Method of fundamental solutions

<h2 id="mfs-formulation">MFS formulation</h2>
Consider a partial differential equation governing certain type of problems

L
        u
        =
        f
        
          (
          
            x
            ,
            y
          
          )
        
        ,
         
         
        
          (
          
            x
            ,
            y
          
          )
        
        ∈
        Ω
        ,
      
    
    {\displaystyle Lu=f\left(x,y\right),\ \ \left(x,y\right)\in \Omega ,}

u
        =
        g
        
          (
          
            x
            ,
            y
          
          )
        
        ,
         
         
        
          (
          
            x
            ,
            y
          
          )
        
        ∈
        ∂
        
          Ω
          
            D
          
        
        ,
      
    
    {\displaystyle u=g\left(x,y\right),\ \ \left(x,y\right)\in \partial \Omega _{D},}

∂
              u
            
            
              ∂
              n
            
          
        
        =
        h
        
          (
          
            x
            ,
            y
          
          )
        
        ,
         
         
        
          (
          
            x
            ,
            y
          
          )
        
        ∈
        ∂
        
          Ω
          
            N
          
        
        ,
      
    
    {\displaystyle {\frac {\partial u}{\partial n}}=h\left(x,y\right),\ \ \left(x,y\right)\in \partial \Omega _{N},}

where 
 
 
 
 L
 
 
 {\displaystyle L}
 
 is the differential partial operator, 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
 represents the computational domain, 
 
 
 
 ∂
 
 Ω
 
 D
 
 
 
 
 {\displaystyle \partial \Omega _{D}}
 
 and 
 
 
 
 ∂
 
 Ω
 
 N
 
 
 
 
 {\displaystyle \partial \Omega _{N}}
 
 denote the Dirichlet and Neumann boundary, respectively, 
 
 
 
 ∂
 
 Ω
 
 D
 
 
 ∪
 ∂
 
 Ω
 
 N
 
 
 =
 ∂
 Ω
 
 
 {\displaystyle \partial \Omega _{D}\cup \partial \Omega _{N}=\partial \Omega }
 
 
and 
 
 
 
 ∂
 
 Ω
 
 D
 
 
 ∩
 ∂
 
 Ω
 
 N
 
 
 =
 ∅
 
 
 {\displaystyle \partial \Omega _{D}\cap \partial \Omega _{N}=\varnothing }
 
.
The MFS employs the fundamental solution of the operator as its basis function to represent the approximation of unknown function u as follows

u
            
            
              ∗
            
          
        
        
          (
          
            x
            ,
            y
          
          )
        
        =
        
          ∑
          
            i
            =
            1
          
          
            N
          
        
        
          α
          
            i
          
        
        ϕ
        
          (
          
            r
            
              i
            
          
          )
        
      
    
    {\displaystyle {{u}^{*}}\left(x,y\right)=\sum \limits _{i=1}^{N}\alpha _{i}\phi \left(r_{i}\right)}

where 
 
 
 
 
 r
 
 i
 
 
 =
 
 ‖
 
 
 (
 
 x
 ,
 y
 
 )
 
 −
 
 (
 
 s
 
 x
 
 i
 
 
 ,
 s
 
 y
 
 i
 
 
 
 )
 
 
 ‖
 
 
 
 {\displaystyle r_{i}=\left\|\left(x,y\right)-\left(sx_{i},sy_{i}\right)\right\|}
 
 denotes the Euclidean distance between collocation points 
 
 
 
 
 (
 
 x
 ,
 y
 
 )
 
 
 
 {\displaystyle \left(x,y\right)}
 
 and source points 
 
 
 
 
 (
 
 s
 
 x
 
 i
 
 
 ,
 s
 
 y
 
 i
 
 
 
 )
 
 
 
 {\displaystyle \left(sx_{i},sy_{i}\right)}
 
, 
 
 
 
 ϕ
 
 (
 ⋅
 )
 
 
 
 {\displaystyle \phi \left(\cdot \right)}
 
 is the fundamental solution which satisfies

L
        ϕ
        =
        δ
        
      
    
    {\displaystyle L\phi =\delta \,}

where 
 
 
 
 δ
 
 
 {\displaystyle \delta }
 
 denotes Dirac delta function, and 
 
 
 
 
 
 
 α
 
 
 i
 
 
 
 
 
 {\displaystyle {{\alpha }_{i}}}
 
 are the unknown coefficients. 
With the source points located outside the physical domain, the MFS avoid the fundamental solution singularity. Substituting the approximation into boundary condition yields the following matrix equation

[
          
            
              
                
                  ϕ
                  
                    (
                    
                      
                        
                        
                          r
                          
                            j
                          
                        
                        |
                      
                      
                        
                          x
                          
                            i
                          
                        
                        ,
                        
                          y
                          
                            i
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      
                        ∂
                        ϕ
                        
                          (
                          
                            
                              
                              
                                r
                                
                                  j
                                
                              
                              |
                            
                            
                              
                                x
                                
                                  k
                                
                              
                              ,
                              
                                y
                                
                                  k
                                
                              
                            
                          
                          )
                        
                      
                      
                        ∂
                        n
                      
                    
                  
                
              
            
          
          ]
        
         
        ⋅
         
        α
        =
        
          (
          
            
              
                
                  g
                  
                    (
                    
                      
                        x
                        
                          i
                        
                      
                      ,
                      
                        y
                        
                          i
                        
                      
                    
                    )
                  
                
              
              
                
                  h
                  
                    (
                    
                      
                        x
                        
                          k
                        
                      
                      ,
                      
                        y
                        
                          k
                        
                      
                    
                    )
                  
                
              
            
          
          )
        
        ,
      
    
    {\displaystyle \left[{\begin{matrix}\phi \left(\left.r_{j}\right|_{x_{i},y_{i}}\right)\\{\frac {\partial \phi \left(\left.r_{j}\right|_{x_{k},y_{k}}\right)}{\partial n}}\\\end{matrix}}\right]\ \cdot \ \alpha =\left({\begin{matrix}g\left(x_{i},y_{i}\right)\\h\left(x_{k},y_{k}\right)\\\end{matrix}}\right),}

where 
 
 
 
 
 (
 
 
 x
 
 i
 
 
 ,
 
 y
 
 i
 
 
 
 )
 
 
 
 {\displaystyle \left(x_{i},y_{i}\right)}
 
 and 
 
 
 
 
 (
 
 
 x
 
 k
 
 
 ,
 
 y
 
 k
 
 
 
 )
 
 
 
 {\displaystyle \left(x_{k},y_{k}\right)}
 
 denote the collocation points, respectively, on Dirichlet and Neumann boundaries. The unknown coefficients 
 
 
 
 
 α
 
 i
 
 
 
 
 {\displaystyle \alpha _{i}}
 
 can uniquely be determined by the above algebraic equation. And then we can evaluate numerical solution at any location in physical domain.

<h2 id="history-and-recent-developments">History and recent developments</h2>
The ideas behind the MFS were developed primarily by V. D. Kupradze and M. A. Alexidze in the late 1950s and early 1960s.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> However, the method was first proposed as a computational technique much later by R. Mathon and R. L. Johnston in the late 1970s,<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> followed by a number of papers by Mathon, Johnston and Graeme Fairweather with applications. The MFS then gradually became a useful tool for the solution of a large variety of physical and engineering problems.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> 
In the 1990s, M. A. Golberg and C. S. Chen extended the MFS to deal with inhomogeneous equations and time-dependent problems, greatly expanding its applicability.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a><a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> Later developments indicated that the MFS can be used to solve partial differential equations with variable coefficients.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> The MFS has proved particularly effective for certain classes of problems such as inverse,<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> unbounded domain, and free-boundary problems.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
Some techniques have been developed to cure the fictitious boundary problem in the MFS, such as the <a href="/facts/Boundary_knot_method/GjAq8xoy">boundary knot method</a>, <a href="/facts/Singular_boundary_method/7tm3TA7H">singular boundary method</a>, and <a href="/facts/Regularized_meshless_method/NsoqVJ6e">regularized meshless method</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Radial_basis_function/QnWM1mBN">Radial basis function</a></li>
<li><a href="/facts/Boundary_element_method/FMIb4mLs">Boundary element method</a></li>
<li><a href="/facts/Boundary_knot_method/GjAq8xoy">Boundary knot method</a></li>
<li><a href="/facts/Boundary_particle_method/csI5W3xM">Boundary particle method</a></li>
<li><a href="/facts/Singular_boundary_method/7tm3TA7H">Singular boundary method</a></li>
<li><a href="/facts/Regularized_meshless_method/NsoqVJ6e">Regularized meshless method</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://web.archive.org/web/20130215105344/http://www.ccms.ac.cn/english/">International Center for Numerical Simulation Software in Engineering & Sciences</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">K. VD, A. MA, The method of functional equations for the approximate solution of certain boundary value problems, USSR Comput Math Math Phys. 4 (1964) 82–126. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">R. Mathon, R.L. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM Journal on Numerical Analysis. (1977) 638–650. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Z. Fu, W. Chen, W. Yang, Winkler plate bending problems by a truly boundary-only boundary particle method, Computational Mechanics. 44 (2009) 757–763. <a href="https://web.archive.org/web/20220508225201/http://em.hhu.edu.cn/chenwen/papers/softmatter/CompMech.pdf" target="_blank">https://web.archive.org/web/20220508225201/http://em.hhu.edu.cn/chenwen/papers/softmatter/CompMech.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">W. Chen, J. Lin, F. Wang, Regularized meshless method for nonhomogeneous problems Archived 2015-06-06 at the Wayback Machine, Engineering Analysis with Boundary Elements. 35 (2011) 253–257. <a href="http://em.hhu.edu.cn/chenwen/papers/softmatter/EABE-RMM.pdf" target="_blank">http://em.hhu.edu.cn/chenwen/papers/softmatter/EABE-RMM.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">W. Chen, F.Z. Wang, A method of fundamental solutions without fictitious boundary Archived 2015-06-06 at the Wayback Machine, Engineering Analysis with Boundary Elements. 34 (2010) 530–532. <a href="http://em.hhu.edu.cn/chenwen/papers/rbf/EABE-SBM.pdf" target="_blank">http://em.hhu.edu.cn/chenwen/papers/rbf/EABE-SBM.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">JIANG Xin-rong, CHEN Wen, Method of fundamental solution and boundary knot method for helmholtz equations: a comparative study, Chinese Journal of Computational Mechanics, 28:3(2011) 338–344 (in Chinese) <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">M.A. Golberg, C.S. Chen, The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations, Boundary Elements Communications. 5 (1994) 57–61. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">M. a. Golberg, C.S. Chen, H. Bowman, H. Power, Some comments on the use of Radial Basis Functions in the Dual Reciprocity Method, Computational Mechanics. 21 (1998) 141–148. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">C.M. Fan, C.S. Chen, J. Monroe, The method of fundamental solutions for solving convection-diffusion equations with variable coefficients, Advances in Applied Mathematics and Mechanics. 1 (2009) 215–230 <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Y.C. Hon, T. Wei, The method of fundamental solution for solving multidimensional inverse heat conduction problems, CMES Comput. Model. Eng. Sci. 7 (2005) 119–132 <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">A.K. G. Fairweather, The method of fundamental solutions for elliptic boundary value problems, Advances in Computational Mathematics. 9 (1998) 69–95. <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
</ol>

Method of fundamental solutions open-in-new

Method of fundamental solutions