telegraph equation

Both the electric voltage and the satisfy the telegraph equation

$f_{xx}^{\prime \prime }-a{f}_{tt}^{\prime \prime }-b{f}_t^{\prime }-cf=0,$(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  $f(x,\mathrm{\hspace{0.17em}0})=f_t^{\prime }(x,\mathrm{\hspace{0.17em}0})=0$  and the boundary conditions  $f(0,t)=g(t)$,  $f(\mathrm{\infty },t)=0$,  then the Laplace transform of the solution function  $f(x,t)$  is

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

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

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

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

$$F_{xx}^{\prime \prime }(x,s)-a[s^{2}F(x,s)-sf(x,\mathrm{\hspace{0.17em}0})-f_t^{\prime }(x,\mathrm{\hspace{0.17em}0})]-b[sF(x,s)-f(x,\mathrm{\hspace{0.17em}0})]-cF(x,s)=0,$$

which due to the initial conditions simplifies to

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

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

$$F(\mathrm{\infty },s)={\int }_0^{\mathrm{\infty }}{e}^{-st}f(\mathrm{\infty },t)𝑑t\equiv 0,$$

whence  $C_1=0$.  Thus the former boundary condition implies

$$C_2=F(0,s)=ℒ\{g(t)\}=G(s).$$

So we obtain the equation (2).

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

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

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