detools::characteristics
-- characteristics of partial differential equationdetools::characteristics
(ldf,s)
determines
the characteristics of the linear differential equation
ldf
. The identifier s
is used as parameter
for the curves.
detools::characteristics(ldf, s, <, init>)
ldf |
- | the differential equation: an element of a domain
generated with the constructor
Dom::LinearDifferentialFunction . |
s |
- | the independent variable: an identifier. |
init |
- | the initial conditions: a list of equations. |
a list of expressions representing the characteristics in parametric form.
detools::charODESystem
,
detools::charSolve
detools::characteristics
tries to determine the
characteristics of a given differential equation. For this purpose, it
first sets up the characteristic system using the method detools::charODESystem
and then
tries to solve it. As the characteristic system is an in general
nonlinear system of ordinary differential equations, this can be a very
hard task.With the following input one can determine the characteristics of the differential equation 2 diff(u,x)+diff(u,y)+3 diff(u,z)-2 u=0.
>> LDF := Dom::LinearDifferentialFunction( Vars = [[x, y, z], u], Rest = [Types = "Indep"]): ldf := LDF( 2*u([x]) + u([y]) + 3*u([z]) - 2*u ): detools::characteristics(ldf, tau)
{[z(tau) = C1 + 3 tau, x(tau) = C2 + 2 tau, y(tau) = C3 + tau, u(tau) = C4 exp(2 tau)]}
The result gives the characteristic curve in parametric
form. The constants C1
, C13
,
C14
, C15
could be fixed by adding some
initial condition. It is easy to see that the basis characteristics,
i.e. the projection on the space of the independent variables
x
, y
, z
, is a straight line and
that the solution grows exponentially on it.