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PDEtools

  

charstrip

  

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

  

splitstrip

  

divide the characteristic strip into uncoupled subsets

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

charstrip(PDE, f, simplifyusingpde)

charstrip(characteristic_strip, f, reverse)

splitstrip(PDE, f, simplifyusingpde)

Parameters

PDE

-

partial differential equation

f

-

indeterminate function

reverse

-

used when passing a characterstic strip to be reversed, to get the PDE behind it

simplifyusingpde

-

optional, can be true (default value) or false, to simplify the characteristic strip using the given PDE aiming at removing as much as possible occurrences of derivatives of f.

Description

• 

These commands work only on first order PDEs, and return the given PDEs characteristic system depending on a parameter _s. Also, when the first argument is a characteristic strip, by passing the keyword reverse, charstrip returns the PDE behind it.

• 

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 (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

withPDEtools:

PDExzfx,y,zfx,y,z+y2yfx,y,z=0

PDE:=xzfx,y,zfx,y,z+y2yfx,y,z=0

(1)

sys0charstripPDE,fx,y,z

sys0:=ⅆⅆ_sf_s=f_s,ⅆⅆ_sx_s=0,ⅆⅆ_sy_s=y_s2,ⅆⅆ_sz_s=x_s

(2)

This system can now be solved using dsolve.

dsolvesys0,f_s,x_s,y_s,z_s,explicit

f_s=_C4ⅇ_s,x_s=_C2,y_s=1_s+_C3,z_s=_C2_s+_C1

(3)

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

sys1splitstripPDE,fx,y,z

sys1:=ⅆⅆ_sf_s=f_s,ⅆⅆ_sy_s=y_s2,ⅆⅆ_sx_s=0,ⅆⅆ_sz_s=x_s

(4)

This makes it possible to solve each ODE separately.

ansmapu→dsolveu,indetsu,Function,explicit,sys1

ans:=f_s=_C1ⅇ_s,y_s=1_s+_C1,x_s=_C2,z_s=_C2_s+_C1

(5)

When computing the characteristic strip, by default the strip is simplified taking the PDE itself into account, aiming at diminishing occurrences of derivatives of the unknown of the problem. This is useful in that it makes easier the integration of the resulting ODE system. On the other hand, there are situations in which this simplification is undesired, for example: when deriving the PDE behind a given characteristic strip. Consider for instance

PDExzx,yyzx,yzx,y=0

PDE:=xzx,yyzx,yzx,y=0

(6)

The corresponding characteristic strip without simplifying it taking PDE into account is given by

charstripPDE,zx,y,simplifyusingpde=false

ⅆⅆ_sx_s=_p2_s,ⅆⅆ_sy_s=_p1_s,ⅆⅆ_sz_s=2_p1_s_p2_s,ⅆⅆ_s_p1_s=_p1_s,ⅆⅆ_s_p2_s=_p2_s &where _p1=xzx,y,_p2=yzx,y

(7)

Departing from this result it is possible to re-obtain PDE (this is possible only when the characteristic strip was constructed without simplifying using the PDE itself, as it is the case above). For that purpose, pass the characteristic strip as first argument and the keyword reverse

charstrip,zx,y,reverse

xzx,yyzx,y+_C1zx,y

(8)

Note the arbitrary constant that appears in this result: for any value of this constant, the characteristic strip is the same.

See Also

dchange

dsolve

dsolve/system

pdetest

PDEtools

pdsolve

 


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