{"id":304,"date":"2024-06-08T18:42:49","date_gmt":"2024-06-08T09:42:49","guid":{"rendered":"https:\/\/www.flumen-jp.com\/wordpress\/?p=304"},"modified":"2025-02-18T17:10:28","modified_gmt":"2025-02-18T08:10:28","slug":"mathematics","status":"publish","type":"post","link":"https:\/\/www.flumen-jp.com\/wordpress\/index.php\/2024\/06\/08\/mathematics\/","title":{"rendered":"\u6570\u5b66"},"content":{"rendered":"\n

\u3000\u3075\u3068\u3057\u305f\u30ad\u30c3\u30ab\u30b1\u304c\u3042\u3063\u3066\u3001\u925b\u76f4\u4e00\u6b21\u5143\u306e\u4e0d\u98fd\u548c\u6d78\u900f\u6d41\u306b\u3064\u3044\u3066\u304a\u52c9\u5f37\u3057\u305f\u3002
\u3000\u30a4\u30f3\u30bf\u30fc\u30cd\u30c3\u30c8\u3067\u8272\u3005\u8abf\u3079\u3066\u3044\u3066\u3001\u5076\u3005\u3001\u4eac\u5927\u9632\u707d\u7814 \u83c5\u539f\u5feb\u6597\u6c0f\u306e\u300e\u4e0d\u98fd\u548c\u6d78\u900f\u6d41\u306e\u89e3\u6790\u624b\u6cd5\u3068\u964d\u96e8\u6d41\u51fa\u30e2\u30c7\u30eb\u3078\u306e\u5fdc\u7528<\/a>\u300f\u3068\u3044\u3046\u8ad6\u6587\u306b\u884c\u304d\u5f53\u305f\u3063\u305f\u3002\u7b2c\uff12\u7ae0\u304c\u300c\u5730\u4e0b\u6c34\u9762\u3092\u6709\u3059\u308b\u571f\u58cc\u306b\u304a\u3051\u308b Richards \u5f0f\u306e\u89e3\u6790\u89e3\u300d\u3068\u306a\u3063\u3066\u304a\u308a\u3001\u8208\u5473\u6df1\u3044\u306e\u3067\u89e3\u6790\u89e3\u306e\u5c0e\u51fa\u3092\u30c8\u30ec\u30fc\u30b9\u3057\u3066\u307f\u305f\u3002
\u3000\u547d\u984c\u306f\u6b21\u306e Richards \u5f0f<\/p>\n\n\n\n

$$
\\frac{\\partial S}{\\partial t} = D\\frac{\\partial^2 S}{\\partial z^2} –
V\\frac{\\partial S}{\\partial z}
\\raise{-12px}{,}\\quad
D = \\frac{k_s}{\\alpha(\\theta_s – \\theta_r)}
\\raise{-12px}{,}\\quad
V = \\frac{k_s}{\\theta_s – \\theta_r}
$$<\/p>\n\n\n\n

\u3092<\/p>\n\n\n\n

$$
S(0, t) = S_U\\,,\\quad
S(L, t) = S_L\\,,\\quad
S(z, 0) = S_0(z)
$$<\/p>\n\n\n\n

\u306e\u5143\u3067\u89e3\u304f\u3001\u3068\u3044\u3046\u3082\u306e\u3067\u3059\u3002
\u3000\u3053\u3053\u306b\u3001\\(t\\)\uff1a\u6642\u9593\u5ea7\u6a19\u3001\\(z\\)\uff1a\u925b\u76f4\u4e0b\u65b9\u3092\u6b63\u3068\u3059\u308b\u7a7a\u9593\u5ea7\u6a19\u3001\\(S\\)\uff1a\u6709\u52b9\u98fd\u548c\u5ea6\u3001\\(\\theta_s\\)\uff1a\u98fd\u548c\u4f53\u7a4d\u542b\u6c34\u7387\u3001\\(\\theta_r\\)\uff1a\u6b8b\u7559\u4f53\u7a4d\u542b\u6c34\u7387\u3001\\(\\alpha\\)\uff1aGardner \u30e2\u30c7\u30eb\u3067\u6c34\u5206\u4fdd\u6301\u66f2\u7dda\u306e\u5f62\u72b6\u3092\u6c7a\u5b9a\u3059\u308b\u30d1\u30e9\u30e1\u30fc\u30bf\u3001\\(k_s\\)\uff1a\u98fd\u548c\u900f\u6c34\u4fc2\u6570\u3001\u3067\u3059\u3002
\u3000\u904e\u7a0b\u3092\u30b9\u30ad\u30c3\u30d7\u3057\u3066\u3001\u4efb\u610f\u306e\u5ea7\u6a19\u3067\u306e\u6d41\u91cf<\/p>\n\n\n\n

$$
q = -\\frac{k_s}{\\alpha} \\frac{\\partial S}{\\partial z} + k_s S
$$<\/p>\n\n\n\n

\u306f<\/p>\n\n\n\n

$$
q =k_s B -k_s \\sum_{n=1}^{\\infty} \\,l_n \\cdot f_n
$$<\/p>\n\n\n\n

$$
l_n = \\frac{2}{L} \\int_0^L \\biggl[\\bigl\\{ S_0(z) – A \\exp\\left( \\alpha z \\right) – B\\bigr\\}\\exp\\left( -\\frac{\\alpha}{2} z \\right) \\sin\\left( \\frac{n \\pi}{L} z \\right)
\\biggr] dz
$$<\/p>\n\n\n\n

$$
f_n=\\exp\\left[-\\left\\{ \\left( \\frac{n \\pi}{L}\\right)^2 + \\frac{\\alpha^2}{4} \\right\\} D\\,t + \\frac{\\alpha}{2} z \\right] \\left[\\frac{1}{2}\\sin\\left( \\frac{n \\pi}{L} z\\right) +\\frac{n \\pi}{\\alpha L} \\cos\\left( \\frac{n \\pi}{L} z\\right)\\right]
$$<\/p>\n\n\n\n

$$
A = \\frac{S_U – S_L}{\\displaystyle{1 – \\exp\\left( \\alpha L \\right)}}
\\raise{-12px}{,} \\quad
B = \\frac{\\displaystyle{S_L – S_U \\exp\\left( \\alpha L \\right)}}{\\displaystyle{{1 – \\exp\\left( \\alpha L \\right)}}}
\\raise{-12px}{,} \\quad
D = \\frac{k_s}{\\alpha(\\theta_s – \\theta_r)}
$$<\/p>\n\n\n\n

\u3067\u4e0e\u3048\u3089\u308c\u307e\u3059\u3002
\u3000\u3053\u308c\u3092 Python \u3067\u66f8\u3044\u3066\uff08\\(\\infty\\) \u306f 500 \u3067\u7f6e\u63db\uff09\u3001\\(S_U=1\\)\u3001\\(S_L=0\\)\u3001\\(S_0(z)=0\\) \u306e\u5143\u3067 \\(z = L\\) \u306b\u304a\u3051\u308b\u6d41\u91cf\u306e\u6642\u9593\u5206\u5e03\u3092\u8a08\u7b97\u3057\u305f\u7d50\u679c\u304c\u4e0b\u56f3\u3067\u3059\u3002\u8ad6\u6587\u306b\u63b2\u8f09\u3055\u308c\u3066\u3044\u308b\u56f3\u3068\u4e00\u81f4\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

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\u3000\u3075\u3068\u3057\u305f\u30ad\u30c3\u30ab\u30b1\u304c\u3042\u3063\u3066\u3001\u925b\u76f4\u4e00\u6b21\u5143\u306e\u4e0d\u98fd\u548c\u6d78\u900f\u6d41\u306b\u3064\u3044\u3066\u304a\u52c9\u5f37\u3057\u305f\u3002\u3000\u30a4\u30f3\u30bf\u30fc\u30cd\u30c3\u30c8\u3067\u8272\u3005\u8abf\u3079\u3066\u3044\u3066\u3001\u5076\u3005\u3001\u4eac\u5927\u9632\u707d\u7814 \u83c5\u539f\u5feb\u6597\u6c0f\u306e\u300e\u4e0d\u98fd\u548c\u6d78\u900f\u6d41\u306e\u89e3\u6790\u624b\u6cd5\u3068\u964d\u96e8\u6d41\u51fa\u30e2\u30c7\u30eb\u3078\u306e\u5fdc\u7528\u300f\u3068\u3044\u3046\u8ad6\u6587\u306b\u884c\u304d\u5f53\u305f\u3063\u305f\u3002\u7b2c\uff12\u7ae0\u304c […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[14],"tags":[45],"class_list":["post-304","post","type-post","status-publish","format-standard","hentry","category-misc","tag-math"],"_links":{"self":[{"href":"https:\/\/www.flumen-jp.com\/wordpress\/index.php\/wp-json\/wp\/v2\/posts\/304","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.flumen-jp.com\/wordpress\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.flumen-jp.com\/wordpress\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.flumen-jp.com\/wordpress\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.flumen-jp.com\/wordpress\/index.php\/wp-json\/wp\/v2\/comments?post=304"}],"version-history":[{"count":18,"href":"https:\/\/www.flumen-jp.com\/wordpress\/index.php\/wp-json\/wp\/v2\/posts\/304\/revisions"}],"predecessor-version":[{"id":327,"href":"https:\/\/www.flumen-jp.com\/wordpress\/index.php\/wp-json\/wp\/v2\/posts\/304\/revisions\/327"}],"wp:attachment":[{"href":"https:\/\/www.flumen-jp.com\/wordpress\/index.php\/wp-json\/wp\/v2\/media?parent=304"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.flumen-jp.com\/wordpress\/index.php\/wp-json\/wp\/v2\/categories?post=304"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.flumen-jp.com\/wordpress\/index.php\/wp-json\/wp\/v2\/tags?post=304"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}