|
In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation. The method is to reduce a partial differential equation to a family of ordinary differential equations along which the solution can be integrated from some initial data given on a suitable hypersurface. ==Characteristics of first-order partial differential equation== For a first-order PDE (partial differential equation), the method of characteristics discovers curves (called characteristic curves or just characteristics) along which the PDE becomes an ordinary differential equation (ODE). Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE. For the sake of motivation, we confine our attention to the case of a function of two independent variables ''x'' and ''y'' for the moment. Consider a quasilinear PDE of the form Suppose that a solution ''z'' is known, and consider the surface graph ''z'' = ''z''(''x'',''y'') in R3. A normal vector to this surface is given by : As a result, equation () is equivalent to the geometrical statement that the vector field : is tangent to the surface ''z'' = ''z''(''x'',''y'') at every point, for the dot product of this vector field with the above normal vector is zero. In other words, the graph of the solution must be a union of integral curves of this vector field. These integral curves are called the characteristic curves of the original partial differential equation. The equations of the characteristic curve may be expressed invariantly by the ''Lagrange-Charpit equations'' : or, if a particular parametrization ''t'' of the curves is fixed, then these equations may be written as a system of ordinary differential equations for ''x''(''t''), ''y''(''t''), ''z''(''t''): : These are the characteristic equations for the original system. 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「In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation. The method is to reduce a partial differential equation to a family of ordinary differential equations along which the solution can be integrated from some initial data given on a suitable hypersurface.==Characteristics of first-order partial differential equation==For a first-order PDE (partial differential equation), the method of characteristics discovers curves (called characteristic curves or just characteristics) along which the PDE becomes an ordinary differential equation (ODE). Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE.For the sake of motivation, we confine our attention to the case of a function of two independent variables ''x'' and ''y'' for the moment. Consider a quasilinear PDE of the formSuppose that a solution ''z'' is known, and consider the surface graph ''z'' = ''z''(''x'',''y'') in R3. A normal vector to this surface is given by:\left(\frac(x,y),\frac(x,y),-1\right).\,As a result, equation () is equivalent to the geometrical statement that the vector field:(a(x,y,z),b(x,y,z),c(x,y,z))\,is tangent to the surface ''z'' = ''z''(''x'',''y'') at every point, for the dot product of this vector field with the above normal vector is zero. In other words, the graph of the solution must be a union of integral curves of this vector field. These integral curves are called the characteristic curves of the original partial differential equation.The equations of the characteristic curve may be expressed invariantly by the ''Lagrange-Charpit equations'':\frac = \frac = \frac,or, if a particular parametrization ''t'' of the curves is fixed, then these equations may be written as a system of ordinary differential equations for ''x''(''t''), ''y''(''t''), ''z''(''t'')::\begin\frac&=&a(x,y,z)\\\frac&=&b(x,y,z)\\\frac&=&c(x,y,z).\endThese are the characteristic equations for the original system.Characteristic equations redirects to this article -->」の詳細全文を読む スポンサード リンク
|