An Implicit Differential Equation Example:

Implicit Mathematical Problems
solved with
Calculus Programming

This section shows how to solve equations of the following form:.

Uxx = f(x, U, Ux, Uxx, ...)

This is a general form of an implicit differential equation. To solve for Uxx in Calculus programming we add the following code right after "model PDE" statement in these examples:

      find Uxx; In ImplicitDE;     ooo     to match error
   end
   model ImplicitDE
     error = Uxx - f(x, U, Ux, Uxx, ooo)
     error = error**2

This 'find' statement will vary its parameters (ie. Uxx) until 'error' equals zero. Thus you will have a value where

Uxx = f(x, U, Ux, Uxx, ooo)

Enjoy learning Calculus-level Programming!

An Implicit Differential Equation Example Source Code:



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




      global all
      problem PoissonsPDE
C ------------------------------------------------------------------------
C --- Calculus Programming example: Poisson's Equation; a PDE (1D) Initial
C --- Value Problem solved.
C ------------------------------------------------------------------------
C
C User parameters ...
!       rho = ...
        e0 = 8.854187817e-12 ! F/m or A2 s4 kg-1m−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
        Uxx = .5		! Initial value ... used in 'find' stmt. below
!       U = ??? ! @ 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, 1x, 10(g14.5, 1x))
      end
      model PDE                         ! Partial Differential Equation
        find Uxx;   in ImplicitDE;    by ajax;   to match error
      end
      model ImplicitDE
        error = (Uxx + rho/e0) * (1.23 + sin( Uxx * pi) - .543) ooo
        error = error**2
      end

An Implicit Differential Equation Example Output:



 o o o

--- AJAX SUMMARY, INVOKED AT ooo FOR MODEL ImplicitDE ----              


   CONVERGENCE CONDITION AFTER  1 ITERATIONS
      UNKNOWNS CONVERGED
      CONSTRAINTS SATISFIED
      ALL SPECIFIED CRITERIA SATISFIED


 LOOP NUMBER .........   [INITIAL]         1
 UNKNOWNS
   Uxx                -5.000000E-01 -1.644488E-01
 CONSTRAINTS
   Error               1.049682E+00  6.404924E-17


---END OF LOOP SUMMARY

 o o o
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