## `detools::characteristics` -- characteristics of partial differential equation

#### Introduction

`detools::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>)`

#### Parameters

`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.

#### Returns

a list of expressions representing the characteristics in parametric form.

#### Related Functions

`detools::charODESystem`, `detools::charSolve`

#### Details

- `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.
- The implemented form of the method of characteristics works only for (quasi-)linear first order equations in one dependent variable.
- With the optional third argument one can prescribe some initial conditions for the characteristic system. There should be exactly one condition for each independent and dependent variable of the original partial differential equation.

#### Example 1

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.

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