Wikipedia comments: "The wave equation is an important second-order linear partial differential equation for the description of waves - as they occur in physics - such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics. Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange."
Do you have a Wave 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.
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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.
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User's Wave Equation Source Code:
For 1-Dimensional Wave Equation use following:
global all
problem WavePDE
C ------------------------------------------------------------------------
C --- Calculus Programming example: Wave Equation; a PDE Initial
C --- Value Problem solved using Method of Lines.
C ------------------------------------------------------------------------
dynamic U, Ut, Utt
C
C User parameters ...
! v = ??? ! velocity?
! ipoints = 21 ! grid pts. over x-axis
! tFinal = 1 ! final time
C
C x-parameter initial settings: x ==> i
! xFinal = 1: ip = ipoints: dx= xFinal/(ipoints-1)
C
C t-parameter initial settings: t ==> m
pi = 4*atan(1): tPrint=tFinal/ip
allot U(ip), Ut(ip), Utt(ip)
C
call tAxis !
end ! Stmt.s not necessary in IVP, but used in BVP versions
model tAxis !
C ... Integrate over t-axis
C
t= 0: tPrt = tPrint: dt = tPrt / 10
Initiate janus; for PDE;
~ equations Utt/Ut, Ut/U; of t; step dt; to tPrt
do while (t .lt. tFinal)
Integrate PDE; by janus
if( t .ge. tPrt) then
print 79, t, (U( ij), ij = 1, ip)
tPrt = tPrt + tPrint
end if
end do
79 format( 1h , f8.4, 3x, 20(g14.5, 1x))
end
model PDE ! Partial Differential Equation
C ! Method of Lines
! U(1) = U0: Ut(1)=0: Utt(1)=0 ! Initial Conditions
do 20 ij = 2, ipoints-1 ! System of ODEs
Uxx = (U(ij+1)-2*U(ij)+U(ij-1))/(dx*dx) !4 2nd order in 'x'
Utt(ij)= v**2 * Uxx ! Wave equation
20 continue
! Ut(ip)= ???: Utt(ip)= ??? ! Final conditions
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 Wave Equation Output:
selected output goes here ...
HTML code for linking to this page:
<a
href="http://goal-driven.net/math-problems/wave-equation.html"><img style="float:left; width:100px" src="http://goal-driven.net/image/fc-compiler-icon.png"/> <strong>Wave Equation</strong> </a>; Simulation to Optimization, Tweak Parameters for Optimal Solution.