find the characteristic strip corresponding to a given first order partial differential equation - Maple Help

Home : Support : Online Help : Mathematics : Differential Equations : PDEtools : PDEtools/charstrip

PDEtools[charstrip] - find the characteristic strip corresponding to a given first order partial differential equation

PDEtools[splitstrip] - divide the characteristic strip into uncoupled subsets

 Calling Sequence charstrip(PDE, f) splitstrip(PDE, f)

Parameters

 PDE - partial differential equation f - indeterminate function

Description

 • These commands work only on first order PDEs, and return the given PDEs characteristic system depending on a parameter _s.
 • The splitstrip command returns the characteristic system divided into subsets, each one containing equations uncoupled with the ODEs of the other subsets. Depending on the PDE, this splitting of the characteristic system may strongly simplify its solving.
 • The Maple command usually used for solving coupled systems of ODEs (that is, the subsets returned by splitstrip) is dsolve. However, dsolve's strength is in solving single ODEs, not systems of ODEs. The PDEtools package, therefore, has an additional command for uncoupling systems of ODEs, which automatically sends the uncoupled ODEs obtained to dsolve, one at a time, to try solving the system sequentially. This often leads to the desired result (see dsolve/system).
 • Furthermore, it is possible to use the characteristic strip method for solving first order PDEs, by calling pdsolve with the option HINT = strip (see pdsolve).
 • This function is part of the PDEtools package, and so it can be used in the form charstrip(..) only after executing the command with(PDEtools). However, it can always be accessed through the long form of the command by using PDEtools[charstrip](..).

Examples

 > $\mathrm{with}\left(\mathrm{PDEtools}\right):$
 > $\mathrm{PDE}:=x\left(\frac{\partial }{\partial z}f\left(x,y,z\right)\right)-f\left(x,y,z\right)+{y}^{2}\left(\frac{\partial }{\partial y}f\left(x,y,z\right)\right)=0$
 ${\mathrm{PDE}}{:=}{x}{}\left(\frac{{\partial }}{{\partial }{z}}{}{f}{}\left({x}{,}{y}{,}{z}\right)\right){-}{f}{}\left({x}{,}{y}{,}{z}\right){+}{{y}}^{{2}}{}\left(\frac{{\partial }}{{\partial }{y}}{}{f}{}\left({x}{,}{y}{,}{z}\right)\right){=}{0}$ (1)
 > $\mathrm{sys0}:=\mathrm{charstrip}\left(\mathrm{PDE},f\left(x,y,z\right)\right)$
 ${\mathrm{sys0}}{:=}\left\{\frac{{ⅆ}}{{ⅆ}{\mathrm{_s}}}{}{f}{}\left({\mathrm{_s}}\right){=}{f}{}\left({\mathrm{_s}}\right){,}\frac{{ⅆ}}{{ⅆ}{\mathrm{_s}}}{}{x}{}\left({\mathrm{_s}}\right){=}{0}{,}\frac{{ⅆ}}{{ⅆ}{\mathrm{_s}}}{}{y}{}\left({\mathrm{_s}}\right){=}{{y}{}\left({\mathrm{_s}}\right)}^{{2}}{,}\frac{{ⅆ}}{{ⅆ}{\mathrm{_s}}}{}{z}{}\left({\mathrm{_s}}\right){=}{x}{}\left({\mathrm{_s}}\right)\right\}$ (2)

This system can now be solved using dsolve.

 > $\mathrm{dsolve}\left(\mathrm{sys0},\left\{f\left(\mathrm{_s}\right),x\left(\mathrm{_s}\right),y\left(\mathrm{_s}\right),z\left(\mathrm{_s}\right)\right\},\mathrm{explicit}\right)$
 $\left\{{f}{}\left({\mathrm{_s}}\right){=}{\mathrm{_C4}}{}{{ⅇ}}^{{\mathrm{_s}}}{,}{x}{}\left({\mathrm{_s}}\right){=}{\mathrm{_C2}}{,}{y}{}\left({\mathrm{_s}}\right){=}\frac{{1}}{{-}{\mathrm{_s}}{+}{\mathrm{_C3}}}{,}{z}{}\left({\mathrm{_s}}\right){=}{\mathrm{_C2}}{}{\mathrm{_s}}{+}{\mathrm{_C1}}\right\}$ (3)

The system was solved by coupling the ODEs above and splitting the set into three subsets, which are mutually uncoupled.

 > $\mathrm{sys1}:=\mathrm{splitstrip}\left(\mathrm{PDE},f\left(x,y,z\right)\right)$
 ${\mathrm{sys1}}{:=}\left\{\left\{\frac{{ⅆ}}{{ⅆ}{\mathrm{_s}}}{}{f}{}\left({\mathrm{_s}}\right){=}{f}{}\left({\mathrm{_s}}\right)\right\}{,}\left\{\frac{{ⅆ}}{{ⅆ}{\mathrm{_s}}}{}{y}{}\left({\mathrm{_s}}\right){=}{{y}{}\left({\mathrm{_s}}\right)}^{{2}}\right\}{,}\left\{\frac{{ⅆ}}{{ⅆ}{\mathrm{_s}}}{}{x}{}\left({\mathrm{_s}}\right){=}{0}{,}\frac{{ⅆ}}{{ⅆ}{\mathrm{_s}}}{}{z}{}\left({\mathrm{_s}}\right){=}{x}{}\left({\mathrm{_s}}\right)\right\}\right\}$ (4)

This makes it possible to solve each ODE separately.

 > $\mathrm{ans}:=\mathrm{map}\left(u→\mathrm{dsolve}\left(u,\mathrm{indets}\left(u,\mathrm{Function}\right),\mathrm{explicit}\right),\mathrm{sys1}\right)$
 ${\mathrm{ans}}{:=}\left\{\left\{{f}{}\left({\mathrm{_s}}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{\mathrm{_s}}}\right\}{,}\left\{{y}{}\left({\mathrm{_s}}\right){=}\frac{{1}}{{-}{\mathrm{_s}}{+}{\mathrm{_C1}}}\right\}{,}\left\{{x}{}\left({\mathrm{_s}}\right){=}{\mathrm{_C2}}{,}{z}{}\left({\mathrm{_s}}\right){=}{\mathrm{_C2}}{}{\mathrm{_s}}{+}{\mathrm{_C1}}\right\}\right\}$ (5)