4 Equations with n Independent Variables



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4 Equations with n Independent Variables

   

Many problems encountered in computational science involve several space variables and possibly a time variable. As indicated in Example 3, it is important for the computational scientist to be aware of the type of equation under consideration. Although the clear trichotomy of types of section 2 is not maintained in this setting, it is still possible to identify equations of elliptic, parabolic and hyperbolic types. The remarks of section 2 regarding algorithms and architectures for problems involving two variables apply equally well to their -variable counterparts.

A general linear PDE of order two in variables has the form

 

If , then the principal part of equation (12) can always be arranged so that ; thus, the matrix can be assumed symmetric. In linear algebra it is shown that every real, symmetric matrix has real eigenvalues. These eigenvalues are the (possibly repeated) zeros of an nth-degree polynomial in , , where is the identity matrix. Let denote the number of positive eigenvalues, and the number of zero eigenvalues (i.e., the multiplicity of the eigenvalue zero), of the matrix . Then equation (12)is:

hyperbolic
if and or and
parabolic
if (equivalently, if )
elliptic
if and or and
ultrahyperbolic
if and

If any of the is nonconstant, the type of equation (12) can vary with position.


 

For the PDE the matrix is

 




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