telegraph equation

Both the electric voltage and the satisfy the telegraph equationMathworldPlanetmath

fxx′′-aftt′′-bft-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 equationMathworldPlanetmath.

If the initial conditionsMathworldPlanetmath are  f(x, 0)=ft(x, 0)=0  and the boundary conditionsf(0,t)=g(t),  f(,t)=0,  then the Laplace transformDlmfMathworldPlanetmath of the solution functionMathworldPlanetmathf(x,t)  is

F(x,s)=G(s)e-xas2+bs+c. (2)

In the special case  b2-4ac=0,  the solution is

f(x,t)=e-bx2ag(t-xa)H(t-xa). (3)

Justification of (2).  Transforming the partial differential equationMathworldPlanetmath (1) (x may be regarded as a parametre) gives

Fxx′′(x,s)-a[s2F(x,s)-sf(x, 0)-ft(x, 0)]-b[sF(x,s)-f(x, 0)]-cF(x,s)=0,

which due to the initial conditions simplifies to


The solution of this ordinary differential equation is


Using the latter boundary condition, we see that


whence  C1=0.  Thus the former boundary condition implies


So we obtain the equation (2).

Justification of (3).  When the discriminant ( of the quadratic equation  as2+bs+c=0  vanishes, the roots ( coincide to  s=-b2a,  and  as2+bs+c=a(s+b2a)2.  Therefore (2) reads


According to the delay theorem, we have


Thus we obtain for -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