User's Poisson Equation


Poisson's Equation
_________

∇2 V = - ρ
ε0
Have a Poisson's equation to solve? Or any other math equations? For the first few months of 2009, we are willing to help you solve them using Calculus-level programming. (Fortran Calculus is scheduled to be online in mid-2009 so take advantage of this free offer will it last.) 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. (For more [~60] example problems to choose from, download FC-Win program.)
All "!" characters in columns 1 or 2 must be deleted before compiling. These "!" were added in order to point out areas needing work.
Next, you may modify a copy of this web page and send it to us for viewing. If accepted, we will post your webpage showing your problem with solution. If you want people to be able to contact you, please include your e-mail address on this web page.
Please mention our www.digitalCalculus.com website to others. Thanks!

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 A2 s4 kg-1m−3 permittivity of free space
!        ipoints = 10          ! grid pts. over x-axis
C
C x-parameter initial settings: x ==> i
!        xFinal =  1:    xPrint = xFinal/ipoints:       ip = ipoints
        pi= 4*atan(1)
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( 1h , f8.4, /2x, 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 A2 s4 kg-1m−3 permittivity of free space
!        ipoints = 10          ! grid pts. over x-axis
!        jpoints = 64          ! grid pts. over y-axis
!        tFinal =  1           ! final time
C
C x-parameter initial settings: x ==> i
!        xFinal =  1:    xPrint = xFinal/ipoints:       ip = ipoints
C
C y-parameter initial settings: y ==> j
!        yFinal =  1:    dy = yFinal/(jpoints-1):       jp = jpoints
        pi= 4*atan(1)
        allot U( jp), Ux( jp), Uxx( jp)
C
        call xAxis      !
      end               ! Stmt.s not necessary in IVP, but used in BVP version
      model xAxis       !
C ... Integrate over x-axis

C settings at x = 0
        do 10 jj = 1, jpoints   ! Initial Conditions, if any
          yPrt = jj * yPrint
!          U(jj) = U0(yPrt):      Ux(jj) = ???:     Uxx(jj) = ???
 10     continue

        x= 0:	xPrt = xPrint:	dx = xPrt / 10
        Initiate merlin;  for PDE;
     ~       equations Uxx/Ux, Ux/U;  of x;  step dx;  to xPrt
        do while (x .lt. xFinal)
          Integrate PDE;  by merlin
          if( x .ge. xPrt) then
            print 79, x, (U(jj), jj = 1, jp)
            xPrt = xPrt + xPrint
          end if
        end do
 79     format( 1h , f8.4, /2x, 10(g14.5, 1x))
      end
      model PDE                         ! Partial Differential Equation
C                                       ! Method of Lines
!        Ux(1)= ???:      Uxx(1)= ???   ! Initial Conditions, if any
        do 40 jj = 2, jpoints-1         ! System of ODEs
          Uyy = (U(jj+1)- 2*U(jj) + U(jj-1))/dy**2
          Uxx(jj)= - rho/e0 - Uyy
 40     continue
      end
      Fmodel U0(yy)       ! Initial starting values @ x = 0
        if( yy .le. 0) then
          U0 = 0
        elseif( yy .lt. .5 ) then
          U0 = (1 - cos( 4 * pi * yy))/2
        else
          U0 = 0
        endif
      end

For 3-dimensional (3D) Poisson Equation use following:

      global all
      problem PoissonsPDE
C ------------------------------------------------------------------------
C --- Calculus Programming example: Poisson's Equation; a PDE (3D) 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 A2 s4 kg-1m−3 permittivity of free space
!        ipoints = 10          ! grid pts. over x-axis
!        jpoints = 50          ! grid pts. over y-axis
!        kpoints = 10          ! grid pts. over z-axis
C
C x-parameter initial settings: x ==> i
!        xFinal =  1:    xPrint = xFinal/ipoints:       ip = ipoints
C
C y-parameter initial settings: y ==> j
!        yFinal =  1:    dy = yFinal/(jpoints-1):       jp = jpoints
C
C z-parameter initial settings: z ==> k
!        zFinal =  1:    dz = zFinal/(kpoints-1):       kp = kpoints
        pi= 4*atan(1)
        allot U(jp,kp), Ux(jp,kp), Uxx(jp,kp)
C
        call xAxis      !
      end               ! Stmt.s not necessary in IVP, but used in BVP version
      model xAxis       !
C ... Integrate over x-axis

C settings at x = 0
        do 10 kk = 1, kpoints	! initial values
          zPrt = kk * zPrint:		jj = 1
!          U(jj,kk) = U0(zPrt):      Ux(jj,kk) = ???:     Uxx(jj,kk) = ???
 10     continue

        x= 0:	  xPrt = xPrint:	dx = xPrt / 10
        Initiate neptune;  for PDE;
     ~       equations Uxx/Ux, Ux/U;  of x;  step dx;  to xPrt
        do while (x .lt. xFinal)
          Integrate PDE;  by neptune
          if( x .ge. xPrt) then
            do 30 jj = 2, jpoints
	      y = (jj - 1) * dy
              print 79, x, y, (U(jj,kk), kk = 1, kp)
 30         continue
            xPrt = xPrt + xPrint
          end if
        end do
 79     format( 1h , f8.4, 1x, f8.4, /2x, 10(g14.5, 1x))
      end
      model PDE                         ! Partial Differential Equation
C                                       ! Method of Lines
!        Ux(1,1)= ???:  Uxx(1,1)= ???   ! Initial Conditions, if any
        do 40 jj = 2, jpoints-1         ! System of ODEs
	  y = (jj - 1) * dy
          do 20 kk = 2, kpoints-1
	    z = (kk - 1) * dz
            Uyy = (U(jj+1,kk)- 2*U(jj,kk) + U(jj-1,kk))/dy**2
            Uzz = (U(jj,kk+1)- 2*U(jj,kk) + U(jj,kk-1))/dz**2
            Uxx(jj)= - rho/e0 - Uyy - Uzz
 20       continue
!          Ux(jj,kp)= ???:  Uxx(jj,kp)= ???   ! Initial Conditions, if any
 40     continue
!        Ux(jp,kp)= ???:    Uxx(jp,kp)= ???   ! Initial Conditions, if any
      end
      Fmodel U0(zz)       ! Initial starting values @ x = 0 & y = 0
!        if( zz .le. 0) then
!          U0 = 0
!        elseif( zz .lt. .5 ) then
!          U0 = (1 - cos( 4 * pi * zz))/2
        else
!          U0 = 0
        endif
      end

User's Poisson Equation Output: 

selected output goes here ...
Visit ODEcalc for an Ordinary Differential Equations Calculator. Try it and see the power of Calculus programming.
HTML code for linking to this page:
<a href="http://www.digitalcalculus.com/math-problems/poisson-equation.html"><img align="middle" width="100" src="http://www.digitalcalculus.com/image/fc-win-icon.gif"/> <strong>Poisson's (Partial Differential) Equation</strong> </a>; Simulation to Optimization, Tweak Parameters for Optimal Solution.

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