telegraph equation
Both the electric voltage and the satisfy the telegraph equation
f′′xx-af′′tt-bf′t-cf=0, | (1) |
where x is distance, t is time and a,b,c are non-negative constants. The equation is a generalised form of the wave equation.
If the initial conditions are f(x, 0)=f′t(x, 0)=0 and the boundary conditions f(0,t)=g(t), f(∞,t)=0, then the Laplace transform
of the solution function
f(x,t) is
F(x,s)=G(s)e-x√as2+bs+c. | (2) |
In the special case b2-4ac=0, the solution is
f(x,t)=e-bx2√ag(t-x√a)H(t-x√a). | (3) |
Justification of (2). Transforming the partial differential equation (1) (x may be regarded as a parametre) gives
F′′xx(x,s)-a[s2F(x,s)-sf(x, 0)-f′t(x, 0)]-b[sF(x,s)-f(x, 0)]-cF(x,s)=0, |
which due to the initial conditions simplifies to
F′′xx(x,s)=(as2+bs+c⏟K2)F(x,s). |
The solution of this ordinary differential equation is
F(x,s)=C1eKx+C2e-Kx. |
Using the latter boundary condition, we see that
F(∞,s)=∫∞0e-stf(∞,t)𝑑t≡0, |
whence C1=0. Thus the former boundary condition implies
C2=F(0,s)=ℒ{g(t)}=G(s). |
So we obtain the equation (2).
Justification of (3). When the discriminant (http://planetmath.org/QuadraticFormula) of the quadratic equation as2+bs+c=0 vanishes, the roots (http://planetmath.org/Equation) coincide to s=-b2a, and as2+bs+c=a(s+b2a)2. Therefore (2) reads
F(x,s)=G(s)a-x√a(s+b2a)=e-bx2√ae-x√asG(s). |
According to the delay theorem, we have
ℒ-1{e-ksG(s)}=g(t-k)H(t-k). |
Thus we obtain for ℒ-1{F(x,s)} the expression of (3).
Title | telegraph equation |
Canonical name | TelegraphEquation |
Date of creation | 2013-03-22 18:03:15 |
Last modified on | 2013-03-22 18:03:15 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 13 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 35L15 |
Classification | msc 35L20 |
Synonym | telegrapher’s equation |
Related topic | HeavisideStepFunction |
Related topic | SecondOrderLinearDifferentialEquation |
Related topic | DelayTheorem |
Related topic | MellinsInverseFormula |
Related topic | TableOfLaplaceTransforms |