detools::characteristics
-- characteristics of partial differential equation
Introductiondetools::characteristics(ldf,s) determines
the characteristics of the linear differential equation
ldf. The identifier s is used as parameter
for the curves.
Call(s)detools::characteristics(ldf, s, <, init>)
Parametersldf |
- | 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. |
Returnsa list of expressions representing the characteristics in parametric form.
Related
Functionsdetools::charODESystem,
detools::charSolve
Detailsdetools::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.
Example
1With 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.