| 释义 |
Partial Differential EquationA partial differential equation (PDE) is an equation involving functions and their Partial Derivatives; for example, the Wave Equation
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Fortunately, partial differential equations of second-order are often amenable to analytical solution. Such PDEs are ofthe form
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If  is a Positive Definite Matrix, i.e., det , the PDE is said to be Elliptic. Laplace's Equation and Poisson's Equation are examples. Boundary conditionsare used to give the constraint  on  , where
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 .
If det , the PDE is said to be Hyperbolic. The Wave Equation is an example of a hyperbolicpartial differential equation. Initial-boundary conditions are used to give
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If det , the PDE is said to be parabolic. The Heat Conduction Equation equation and otherdiffusion equations  are examples. Initial-boundary conditions are used to give
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References
 Partial Differential EquationsArfken, G. ``Partial Differential Equations of Theoretical Physics.'' §8.1 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 437-440, 1985. Bateman, H. Partial Differential Equations of Mathematical Physics. New York: Dover, 1944. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Partial Differential Equations.'' Ch. 19 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 818-880, 1992. Sobolev, S. L. Partial Differential Equations of Mathematical Physics. New York: Dover, 1989. Sommerfeld, A. Partial Differential Equations in Physics. New York: Academic Press, 1964. Webster, A. G. Partial Differential Equations of Mathematical Physics, 2nd corr. ed. New York: Dover, 1955.
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