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A.4 Derivation of the solution for a one-dimensional infinite cable
Starting with the differential equation
V + ^∂v 1 ∂2v _ .
rm + ∂t ra ∂x2 ^l
to find the impulse response (Green’s function) use an impulsive input i in space and time
V ∂v 1 ∂2v
-+c^~-^=δ^δ^
rm ∂t ra OX2
Take the Fourier transform with respect to space
( V ∂v 1 ∂2v 1
I rm ∂t ra ∂x2 J
V ∂v 1 2 1 1
— + C—----(jω) V— = δ(t) .—
rm ∂t ra ra √27Γ
∂υ / 1
∂t ∖rm
,2
1
z2π
Solve first order ODE with respect to time by finding integrating factor μ, multiplying both
sides by μ and integrating
Z1 Z 1 ω2∖
μ = exp — — + — i
∖ C ∖ τm τa /
1
Tm
,2
1 / X Z 1 ʌ Z ω2
u ---=u{t) exp —11 exp ——t
CV2^ ∖ Crrn ) ∖Cra
Take the inverse Fourier transform
/ ω2
raC / Cra 2
---exp —■—x
2t ∖ 4t
according to Wolfram Mathematica, so...
'v
∖ Z Cra 9
~t)exp("^4Γx