# telegraph equation

Both the electric voltage and the satisfy the

 $\displaystyle f_{xx}^{\prime\prime}-af_{tt}^{\prime\prime}-bf_{t}^{\prime}-cf=0,$ (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  $f(x,\,0)=f_{t}^{\prime}(x,\,0)=0$  and the boundary conditions$f(0,\,t)=g(t)$,  $f(\infty,\,t)=0$,  then the Laplace transform of the solution function$f(x,\,t)$  is

 $\displaystyle F(x,\,s)=G(s)e^{-x\sqrt{as^{2}+bs+c}}.$ (2)

In the special case  $b^{2}-4ac=0$,  the solution is

 $\displaystyle f(x,\,t)=e^{-\frac{bx}{2\sqrt{a}}}g(t-x\sqrt{a})H(t-x\sqrt{a}).$ (3)

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

 $F_{xx}^{\prime\prime}(x,\,s)-a[s^{2}F(x,\,s)-sf(x,\,0)-f_{t}^{\prime}(x,\,0)]-% b[sF(x,\,s)-f(x,\,0)]-cF(x,\,s)=0,$

which due to the initial conditions simplifies to

 $F_{xx}^{\prime\prime}(x,\,s)=(\underbrace{as^{2}+bs+c}_{K^{2}})F(x,\,s).$

The solution of this ordinary differential equation is

 $F(x,\,s)=C_{1}e^{Kx}+C_{2}e^{-Kx}.$

Using the latter boundary condition, we see that

 $F(\infty,\,s)=\int_{0}^{\infty}e^{-st}f(\infty,\,t)\,dt\equiv 0,$

whence  $C_{1}=0$.  Thus the former boundary condition implies

 $C_{2}=F(0,\,s)=\mathcal{L}\{g(t)\}=G(s).$

So we obtain the equation (2).

Justification of (3).  When the discriminant (http://planetmath.org/QuadraticFormula) of the quadratic equation  $as^{2}\!+\!bs\!+\!c=0$  vanishes, the roots (http://planetmath.org/Equation) coincide to  $s=-\frac{b}{2a}$,  and  $as^{2}\!+\!bs\!+\!c=a(s+\frac{b}{2a})^{2}$.  Therefore (2) reads

 $F(x,\,s)=G(s)a^{-x\sqrt{a}(s+\frac{b}{2a})}=e^{-\frac{bx}{2\sqrt{a}}}e^{-x% \sqrt{a}s}G(s).$

According to the delay theorem, we have

 $\mathcal{L}^{-1}\{e^{-ks}G(s)\}=g(t-k)H(t-k).$

Thus we obtain for $\mathcal{L}^{-1}\{F(x,\,s)\}$ 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