On my 4 years  very cheap old laptop on Maple 2018(Windows),take  the first evaulation about 4 minutes and 34 second.

SUM1_vers.mw

@tomleslie

Nice answer ,one vote from me. :) 

What do you mean by Laplace ?

@Ridz 

If I answered correctly to yours question please give me thumbs-up (upvote) ? 

@Ridz 

The_Hard_Spring_by_Maple_2018_ver_2.mw

Answer you can find in the book: 

Nonlinear Physics with Maple for Scientists and Engineers Richard H. Enns, George McGuire 2001 year on page 182.

Book you can find: https://books.google.com.

You want to solve PDE or ODE? Decide?

https://www.mapleprimes.com/questions/202895-3-Parameter-ODE-Bifurcation-Values

conjugate(x) = abs(x)^2/x

@Preben Alsholm 

Thanks for info.Yes I'm also got lost.

Regards,MI

@9009134

I used the talbot method or another methods for numerically inverse laplace transform but i gain different result.!!!

And I do not know why this is happening !

Strange, mysterious things happen. 

FracPDE_vers_3.mw

@Preben Alsholm 

Yes I made a mistake.Thanks.

@9009134 

I don't know why Talbot method fail in yours case.I not a expert. Another  numeric Inverse Laplace method you can find here:https://www.google.com

@9009134 

It's all correct,but in your case for which the Talbot method fails. Only for short time we obtain a good approximation.

@9009134 

I have Maple 2018 on Windows 8.1 64 bit.

I don't remember refrence I only know a method:STEHFEST

Only found:

http://www.esd.mun.ca/~farq/PDFs/JianhuiLi_SEG_2015.pdf

http://www.math.yorku.ca/~akuznets/Gaver_Stehfest.pdf

AN IMPROVED TALBOT METHOD FOR NUMERICAL LAPLACE TRANSFORM INVERSION

from here:

https://arxiv.org/pdf/1304.2505.pdf