- #1
bolbteppa
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What's really going on as regards characterstics?
For second order PDE's, in this video in which characteristics were defined as lines in the domain along which the highest order partial derivatives were discontinuous, then using this idea, one can naturally use matrices & determinants to derive the conditions under which the second order terms are equal to zero - deriving the [itex]B^2 - 4AC[/itex] classification of partial differential equations.
But then in this note characteristics were those curves along which the coefficient of the first term vanishes, so that a power series solution may be constructed (is that tiny discussion given an actual proof of the Cauchy-Kovalevsky theorem, just without ε-δ?'s?). Here we apparently have two conflicting descriptions of what a characteristic is, definitions that don't seem to merge as equivalent.
Thirdly, in this note characteristics are shown to arise via a change of variables aimed at eliminating higher order derivatives, how does a change of variables relate to all of this? I imagine it's something to do with choosing coordinates so that the characteristics in the domain look nicer, but I have no concrete idea as to how it applies geometrically...
Finally, how does all of this, whatever the right way of looking at it actually is, relate to characteristics of first order PDE's exactly? On pages 5 - 6 a nice geometric interpretation of first order characteristics is given, but it looks like it has nothing to do with what has been described above.
What's going on?
For second order PDE's, in this video in which characteristics were defined as lines in the domain along which the highest order partial derivatives were discontinuous, then using this idea, one can naturally use matrices & determinants to derive the conditions under which the second order terms are equal to zero - deriving the [itex]B^2 - 4AC[/itex] classification of partial differential equations.
For those who are interested, the method is that given
[itex] \mathcal{L}[\phi] = A \phi_{xx} + B \phi_{xy} + C \phi_{yy} + H(x,y,\phi, \phi_x, \phi_y) = 0[/itex]
we basically want [itex] \vec{v} = (\frac{\partial ^2 \phi}{\partial x^2},\frac{\partial ^2 \phi}{\partial x \partial y},\frac{\partial ^2 \phi}{\partial y^2}) = (0,0,0) [/itex] & since this is a vector, we construct an equation of the form [itex] \mathcal{T}(\vec{v}) = \vec{w} [/itex] & examine the case where a solution [itex] \vec{v}[/itex] doesn't exist, i.e. a case where an infinite number of solutions exist.
Using
[tex] d(\frac{\partial \phi}{\partial x}) = \frac{\partial}{\partial x} \frac{\partial \phi}{\partial x}dx + \frac{\partial }{\partial y} \frac{\partial \phi}{\partial x} dy = \frac{\partial ^2 \phi}{\partial x^2 }dx + \frac{\partial ^2 \phi}{\partial y \partial x} dy [/tex]
[tex] d(\frac{\partial \phi}{\partial y}) = \frac{\partial}{\partial x} \frac{\partial \phi}{\partial y}dx + \frac{\partial }{\partial y} \frac{\partial \phi}{\partial y} dy = \frac{\partial ^2 \phi}{\partial x\partial y }dx + \frac{\partial ^2 \phi}{ \partial ^2 y} dy [/tex]
we arrive at
[tex] \mathcal{T}(\vec{v}) = \vec{w} \rightarrow \left[ \begin{matrix} A & B & C \\ dx & dy & 0 \\ 0 & dx & dy \end{matrix} \right] \left[ \begin{matrix} \phi_{xx} \\ \phi_{xy} \\ \phi_{yy} \end{matrix} \right] = \left[ \begin{matrix} -H \\ d( \frac{\partial \phi}{\partial x}) \\ d( \frac{\partial \phi}{\partial y}) \end{matrix} \right] [/tex]
so that [itex] \det{\mathcal{T}} = Ady^2 - Bdxdy + Cdx^2 = 0 \rightarrow A(\frac{dy}{dx})^2 - B(\frac{dy}{dx}) + C = 0[/itex] implies [itex] \frac{dy}{dx} = \frac{B \pm \sqrt{B^2 - 4AC}}{2A} [/itex]
These are the differential equations of the characteristics, splitting into three cases: two real equations, one real equation or two complex equations... Along the characteristics we then have to solve [itex] H(x,y,\phi, \phi_x, \phi_y) = 0 [/itex], & my guess is that the solution for this may not be the same solution for the general second order PDE, thus there will be discontinuities in the solution?
[itex] \mathcal{L}[\phi] = A \phi_{xx} + B \phi_{xy} + C \phi_{yy} + H(x,y,\phi, \phi_x, \phi_y) = 0[/itex]
we basically want [itex] \vec{v} = (\frac{\partial ^2 \phi}{\partial x^2},\frac{\partial ^2 \phi}{\partial x \partial y},\frac{\partial ^2 \phi}{\partial y^2}) = (0,0,0) [/itex] & since this is a vector, we construct an equation of the form [itex] \mathcal{T}(\vec{v}) = \vec{w} [/itex] & examine the case where a solution [itex] \vec{v}[/itex] doesn't exist, i.e. a case where an infinite number of solutions exist.
Using
[tex] d(\frac{\partial \phi}{\partial x}) = \frac{\partial}{\partial x} \frac{\partial \phi}{\partial x}dx + \frac{\partial }{\partial y} \frac{\partial \phi}{\partial x} dy = \frac{\partial ^2 \phi}{\partial x^2 }dx + \frac{\partial ^2 \phi}{\partial y \partial x} dy [/tex]
[tex] d(\frac{\partial \phi}{\partial y}) = \frac{\partial}{\partial x} \frac{\partial \phi}{\partial y}dx + \frac{\partial }{\partial y} \frac{\partial \phi}{\partial y} dy = \frac{\partial ^2 \phi}{\partial x\partial y }dx + \frac{\partial ^2 \phi}{ \partial ^2 y} dy [/tex]
we arrive at
[tex] \mathcal{T}(\vec{v}) = \vec{w} \rightarrow \left[ \begin{matrix} A & B & C \\ dx & dy & 0 \\ 0 & dx & dy \end{matrix} \right] \left[ \begin{matrix} \phi_{xx} \\ \phi_{xy} \\ \phi_{yy} \end{matrix} \right] = \left[ \begin{matrix} -H \\ d( \frac{\partial \phi}{\partial x}) \\ d( \frac{\partial \phi}{\partial y}) \end{matrix} \right] [/tex]
so that [itex] \det{\mathcal{T}} = Ady^2 - Bdxdy + Cdx^2 = 0 \rightarrow A(\frac{dy}{dx})^2 - B(\frac{dy}{dx}) + C = 0[/itex] implies [itex] \frac{dy}{dx} = \frac{B \pm \sqrt{B^2 - 4AC}}{2A} [/itex]
These are the differential equations of the characteristics, splitting into three cases: two real equations, one real equation or two complex equations... Along the characteristics we then have to solve [itex] H(x,y,\phi, \phi_x, \phi_y) = 0 [/itex], & my guess is that the solution for this may not be the same solution for the general second order PDE, thus there will be discontinuities in the solution?
But then in this note characteristics were those curves along which the coefficient of the first term vanishes, so that a power series solution may be constructed (is that tiny discussion given an actual proof of the Cauchy-Kovalevsky theorem, just without ε-δ?'s?). Here we apparently have two conflicting descriptions of what a characteristic is, definitions that don't seem to merge as equivalent.
Thirdly, in this note characteristics are shown to arise via a change of variables aimed at eliminating higher order derivatives, how does a change of variables relate to all of this? I imagine it's something to do with choosing coordinates so that the characteristics in the domain look nicer, but I have no concrete idea as to how it applies geometrically...
Finally, how does all of this, whatever the right way of looking at it actually is, relate to characteristics of first order PDE's exactly? On pages 5 - 6 a nice geometric interpretation of first order characteristics is given, but it looks like it has nothing to do with what has been described above.
What's going on?