\u3057\u3066 (2) \u3092\u89e3\u304f\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f5c\u6210\u305b\u3088\u3002
(![\"N\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" "\\"Rendered")
\u306f![\"100\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-2c4b0a8a79b7bd0e6e72a5fca0be9982_l3.png\" "\\"Rendered")
\u7a0b\u5ea6\u3001![\"T \sim 100\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-9b522c2bf5b771d20106656455b6cc3b_l3.png\" "\\"Rendered")
\u3067\u3068\u306a\u308b\u3088\u3046\u306b![\"M\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-10ebb71bad275c1815a8f2a8c5dea0be_l3.png\" "\\"Rendered")
\u3092\u9069\u5ea6\u306b\u9078\u3073, \u63cf\u5199\u983b\u5ea6\u3082\u9069\u5ea6\u306b\u9078\u3076\u4e8b\u3002)<\/p>\r\n
![\"\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/uploads\/2020\/05\/kadai1_anim-1.gif\")
<\/p>\r\n\r\n\r\n
\r\n<\/span>\u8ab2\u984c2<\/u><\/strong><\/span><\/h4>\r\n\r\n\r\n\r\n![\"L=1, c=1, u(0,t)=u(1,t)=0\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-db49f8a1dbb4dec6dd4c85ab4c12ce8f_l3.png\" "\\"Rendered")
(![\"N = 100\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-bffab0195f6d8d6fda24a35dc3032de6_l3.png\" "\\"Rendered")
\u7a0b\u5ea6) \u306f\u305d\u306e\u307e\u307e\u3067\u3001![\"f(x)\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" "\\"Rendered")
\u304a\u3088\u3073![\"g(x)\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-6a5054ad96fb0978182340eb214bb125_l3.png\" "\\"Rendered")
\u3092\u4ed6\u306e\u3082\u306b\u304b\u3048\u3066\u307f\u3088\u3002\u4f8b\u3048\u3070\u95a2\u6570![\"g(x)=0\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-5022143360c0705e4e9fc41949227d6e_l3.png\" "\\"Rendered")
\u3001\u95a2\u6570\u3092<\/p>\r\n\r\n\r\n\r\n
![\"f(x) = \exp(-100(x-0.4)^2)\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-d134a631903a22d880a941a5913920cd_l3.png\" "\\"Rendered")
<\/p>\r\n
![\"\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/uploads\/2020\/05\/peak1.png\")
<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
\u3068\u3057\u305f\u308a<\/p>\r\n\r\n\r\n\r\n
![\"f(x) = \exp(-100(x-0.4)^2) - \exp(-50(x-0.7)^2) + \exp(-10(x-0.5)^2)\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-431f1cd3973668cb846c2f3a08c8f303_l3.png\" "\\"Rendered")
<\/p>\r\n
![\"\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/uploads\/2020\/05\/peak2.png\")
<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
\u3068\u3059\u308b\u7b49\u3002
\uff08\u5c40\u5728\u5316\u3057\u305f\u521d\u671f\u5024\u304b\u3089\u306f\u3058\u3081\u308b\u3068\u3001\u6ce2\u306e\u4f1d\u64ad\u304c\u89b3\u5bdf\u3055\u308c\u308b\u3002\uff09<\/p>\r\n\r\n\r\n
\r\n<\/span>\u8ab2\u984c3<\/u><\/strong><\/span><\/h4>\r\n\r\n\r\n\r\n\u5883\u754c\u6761\u4ef6\u3092\u3001Neumann\u5883\u754c\u6761\u4ef6\u3084\u5468\u671f\u5883\u754c\u6761\u4ef6\u3092\u9069\u5fdc\u3057\u305f\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f5c\u6210\u3057\u3001\u5b9f\u969b\u306b\u6570\u5024\u8a08\u7b97\u3057\u3066\u307f\u3088\u3002<\/p>\r\n\r\n\r\n\r\n
\u5468\u671f\u5883\u754c\u6761\u4ef6![\"u(x,t)=u(x+1,t)\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-de63cf03a80f7009883ad382e9fbee01_l3.png\" "\\"Rendered")
\u306e\u3082\u3068\u3001\u9032\u884c\u6ce2\u89e3![\"\sin 2\pi (x-t)\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-fccf64b8285321903f94019d49aba692_l3.png\" "\\"Rendered")
\u3092\u5b9f\u73fe\u3059\u308b\u305f\u3081\u306b\u306f\u3001\u3069\u306e\u3088\u3046\u306a\u30d1\u30e9\u30e1\u30fc\u30bf\u3067\u3001\u3069\u306e\u3088\u3046\u306a\u521d\u671f\u5024\u304b\u3089\u8a08\u7b97\u3059\u308b\u5fc5\u8981\u304c\u6709\u308b\u304b\u3002<\/p>\r\n\r\n\r\n
\r\n<\/span>\u767a\u5c55\u8ab2\u984c1<\/u><\/strong><\/span><\/h4>\r\n\r\n\r\n\r\n\u5f26\u306e\u632f\u52d5\u3067\u306f\u306a\u304f\u3001\u819c\u306e\u632f\u52d5\u3092\u8003\u3048\u305f\u3044\u3002\u3059\u306a\u308f\u3061\u3001\uff11\u6b21\u5143\u6ce2\u52d5\u65b9\u7a0b\u5f0f\u3067\u306f\u306a\u304f\uff12\u6b21\u5143\u6ce2\u52d5\u65b9\u7a0b\u5f0f\u3092\u8003\u3048\u308b\u3002\u3069\u306e\u3088\u3046\u306a\u65b9\u7a0b\u5f0f\u306b\u306a\u308b\u3067\u3042\u308d\u3046\u304b\uff1f<\/p>\r\n\r\n\r\n
\r\n<\/span>\u767a\u5c55\u8ab2\u984c2<\/u><\/strong><\/span><\/h4>\r\n\r\n\r\n\r\n\uff12\u6b21\u5143\u6ce2\u52d5\u65b9\u7a0b\u5f0f\u3092\u5dee\u5206\u5316\u3057\u3001\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f5c\u6210\u3057\u3001\u6570\u5024\u8a08\u7b97\u305b\u3088\u3002<\/p>\r\n
Python\u3067\u306e3\u6b21\u5143\u30d7\u30ed\u30c3\u30c8\u306e\u65b9\u6cd5\u306f\u5404\u81ea\u8abf\u3079\u3066\u307f\u3066\u304f\u3060\u3055\u3044\u3002<\/pre>\r\n\r\n\r\n\r\n\r\n\r\n<\/h3>\r\n
\r\n<\/span>\u88dc\u8db3\uff1a\u5dee\u5206\u8fd1\u4f3c\u306b\u3064\u3044\u3066<\/span><\/h2>\r\n![\"\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/uploads\/2020\/05\/slide3.gif\")
<\/p>\r\n
![\"\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/uploads\/2020\/05\/slide4.gif\")
<\/p>\r\n
![\"\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/uploads\/2020\/05\/slide5.gif\")
<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
<\/span>\u5c0e\u95a2\u6570\u306e\u5dee\u5206\u8fd1\u4f3c\u306b\u3064\u3044\u3066<\/span><\/h2>\r\n\r\n\r\n\r\n\u95a2\u6570 u(x) \u3092 x \u306e\u307e\u308f\u308a\u3067Taylor\u5c55\u958b\u3059\u308b\u3068<\/p>\r\n\r\n\r\n\r\n
(S1) <\/span> <\/span>![\"\begin{equation*}u(x+d) = u(x) + d\dot{u}(x) + \frac{1}{2}d^2 \ddot{u}(x) + \frac{1}{6}d^3 \dddot{u}(x) + O(d^4)\end{equation*}\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-c4283c033db6d66d80a75f3130b34e4c_l3.png\" "\\"Rendered")
<\/p><\/p>\r\n\r\n\r\n\r\n (S2) <\/span> <\/span>![\"\begin{equation*}u(x-d) = u(x) - d\dot{u}(x) + \frac{1}{2}d^2 \ddot{u}(x) - \frac{1}{6}d^3 \dddot{u}(x) + O(d^4)\end{equation*}\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-bef4f133bb74477fb9bbf49d4b941818_l3.png\" "\\"Rendered")
<\/p><\/p>\r\n\r\n\r\n\r\n\u3068\u306a\u308a\u3001\u4e21\u8fba\u3092\u52a0\u3048\u308b\u3068\u3001<\/p>\r\n\r\n\r\n\r\n
(S3) <\/span> <\/span>![\"\begin{equation*}u(x+d) + u(x-d) = 2u(x) + d^2 \dot{u}(x) + O(d^4)\end{equation*}\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-4d3a3f5c42d903cc8055510173255cb6_l3.png\" "\\"Rendered")
<\/p><\/p>\r\n\r\n\r\n\r\n\u3068\u306a\u308a\u307e\u3059\u3002O(d4<\/sup>)\u3092\u7121\u8996\u3059\u308b\u3068\u3001<\/p>\r\n\r\n\r\n\r\n (S4) <\/span> <\/span>![\"\begin{equation*}\ddot{u} \approx \frac{1}{d^2} \{ u(x+d) - 2u(x) + u(x-d) \}\end{equation*}\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-60996e778bf042b92f1fc051cc43b1af_l3.png\" "\\"Rendered")
<\/p><\/p>\r\n\r\n\r\n\r\n\u3053\u306e\u5f0f\u306e\u53f3\u8fba\u306e\u8aa4\u5dee\u306f![\"d^2\"](\"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wordpress\/wp-content\/ql-cache\/quicklatex.com-ada9e1fdde16b82afde9c527bf2560b4_l3.png\" "\\"Rendered")
\u7a0b\u5ea6\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\r\n\r\n\r\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"parent":2139,"menu_order":15,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"jetpack_post_was_ever_published":false,"footnotes":""},"class_list":["post-2160","page","type-page","status-publish","hentry"],"aioseo_notices":[],"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/PaSLRw-yQ","_links":{"self":[{"href":"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wp-json\/wp\/v2\/pages\/2160","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wp-json\/wp\/v2\/comments?post=2160"}],"version-history":[{"count":28,"href":"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wp-json\/wp\/v2\/pages\/2160\/revisions"}],"predecessor-version":[{"id":3507,"href":"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wp-json\/wp\/v2\/pages\/2160\/revisions\/3507"}],"up":[{"embeddable":true,"href":"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wp-json\/wp\/v2\/pages\/2139"}],"wp:attachment":[{"href":"https:\/\/home.hiroshima-u.ac.jp\/~mafujii\/wp-json\/wp\/v2\/media?parent=2160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}