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LieAlgebrasOfVectorFields

  

IsLinearizable

  

Checking if an ODE system can be transformed into a linear one

 

Calling Sequence

Parameters

Description

Calling Sequence

IsLinearizable(DEs, V)

Parameters

DEs

-

an equation or a list of differential equations

V

-

a VectorField object

Description

• 

The command IsLinearizable(...) checks if an ordinary differential equations (ODEs) system can be transformed to a linear ODE by a point transformation. In other words, let S be a single ODE system with a single dependent variable u and independent variable x. Then the method returns true if there exists an invertible transformation x=ψz,w,u=φz,w to a single linear ODE, for some smooth function ψ and φ, and return false otherwise.

• 

The second input argument is a VectorField object where the first argument ODEs is associated with. For more detail about how to construct a VectorField object, see LieAlgebrasOfVectorFields[VectorField]

• 

This command is part of the LieAlgebrasOfVectorFields package. For more detail, see Overview of the LieAlgebrasOfVectorFields package.

• 

This command can be used in the form IsLinearizable(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used in the form :-LieAlgebrasOfVectorFields:-IsLinearizable(...).

with(LieAlgebrasOfVectorFields);

Differential,DisplayStructure,Distribution,EliminationLAVF,EliminationSystem,IDBasis,IsLinearizable,LAVF,LHLibrary,LHPDE,LHPDO,OneForm,SymmetryLAVF,VFPDO,VectorField

(1)

Typesetting:-Settings(userep=true);

false

(2)

Typesetting:-Suppress([xi(x,y),eta(x,y)]);

V := VectorField(xi(x,u)*D[x] + eta(x,u)*D[u], space = [x,u]);

Vξx,ux+ηx,uu

(3)

ODE[1] := diff(u(x),x,x,x) + u(x)*diff(u(x),x,x)^2 + 2*u(x) = 0;

ODE1ⅆ3ⅆx3ux+uxⅆ2ⅆx2ux2+2ux=0

(4)

L := IsLinearizable(ODE[1], V);

Lfalse

(5)

ODE[2] := 2*x^2*u(x)*diff(u(x),x,x,x,x) + x^2*u(x)^2 + 8*x^2*diff(u(x),x)*diff(u(x),x,x,x) + 16*x*u(x)*diff(u(x),x,x,x) + 6*x^2*diff(u(x),x,x)^2 + 48*x*diff(u(x),x)*diff(u(x),x,x) + 24*u(x)*diff(u(x),x,x) + 24*diff(u(x),x)^2 = 0;

ODE22x2uxⅆ4ⅆx4ux+x2ux2+8x2ⅆⅆxuxⅆ3ⅆx3ux+16xuxⅆ3ⅆx3ux+6x2ⅆ2ⅆx2ux2+48xⅆⅆxuxⅆ2ⅆx2ux+24uxⅆ2ⅆx2ux+24ⅆⅆxux2=0

(6)

IsLinearizable(ODE[2], V);

true

(7)

ODE[3] := diff(u(x), x, x, x) + 3*diff(u(x), x)*(diff(u(x), x, x) - diff(u(x), x))/u(x) - 3*diff(u(x), x, x) + 2*diff(u(x), x) - u(x) = 0;

ODE3ⅆ3ⅆx3ux+3ⅆⅆxuxⅆ2ⅆx2uxⅆⅆxuxux3ⅆ2ⅆx2ux+2ⅆⅆxuxux=0

(8)

IsLinearizable(ODE[3], V);

true

(9)

FalknerEq := diff(u(x), x, x, x) + u(x)*diff(u(x), x, x) + beta*(1 - diff(u(x), x, x)^2) = 0;

FalknerEqⅆ3ⅆx3ux+uxⅆ2ⅆx2ux+β1ⅆ2ⅆx2ux2=0

(10)

IsLinearizable(FalknerEq, V);

false

(11)

See Also

LieAlgebrasOfVectorFields (Package overview)

LAVF (Object overview)