Green's function for the three-variable Laplace equation

<p>In <a href="/facts/Physics/a7aH2y08">physics</a>, the Green's function (or <a href="/facts/Fundamental_solution/hdu8vXJa">fundamental solution</a>) for the Laplacian (or Laplace operator) in three variables is used to describe the response of a particular type of physical system to a <a href="/facts/Point_source/AHmNlt03">point source</a>. In particular, this <a href="/facts/Green%2527s_function/rKxKKPcp">Green's function</a> arises in systems that can be described by <a href="/facts/Poisson%2527s_equation/Pr8zkCXB">Poisson's equation</a>, a <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equation</a> (PDE) of the form

∇
          
            2
          
        
        u
        (
        
          x
        
        )
        =
        f
        (
        
          x
        
        )
      
    
    {\displaystyle \nabla ^{2}u(\mathbf {x} )=f(\mathbf {x} )}

where 
  
    
      
        
          ∇
          
            2
          
        
      
    
    {\displaystyle \nabla ^{2}}
  
 is the <a href="/facts/Laplace_operator/kov9u3PT">Laplace operator</a> in 
  
    
      
        
          
            R
          
          
            3
          
        
      
    
    {\displaystyle \mathbb {R} ^{3}}
  
, 
  
    
      
        f
        (
        
          x
        
        )
      
    
    {\displaystyle f(\mathbf {x} )}
  
 is the source term of the system, and 
  
    
      
        u
        (
        
          x
        
        )
      
    
    {\displaystyle u(\mathbf {x} )}
  
 is the solution to the equation. Because 
  
    
      
        
          ∇
          
            2
          
        
      
    
    {\displaystyle \nabla ^{2}}
  
 is a linear <a href="/facts/Differential_operator/Nr15J0IE">differential operator</a>, the solution 
  
    
      
        u
        (
        
          x
        
        )
      
    
    {\displaystyle u(\mathbf {x} )}
  
 to a general system of this type can be written as an integral over a distribution of source given by 
  
    
      
        f
        (
        
          x
        
        )
      
    
    {\displaystyle f(\mathbf {x} )}
  
:

u
        (
        
          x
        
        )
        =
        ∫
        G
        (
        
          x
        
        ,
        
          
            x
            ′
          
        
        )
        f
        (
        
          
            x
            ′
          
        
        )
        d
        
          
            x
          
          ′
        
      
    
    {\displaystyle u(\mathbf {x} )=\int G(\mathbf {x} ,\mathbf {x'} )f(\mathbf {x'} )d\mathbf {x} '}

where the <a href="/facts/Green%2527s_function/rKxKKPcp">Green's function</a> for Laplacian in three variables 
  
    
      
        G
        (
        
          x
        
        ,
        
          
            x
            ′
          
        
        )
      
    
    {\displaystyle G(\mathbf {x} ,\mathbf {x'} )}
  
 describes the response of the system at the point 
  
    
      
        
          x
        
      
    
    {\displaystyle \mathbf {x} }
  
 to a point source located at 
  
    
      
        
          
            x
            ′
          
        
      
    
    {\displaystyle \mathbf {x'} }
  
:

∇
          
            2
          
        
        G
        (
        
          x
        
        ,
        
          
            x
            ′
          
        
        )
        =
        δ
        (
        
          x
        
        −
        
          
            x
            ′
          
        
        )
      
    
    {\displaystyle \nabla ^{2}G(\mathbf {x} ,\mathbf {x'} )=\delta (\mathbf {x} -\mathbf {x'} )}

and the point source is given by 
  
    
      
        δ
        (
        
          x
        
        −
        
          
            x
            ′
          
        
        )
      
    
    {\displaystyle \delta (\mathbf {x} -\mathbf {x'} )}
  
, the <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a>.
</p>

Green's function for the three-variable Laplace equation open-in-new

Green's function for the three-variable Laplace equation