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PDEtools

  

mapde

  

map a PDE into a PDE with a different format

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

mapde(PDE, into, f)

Parameters

PDE

-

partial differential equation

into

-

string, one of noF, homo, ccoeff, canom, and canop

f

-

(optional) name of the indeterminate function

Description

• 

Given a PDE, mapde's main goal is to map it into another PDE with a different format, perhaps easier to solve. (Of the options mentioned above, homo and ccoeff are not yet implemented.)

• 

Description of options

noF

no explicit dependence on the indeterminate function

homo

homogeneous PDE

ccoeff

DE with constant coefficients for the higher derivatives

canom

canonical form with only one mixed derivative

canop

canonical form with only pure (no mixed) derivatives

• 

The options canom and canop only work with linear second order PDEs with only two differentiation variables. The mapde command always succeeds as long as the coefficients are constant. With variable coefficients, mapde succeeds only sometimes; in those cases, the mapping is realized through a nonlinear transformation which appears as the algebraic inversion of the solution of a pair of auxiliary first order PDEs. However, the system does not always succeed in solving that auxiliary problem.

• 

The option noF, at present, only works when the received PDE is a homogeneous function of the indeterminate function and its derivatives. In these cases, mapde introduces the indeterminate function as equal to the exponential of another (auxiliary) function.

• 

This function is part of the PDEtools package, and so it can be used in the form mapde(..) only after executing the command with(PDEtools). However, it can always be accessed through the long form of the command by using PDEtools[mapde](..).

Examples

withPDEtools:

PDExfx,y,z3=fx,y,z2y2fx,y,zzfx,y,z

PDE:=xfx,y,z3=fx,y,z2y2fx,y,zzfx,y,z

(1)

pdsolvePDE

fx,y,z=_F1x_F2y_F3z &where ⅆⅆx_F1x3=_c1_F1x3,ⅆ2ⅆy2_F2y=_c2_F2y,ⅆⅆz_F3z=_c1_F3z_c2

(2)

Here is the solution after mapping the PDE into 'noF'.

PDE1mapdePDE,noF

PDE1:=x_F1x,y,z3y_F1x,y,z2z_F1x,y,z2y2_F1x,y,zz_F1x,y,z=0 &where _F1x,y,z=lnfx,y,z

(3)

anspdsolveop1,PDE1

ans:=_F1x,y,z=_F2x+_F3y+_F4z &where ⅆⅆx_F2x=_c1,ⅆ2ⅆy2_F3y=_c13_c3ⅆⅆy_F3y2,ⅆⅆz_F4z=_c3

(4)

buildans

_F1x,y,z=_c13/2y_c3+_c1xln2+12ln_c3ⅇ2_c13/2y_c3_C2_C32_c13+_c3z+_C4+_C1

(5)

The solution to the original PDE can be obtained by taking the exponential of the above.

PDEa2x2fx,y+2b2xyfx,y+c2y2fx,y

PDE:=a2x2fx,y+2b2yxfx,y+c2y2fx,y

(6)

mapdePDE,canop

b2a+c2_ξ12f_ξ1,_ξ2+a2_ξ22f_ξ1,_ξ2 &where _ξ1=ybxa,_ξ2=x

(7)

mapdePDE,canom

4ac+4b22_ξ2_ξ1f_ξ1,_ξ2 &where _ξ1=122ac+b22bxa+y,_ξ2=124ac+b2b4ac+4b2+1x12yaac+b2

(8)

Case of variable coefficients (an easy example)

PDE1subsa=x2,c=y2,b=yx,PDE

PDE1:=x22x2fx,y+2xy2yxfx,y+y22y2fx,y

(9)

Here is the solution after mapping the PDE into canop.  Note that the transformation is nonlinear.

PDE2mapdePDE1,canop

PDE2:=_ξ122_ξ12f_ξ1,_ξ2 &where _ξ1=x,_ξ2=yx

(10)

pdsolveop1,PDE2

f_ξ1,_ξ2=_F1_ξ2_ξ1+_F2_ξ2

(11)

See Also

build

dchange

PDEtools

pdsolve

 


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