Diego Ramos
Universidad del Valle
Cali - Colombia
◼
  • The following problem is taken from Introduction to Partial Differential Equations by Peter J. Olver. The proposed solution makes use of Wolfram Language for symbolic computations.
    • Problem
    Find all (real) solutions of the two-dimensional Laplace equation
    u
    xx
    +
    u
    yy
    =0
    of the form
    u=logp(x,y)
    , where
    p(x,y)
    is a quadratic polynomial.
    Define
    u(x,y)
    with the prescribed form:
    In[]:=
    Clear[p];​​u[x,y]:=Log[p[x,y]];
    Find the laplacian equation for
    u(x,y):
    In[]:=
    Simplify[Laplacian[u[x,y],{x,y}]]0
    Out[]=
    -
    2
    (0,1)
    p
    [x,y]
    +
    2
    (1,0)
    p
    [x,y]
    -p[x,y]
    (0,2)
    p
    [x,y]+
    (2,0)
    p
    [x,y]
    2
    p[x,y]
    0
    Multiply both sides by
    2
    p(x,y)
    In[]:=
    LaplacianEquation=Simplify[
    2
    p[x,y]
    Laplacian[u[x,y],{x,y}]0]
    Out[]=
    2
    (0,1)
    p
    [x,y]
    +
    2
    (1,0)
    p
    [x,y]
    p[x,y]
    (0,2)
    p
    [x,y]+
    (2,0)
    p
    [x,y]
    Observe that the polynomial
    p(x,y)
    must safisty the partial differential equation
    2
    ∂p
    ∂x
    +
    2
    ∂p
    ∂y
    =p(x,y) Δp(x,y)
    (
    1
    )
    Then, if
    p(x,y)=a+bx+cy+d
    2
    x
    +e
    2
    y
    +fxy
    is quadric polynomial we can solve this equation finding the coefficientes of
    p(x,y)
    .
    Define the quadric polynomial:
    In[]:=
    p[x,y]=a+bx+cy+dx^2+ey^2+fxy;
    Obtain the equation (1) in terms of the coeffients of
    p(x,y):
    In[]:=
    Equation1=Collect[Expand[D[p[x,y],x]^2+D[p[x,y],y]^2==p[x,y]Laplacian[p[x,y],{x,y}]],{x,y,xy}]
    Out[]=
    2
    b
    +
    2
    c
    +(4bd+2cf)x+(4
    2
    d
    +
    2
    f
    )
    2
    x
    +(4ce+2bf)y+(4df+4ef)xy+(4
    2
    e
    +
    2
    f
    )
    2
    y
    2ad+2ae+(2bd+2be)x+(2
    2
    d
    +2de)
    2
    x
    +(2cd+2ce)y+(2df+2ef)xy+(2de+2
    2
    e
    )
    2
    y
    From equation (1) extract the coefficients from each side:
    In[]:=
    LeftCoefficients=CoefficientRules[Equation1[[1]],{x,y}]
    Out[]=
    {{2,0}4
    2
    d
    +
    2
    f
    ,{1,1}4df+4ef,{1,0}4bd+2cf,{0,2}4
    2
    e
    +
    2
    f
    ,{0,1}4ce+2bf,{0,0}
    2
    b
    +
    2
    c
    }
    In[]:=
    RightCoefficients=CoefficientRules[Equation1[[2]],{x,y}]
    Out[]=
    {{2,0}2
    2
    d
    +2de,{1,1}2df+2ef,{1,0}2bd+2be,{0,2}2de+2
    2
    e
    ,{0,1}2cd+2ce,{0,0}2ad+2ae}
    Equate the corresponding coefficients:
    In[]:=
    CoefficientEquation=Thread[Values@LeftCoefficientsValues@RightCoefficients]
    Out[]=
    {4
    2
    d
    +
    2
    f
    2
    2
    d
    +2de,4df+4ef2df+2ef,4bd+2cf2bd+2be,4
    2
    e
    +
    2
    f
    2de+2
    2
    e
    ,4ce+2bf2cd+2ce,
    2
    b
    +
    2
    c
    2ad+2ae}
    Solve the coefficient equation:
    In[]:=
    solutions=Solve[CoefficientEquation,{a,b,c,d,e,f}]
    Out[]=
    {c-b,e-d,f-2d},{cb,e-d,f2d},a
    2
    b
    +
    2
    c
    4d
    ,ed,f0,{c-b,d0,e0,f0},{cb,d0,e0,f0}
    The unique solution that can be stablished for real coefficients is:
    In[]:=
    solutions[[3]]
    Out[]=
    a
    2
    b
    +
    2
    c
    4d
    ,ed,f0
    Then, the polinomial
    p(x,y)
    must be of the form
    p(x,y)=
    2
    α
    +
    2
    β
    4λ
    +αx+βy+λ
    2
    x
    +λ
    2
    y
    .
    Verify that the quadric polynomial
    p(x,y)
    satisfy the laplace equation:
    In[]:=
    LaplacianLog
    2
    α
    +
    2
    β
    4λ
    +αx+βy+λ
    2
    x
    +λ
    2
    y
    ,{x,y}//Simplify
    Out[]=
    0