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单词 ExampleOfRewritingADifferentialEquationAsAPfaffianSystem
释义

example of rewriting a differential equation as a Pfaffian system


To show how one may reformulate a differential equationMathworldPlanetmath as Pfaff’s problem for a set of differential formsMathworldPlanetmath, consider the wave equationMathworldPlanetmath

2ut2=2ux2+2uy2

The first step is to rewrite the equation as a system of first-order equations

at-bx-cy=0
ut-a=0
ux-b=0
uy-c=0

To translateMathworldPlanetmath these equations into the language of differential forms, we shall use the fact that

du=utdt+uxdx+uydy

from which it follows that

dudxdy=utdtdxdy
dudydt=uxdtdxdy
dudtdx=uydtdxdy

We can do likewise with a or b or c in the place of u; there is no point in repeating the formulas for each of these variables.

Multiplying the differential equations through by the form dtdxdy and using the above identities to eliminate partial derivativesMathworldPlanetmath, we obtain the following system of differential forms:

dadxdy-dbdydt-dcdtdx
dudxdy-adtdxdy
dudydt-bdtdxdy
dudtdx-cdtdxdy

From the way these forms were constructed, it is clear that a three dimensional surface in the seven dimensional space with coordinatesPlanetmathPlanetmath x,y,t,a,b,c,u which solves Pfaff’s problem and can be parameterized by x,y,t corresponds to the graph of a solution to the system of differential equations, and hence to a solution of the wave equation.

Note: These considerations are purely local. The global topology of the seven-dimensional space will depend on the domain on which the original wave equation was formulated and on the boundary conditionsMathworldPlanetmath.

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更新时间:2025/5/4 9:37:37