       InverseLaplaceTransform

• To: mathgroup at smc.vnet.net
• Subject: [mg29576] InverseLaplaceTransform
• From: "Toshiyuki \(Toshi\) Meshii" <meshii at mech.fukui-u.ac.jp>
• Date: Wed, 27 Jun 2001 05:12:14 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Hello,

I am trying to obtain an inverse-Laplace transformation solution, which is
known to exist.
That is,

A = (h*Tf*BesselK[1, b*q])/(s*BesselI[1, b*q]*(h*BesselK[0, a*q] +
q*BesselK[1, a*q]) + s*(h*BesselI[0, a*q] - q*BesselI[1, a*q])*
BesselK[1, b*q]); B = (h*Tf*BesselI[1, b*q])/
(s*BesselI[1, b*q]*(h*BesselK[0, a*q] + q*BesselK[1, a*q]) +
s*(h*BesselI[0, a*q] - q*BesselI[1, a*q])*BesselK[1, b*q]);

q = Sqrt[s/k];

TT[r_] = A*BesselI[0, q*r] + B*BesselK[0, q*r];

InverseLaplaceTransform[TT[r], s, t]

where 0<a<r<b and constants k, h and Tf are positive.
Mathematica seems not to be able to obtain the problem directly.
Does anyone know a smart way to handle the problem?

It is known that the integral for inverse transform has a pole at
s = 0
s = -k * xn^2
where +xn and -xn are all real and simple roots of

(x*BesselJ(1, x*a) + h*BesselJ(0, x*a) )* BesselY(1, x*b)
== (x*BesselY(1, x*a) + h*BesselY(0, x*a) )* BesselJ(1, x*b)

and the residues exist.

-Toshi

```

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