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131

A.4 Derivation of the solution for a one-dimensional infinite cable

Starting with the differential equation

V ₊ ^∂v 1 ∂²v _ .
r_m ⁺ ∂t r_a ∂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^
r_m    ∂t r_a OX²

Take the Fourier transform with respect to space

( V     ∂v 1 ∂²v 1

I r_m     ∂t r_a ∂x² J

V ∂v 1        ₂ 1          1

— + C—----(jω) V— = δ(t) .—

r_m ∂t r_a         r_a √27Γ

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 ∖ τ_m τ_a /

1

Tm

,2

1     / X Z 1 ʌ Z ω²

u  ---=u{t) exp   —11 exp ——t

CV2^ ∖ Cr_rn ) ∖Cr_a

Take the inverse Fourier transform

r_aC    / Cr_{a 2}

---exp —■—x

2t     ∖ 4t

according to Wolfram Mathematica, so...

∖ Z Cr_{a 9}
~^t)^exp("^4Γ^x

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