Boundary Value Problems (BVPs)
solved with
Calculus Programming
_________
Wikipedia comments: "Boundary value problems (BVPs) are similar to initial-value-problems (IVPs). A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value)."
This section shows how to solve equations of the following form:.
Uxx = f(x, U, Ux, ...)
This is a general form of a differential equation. To solve with the boundary values/conditions, Udesired & Uend, in Calculus programming we replace the following code for the "call xAxis" statement in these examples:
Udesired = 1.23e-3: Uend = 9.87e-3 ! Initial Conditions (BCs) error = 0: U0 = 1 find U0 In xAxis ooo to match error
plus, add some error statements thru out
ooo
error = error + (U - ???)**2
This 'find' statement will vary the 'U0' parameter until 'error' equals zero. Thus meeting your boundary conditions.
Enjoy learning Calculus-level Programming!
Example Boundary Value Problem Source
Code:
A Boundary Value Problem:
global all
problem nonLinPDE
C ------------------------------------------------------------------------
C --- Calculus Programming example: non-linear PDE (1D) Boundary
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: pi= 4*atan(1)
C
Udesired = 1.23e-3: Uend = 9.87e-3 ! Initial Conditions (BCs)
error = 0: U0 = 1
find U0; in xAxis; by mars; to match error
end
model xAxis
C ... Integrate over x-axis
C
x= 0: xPrt = xPrint: dx = xPrt / 10
Initiate janus; for PDE;
~ equations Uxx/Ux, Ux/U; of x; step dx; to xPrt
! Ux = ??? ! @ x = 0 ... Ux at x=0 to be found as a BC
U = U0
do while (x .lt. xFinal)
Integrate PDE; by janus
if( x .ge. xPrt) then
print 79, x, U, Ux, Uxx
xPrt = xPrt + xPrint
if( x .eq. ???) then
error = error + (U - Udesired)**2
end if
end if
end do
error = error + (U - Uend)**2
79 format( 1h , f8.4, 2x, 10(g14.5, 2x))
end
model PDE ! Partial Differential Equation
Uxx = -rho/e0 * (1.23 + sin( Ux * pi) - .543) ooo ! your Diff Eq goes here
end
Example Boundary Value Problem Output:
selected output goes here ...
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