ANTITONIC PARTIAL DIFFERENTIAL EQUATIONS

A prototype of the antitonic process is the simple algebraic formula: xy = 1.

This, of course, describes a continuous aantitonic process -- what, in another file at this Website, is described as a cantitonic, as distinguished from a dantitonic, for a discrete/discontinuous process. But the prototype above is sufficient for both.

We can connect this with types of partial differential equations.

The general algebraic equation in one functand has the form: 
	ax2 + bx + c = 0
This has the solution: x = (-b ±(b2 - 4ac))/2a.

The discriminant of this equation -- namely, b2 - 4ac -- discriminates between (or distinguishes) the various types of solution:

  • If the discriminant is zero -- that is, if b2 = 4ac -- then the equation has only a single solution, x = -b/2a ; but if the discriminant is nonzero, then the equation has two distinct solutions.
  • If the discriminant is positive -- that is, if b2 - 4ac > 0 -- then the equation has two distinct real number solutions.
  • If the discriminant is negative -- that is, if b2 - 4ac < 0 -- then the equation has two distinct complex number solutions
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The general second-degree analytic equation in two functands has the form: 
	ax2 + bxy + cy2 + dx + ey + f = 0
Taking the above discriminant as a model, a similar form derived from the general second degree analytic equation can be used to determine the geometric nature of the equation, that is,
  • if b2 - 4ac = 0, the curve is a parabola;
  • if b2 - 4ac < 0, the curve is an ellipse;
  • if b2 - 4ac > 0, the curve is a hyperbola
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The general second=order partial differential equation (pde) in two functands has the form: 
	auxx + 2buxy + cuyy + dux + fuy + fu = g
Similarly, the discriminant can be used to discrimiinate between different types of partial differential equations:
  • if b2 - 4ac = 0, the pde is parabolic;
  • if b2 - 4ac < 0, the pde is elliptic;
  • if b2 - 4ac > 0, the pde is hyperbolic
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It is noted in the literature that only the hyperbolic pde provides the form for a wave equation. That is, corresponding to the algebraic antitone, we have the analytic antitone for modeling waves.