@Preben Alsholm  Yes I have the same problem, I am sure my equations are correct. I think my inverse laplace transform calculations are not correct. That is why I wanted not to share my code. Because I know that is not the way to compute.

@Preben Alsholm  sorry  for that, I just shared my worksheet.

@Axel Vogt Thank you, I realised my mistake. Now I am sure I have the correct roots.

@Carl Love  Thanks for your remark!

@Axel Vogt  Edit: only 0. + 0. I and   0.0388684654358756 + 0.0388684654358756 I  are satisfying the equation (almost zero). The interesting thing is I use these initial 5 roots in my previous post (see the following link):

http://www.mapleprimes.com/questions/216033-How-Do-I-Plot-A-Solution-With-Series

Only those 3 roots give me the  mode 1,2 and 3, but they do not satisfy the eq. The rest does not converge. I do not understand...Either roots are missing or the codes in previous post are wrong!

@Axel Vogt Thank you for detailed answer. I found this approach is really interesting. I would like to consider the first 5 roots. That is why, I just played around with your code. I gave different values to j. I obtain the following values:

             j=   -1 -> beta=  0.0186050075840613 + 0.395348060843754 I,     0. + 0. I, 

                                    0.0388684654358756 + 0.0388684654358756 I,     0.0336721576087020 + 0.718340271278585 I

             j=   0 -> beta=   0.395348060843754 + 0.0186050075840608 I
     

             j=  1 -> beta=   0.718340271278585 + 0.0336721576087003 I

             j=   2 -> beta=   0.99999998630970 + 0.99999998630970 I,       1.04526212469817 + 0.0428592715037274 I

            j=   3 -> beta= 1.37292217295828 + 0.0457415918100320 I

 
            j=   4 -> beta= 1.69789407914236 + 0.0441555684962661 I

When I substitute 0.395348060843754 + 0.0186050075840608 I,    0.718340271278585 + 0.0336721576087003 I,   and   1.04526212469817 + 0.0428592715037274 I into the equation. I do not see those roots  satisfy the eq=0.  Should I neglect them?
 

@Carl Love Thank! I will check my other calculations then

@Axel Vogt May you share the worksheet, please?

@Carl Love Hi! From your previous comments to other similar questions, I already apply RootFinding:-Analytic. Check the worksheet finalroots.mw . I compare the positive real part of the roots with my analytic roots, which you can see in the worksheet. Initial roots which is less than 2 are not so similar with analytic roots. That is why I think that I might miss some roots. So I decided to solve a nonlinear system. I think that way is also difficult to apply.

@one man  I could not plot it, do I need additional package for it? fsolve gives only one root, which I need others as well.

@tomleslie  I got it! You saved my time, thanks! I agree with you about B-list. B should be the derivative of beta, but beta is not in p form anymore. I need to find it out how to rewrite B-list. Thanks again!

@tomleslie  Now it is more clear for me, thank you! I have one more question! I realized I made a mistake with defining beta function. P:=P1+i*P2 and beta:=beta1+i*beta2 for P1,P2, beta1 and beta2 are element of Real Numbers. P=i*beta^2. How can I define beta function in the maple code ? The beta definition in the maple file gives negative beta values. All beta values should be positive. By solving nonlinear system, I obtain the following beta definition. I cannot write beta in only p form anymore. Can I define P1 and P2 vectors like P:=[] before and write beta in seq form?

@vv  I knew my codes were not efficient. Thanks for your comments!

@tomleslie Thank you for your help! At least I understood how I can write my series functions. Due to B (derivate of beta), we have division by zero problem. I decided to substitue beta function into U(x,t) and rewriting with respect to Pn. I think I was confused a bit with [j+1] and [j] in the last second equation. I played around with your file. May you have a look the following file, why it does not work again? Thanks!

plot3.mw

@Preben Alsholm Thank you for your comment. It is a typo. It should be t, I edited the file and reloaded.