Wikipedia comments: "The telegrapher's equations (or just telegraph equations) are a pair of linear differential equations which describe the voltage and current on an electrical transmission line with distance and time. The equations come from Oliver Heaviside who in the 1880s developed the transmission line model, which is described in this article. The model demonstrates that the electromagnetic waves can be reflected on the wire, and that wave patterns can appear along the line. The theory applies to transmission lines of all frequencies including high-frequency transmission lines (such as telegraph wires and radio frequency conductors), audio frequency (such as telephone lines), low frequency (such as power lines) and direct current."
c2 ∇2
U = Utt + (α + β) Ut + αβU
Do you have a Telegrapher'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 Telegrapher's Equation Source Code:
For 1-Dimensional Telegrapher's Equation use following:
global all
problem TelegraphPDE
C ------------------------------------------------------------------------
C --- Calculus Programming example: Telegrapher's Equation; a PDE Initial
C --- Value Problem solved using Method of Lines.
C ------------------------------------------------------------------------
dynamic U, Ut, Utt
C
C User parameters ...
! alpha = ...
! beta = ...
! c = ...
! ipoints = 20 ! grid pts. over x-axis
! tFinal = 1 ! final time
C
C x-parameter initial settings: x ==> i
pi= 4*atan(1): ip = ipoints: dx = xFinal/ipoints
C
C t-parameter initial settings: t ==> m
! tFinal = 1: tPrint = tFinal/20
allot U( ip), Ut( ip), Utt( ip)
C
call tAxis !
end ! Stmt.s not necessary in IVP, but used in BVP version
model tAxis !
C ... Integrate over t-axis
C
t= 0: tPrt = tPrint: dt = tPrt / 10
Initiate gemini; for PDE;
~ equations Utt/Ut, Ut/U; of t; step dt; to tPrt
do while (t .lt. tFinal)
Integrate PDE; by gemini
if( t .ge. tPrt) then
print 79, t, (U(ii), ii = 1, ip)
tPrt = tPrt + tPrint
end if
end do
79 format( 1x,f8.4,20(g14.5, 1x))
end
model PDE ! Partial Differential Equation
C ! Method of Lines
! U(1)=U0(t): Ut(1)=0: Utt(1)=0 ! Initial Conditions
do 20 ii = 2, ipoints-1 ! System of ODEs
Uxx = (U(ii+1)-2*U(ii)+U(ii-1))/(dx*dx) !4 2nd order in 'x'
Utt(ii)= c**2 * Uxx - (alpha+beta)* Ut - alpha*beta*U
20 continue
! Ut(ip)= ???: Utt(ip)= ??? ! Final Conditions, if any
end
Fmodel U0(xx) ! Initial starting values @ t = 0
! if( xx .le. 0) then
! U0 = 0
! elseif( xx .lt. .5 ) then
! U0 = ... f(xx)
else
! U0 = 0
endif
end
User's Telegrapher's Equation Output:
selected output goes here ...
HTML code for linking to this page:
<a
href="http://goal-driven.net/math-problems/telegraph-equation.html"><img style="float:left; width:100px" src="http://goal-driven.net/image/fc-compiler-icon.png"/> <strong>Telegrapher's Equation</strong> </a>; Simulation to Optimization, Tweak Parameters for Optimal Solution.
