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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク First-order partial differential equation ： ウィキペディア英語版 First-order partial differential equation In mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of ''n'' variables. The equation takes the form :$F(x\_1,\backslash ldots,x\_n,u,u\_,\backslash ldots\; u\_)\; =0.\; \backslash ,$ Such equations arise in the construction of characteristic surfaces for hyperbolic partial differential equations, in the calculus of variations, in some geometrical problems, and in simple models for gas dynamics whose solution involves the method of characteristics. If a family of solutions of a single first-order partial differential equation can be found, then additional solutions may be obtained by forming envelopes of solutions in that family. In a related procedure, general solutions may be obtained by integrating families of ordinary differential equations. ==Characteristic surfaces for the wave equation== Characteristic surfaces for the wave equation are level surfaces for solutions of the equation :$u\_t^2\; =\; c^2\; \backslash left(u\_x^2\; +u\_y^2\; +\; u\_z^2\; \backslash right).\; \backslash ,$ There is little loss of generality if we set $u\_t\; =1$: in that case ''u'' satisfies :$u\_x^2\; +\; u\_y^2\; +\; u\_z^2=\; \backslash frac.\; \backslash ,$ In vector notation, let :$\backslash vec\; x\; =\; (x,y,z)\; \backslash quad\; \backslash hbox\; \backslash quad\; \backslash vec\; p\; =\; (u\_x,\; u\_y,\; u\_z).\backslash ,$ A family of solutions with planes as level surfaces is given by :$u(\backslash vec\; x)\; =\; \backslash vec\; p\; \backslash cdot\; (\backslash vec\; x\; -\; \backslash vec),\; \backslash ,$ where :$|\; \backslash vec\; p\; \backslash ,|\; =\; \backslash frac,\; \backslash quad\; \backslash text\; \backslash quad\; \backslash vec\; \backslash quad\; \backslash text.\backslash ,$ If ''x'' and ''x''0 are held fixed, the envelope of these solutions is obtained by finding a point on the sphere of radius 1/''c'' where the value of ''u'' is stationary. This is true if $\backslash vec\; p$ is parallel to $\backslash vec\; x\; -\; \backslash vec$. Hence the envelope has equation :$u(\backslash vec\; x)\; =\; \backslash pm\; \backslash frac\; |\; \backslash vec\; x\; -\backslash vec\; \backslash ,|.$ These solutions correspond to spheres whose radius grows or shrinks with velocity ''c''. These are light cones in space-time. The initial value problem for this equation consists in specifying a level surface ''S'' where ''u''=0 for ''t''=0. The solution is obtained by taking the envelope of all the spheres with centers on ''S'', whose radii grow with velocity ''c''. This envelope is obtained by requiring that :$\backslash frac\; |\; \backslash vec\; x\; -\; \backslash vec\backslash ,\; |\; \backslash quad\; \backslash hbox\; \backslash quad\; \backslash vec\; \backslash in\; S.\; \backslash ,$ This condition will be satisfied if $|\; \backslash vec\; x\; -\; \backslash vec\backslash ,\; |$ is normal to ''S''. Thus the envelope corresponds to motion with velocity ''c'' along each normal to ''S''. This is the Huygens' construction of wave fronts: each point on ''S'' emits a spherical wave at time ''t''=0, and the wave front at a later time ''t'' is the envelope of these spherical waves. The normals to ''S'' are the light rays. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「First-order partial differential equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.