1.1 Partial Differential Equations-Some Basics
7
wave number veetor k, or equivalently, if for every k sueh that [k] = I there exist bounded matriees T- 1 (k) and T(k) sueh that D(k) = T - 1 (k)P(k)T(k) is a
diagonal matrix with real eigenvalues.
The definition of a hyperbolie system in several spaee dimensions is extended
to the ease where the eoeffieient matriees in (1.10) are smooth funetions of x
and t by requiring that at every point (x, t) throughout some domain R there exist
bounded matriees T- 1 (k , x, t) and T(k, x, t) sueh that for an real veetors k ofunit
length, T- 1 (k, x, t)P(k, x, t)T(k, x, t) is a diagonal matrix with real eigenvalues
(Gustafsson et aI. 1995, p. 221) . Sinee an symmetrie matriees may be transformed
to real-valued diagonal matriees, the matrix P will be symmetrie and the original
system (1.10) will be hyperbolie if an the eoefficient matrices A q are symmetrie.
The easiest way to show that many multidimensional systems are hyperbolie is
to transform them to equivalent systems in whieh all the eoefficient matriees are
symmetrie.
1.1.2 Linear Second-Order Equations in Two Independent
Variables
Not all waves are solutions to hyperbolie equations. Hyperbolie equations ean
be eompared with two other fundamental types of partial differential equations,
parabolic and elliptic equations, by eonsidering the general family of linear
seeond-order partial differential equations in two independent variables
a(x, y)u x x + 2b(x, y)u x y + c(x, y)u yy + L(x, y, u , U x , u y ) = O. (1.11)
In the preeeding, the subseripts denote partial derivatives, and L is a linear funetion of u, U x, and uy whose eoeffieients may depend on x and y, New independent
variables TJ and ean be defined that transform (1.11) into one of three eanonieal
forms . The particular form that ean beaehieved depends on the number of families
of eharaeteristic eurves associated with (1.11). In those regions of the x-y plane
where b
2 - ac > 0 there are two independent families of eharaeteristic eurves ;
the equation is hyperbolie, and it ean be transformed to the eanonical form
(1.12)
There is one family of eharaeteristie eurves and the equation is parabolie in those
regions where b 2 - ac = 0, in whieh ease (1.11) ean be transformed to
(1.13)
In those regions where b
2 - ac < 0, there are no real-valuedcharacteristic eurves;
the equation is elliptic, and it transforms to
(1.14)
In eaeh ofthe preeeding, Lo, TJ , u , u1/) is a linear funetion of u,
and u1/
with eoefficients that may depend on and TJ.
7
wave number veetor k, or equivalently, if for every k sueh that [k] = I there exist bounded matriees T- 1 (k) and T(k) sueh that D(k) = T - 1 (k)P(k)T(k) is a
diagonal matrix with real eigenvalues.
The definition of a hyperbolie system in several spaee dimensions is extended
to the ease where the eoeffieient matriees in (1.10) are smooth funetions of x
and t by requiring that at every point (x, t) throughout some domain R there exist
bounded matriees T- 1 (k , x, t) and T(k, x, t) sueh that for an real veetors k ofunit
length, T- 1 (k, x, t)P(k, x, t)T(k, x, t) is a diagonal matrix with real eigenvalues
(Gustafsson et aI. 1995, p. 221) . Sinee an symmetrie matriees may be transformed
to real-valued diagonal matriees, the matrix P will be symmetrie and the original
system (1.10) will be hyperbolie if an the eoefficient matrices A q are symmetrie.
The easiest way to show that many multidimensional systems are hyperbolie is
to transform them to equivalent systems in whieh all the eoefficient matriees are
symmetrie.
1.1.2 Linear Second-Order Equations in Two Independent
Variables
Not all waves are solutions to hyperbolie equations. Hyperbolie equations ean
be eompared with two other fundamental types of partial differential equations,
parabolic and elliptic equations, by eonsidering the general family of linear
seeond-order partial differential equations in two independent variables
a(x, y)u x x + 2b(x, y)u x y + c(x, y)u yy + L(x, y, u , U x , u y ) = O. (1.11)
In the preeeding, the subseripts denote partial derivatives, and L is a linear funetion of u, U x, and uy whose eoeffieients may depend on x and y, New independent
variables TJ and ean be defined that transform (1.11) into one of three eanonieal
forms . The particular form that ean beaehieved depends on the number of families
of eharaeteristic eurves associated with (1.11). In those regions of the x-y plane
where b
2 - ac > 0 there are two independent families of eharaeteristic eurves ;
the equation is hyperbolie, and it ean be transformed to the eanonical form
(1.12)
There is one family of eharaeteristie eurves and the equation is parabolie in those
regions where b 2 - ac = 0, in whieh ease (1.11) ean be transformed to
(1.13)
In those regions where b
2 - ac < 0, there are no real-valuedcharacteristic eurves;
the equation is elliptic, and it transforms to
(1.14)
In eaeh ofthe preeeding, Lo, TJ , u , u1/) is a linear funetion of u,
and u1/
with eoefficients that may depend on and TJ.