User's Telegrapher's Equation

Telegrapher's Equation
_________

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."

c22 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.

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.

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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://fortranCalculus.info/math-problems/telegraph-equation.html"><img style="float:left; width:100px" src="http://fortranCalculus.info/image/fc-compiler-icon.png"/> <strong>Telegrapher's Equation</strong> </a>; Simulation to Optimization, Tweak Parameters for Optimal Solution.

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