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Volterra equation, asymptotic behaviour

Dear all,

I want to solve the Volterra integral equation (of 2nd kind). But I only need to solve it analytically for large times "tau", i.e. I only need the asymptotic behaviour as "tau -> infinity".

By simple algebra, I obtain an approximative analytical expression in this limit. However, this expression is not in agreement with the exact numerical solution of the relevant Volterra equation. Please see PDF-file attached!

Can anyone help me finding a right asymptotic expression for "c(tau)"?

I appreciate any feedback!

Al the best,

perr

PS:

I have also tried to solve this applying Laplace transformation. However, I'm not sure that this method will help me here: Suppose that the solution "c(tau)" scales as "1/tau", I am not sure how to get a "1/tau" term from Laplace transform analysis since the Laplace transform of "1/t" diverges.