Wikipedia comments: "In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in electrostatics, mechanical engineering and theoretical physics. Commonly used to model diffusion, it is named after the French mathematician, geometer, and physicist Simeon Denis Poisson."

∇^{2} V = - ρ / ε_{0}

Have a Poisson's equation to solve? Or any other math equations? For the next few months of 2014, we are willing to help you solve them using Calculus-level programming. To start, copy and modify the source code below in a file we'll call it {abc}{123}.fc where {abc} = your initials and {123} = any number or id; 8 characters max. for filename. Edit your {abc}{123}.fc file, especially lines starting with a "!" character.
E-mail us your {abc}{123}.fc file. We will compile & execute it and send you the output file. Try compiling you own code with our FC-Compiler program. Download and install FC-Compiler with more [~60] example problems to choose from. Learn how easy it works.

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User's Poisson Equation Source Code:

For 1-dimensional (1D) Poisson Equation use following:

global all
problem PoissonsPDE
C ------------------------------------------------------------------------
C --- Calculus Programming example: Poisson's Equation; a PDE (1D) Initial
C --- Value Problem solved using Method of Lines.
C ------------------------------------------------------------------------
C
C User parameters ...
! rho = ...
e0 = 8.854187817e-12 ! F/m or A^{2 }s^{4} kg^{-1}m^{−3} permittivity of free space
C
C x-parameter initial settings: x ==> i
xFinal= 1: xPrint = xFinal/20
C
call xAxis !
end ! Stmt.s not necessary in IVP, but used in BVP versions
model xAxis !
C ... Integrate over x-axis
C
x= 0: xPrt = xPrint: dx = xPrt / 10
! U = ??? ! @ x = 0 ... initial value
! Ux = ??? ! @ x = 0
Initiate janus; for PDE;
~ equations Uxx/Ux, Ux/U; of x; step dx; to xPrt
do while (x .lt. xFinal)
Integrate PDE; by janus
if( x .ge. xPrt) then
print 79, x, U, Ux, Uxx
xPrt = xPrt + xPrint
end if
end do
79 format( 1x, f8.4, 1x,10(g14.5, 2x))
end
model PDE ! Partial Differential Equation
Uxx = - rho/e0
end

For 2-dimensional (2D) Poisson Equation use following:

global all
problem PoissonsPDE
C ------------------------------------------------------------------------
C --- Calculus Programming example: Poisson's Equation; a PDE (2D) Initial
C --- Value Problem solved using Method of Lines.
C ------------------------------------------------------------------------
dynamic U, Ux, Uxx
C
C User parameters ...
! rho = ...
e0 = 8.854187817e-12 ! F/m or A^{2 }s^{4} kg^{-1}m^{−3} permittivity of free space
ipoints=20 ! grid pts. over x-axis
C
C x-parameter initial settings: x ==> i
xFinal = 1: ip = ipoints: xPrint = xFinal/20
allot U(ip), Ux(ip), Uxx(ip)
C
call xAxis !
end ! Stmt.s not necessary in IVP, but used in BVP versions
model xAxis !
C ... Integrate over x-axis
C
x= 0: xPrt = xPrint: dx = xPrt / 10
! U = ??? ! @ x = 0 ... initial value
! Ux = ??? ! @ x = 0
Initiate janus; for PDE;
~ equations Uxx/Ux, Ux/U; of x; step dx; to xPrt
do while (x .lt. xFinal)
Integrate PDE; by janus
if( x .ge. xPrt) then
print 79, x, U, Ux, Uxx
xPrt = xPrt + xPrint
end if
end do
79 format( 1x,f8.4,1x,20(g14.5,1x))
end
model PDE ! Partial Differential Equation
! U(1) = U0: Ux(1)=0: Uxx(1)=0 ! Initial Conditions
do 20 ij = 2, ipoints-1 ! System of ODEs
Uyy = (U(ij+1)-2*U(ij)+U(ij-1))/(dy*dy) !4 2nd order in 'y'
Uxx(ij)= - rho/e0 - Uyy ! Poisson's PDE
20 continue
! Ux(ip)= ???: Uxx(ip)= ??? ! Final conditions (ie. BC)
end

User's Poisson Equation Output:

selected output goes here ...

HTML code for linking to this page:

<a
href="http://goal-driven.net/math-problems/poisson-equation.html"><img style="float:left; width:100px"
src="http://goal-driven.net/image/fc-compiler-icon.png"/>
<strong>Poisson's (Partial Differential) Equation</strong>
</a>; Simulation to Optimization, Tweak Parameters for Optimal
Solution.