@tomleslie

I thank you for your comments and answering to my question.

Unfortunately I didn't write the fourth boundary conditions for you.
The four boundary conditions are in here:

V(r1,z)= V(r2,z)=0
V(r,0)= V(r,b)=0

and this point that the coordinates are

  r1 < r <r2   &    0< z < b

And I want to tell you I don't have the Matlab Code I just olnly the final solution attached in the Maple primes websie.

@Carl Love 

Thanks a lot Mr. Carl.

God bless you

@tomleslie 

Ok, thanks. But I thought Maple could give the right answer instead of  F_1(x+t)+ F_2(t-x)+ cos(x).

Is the answer correct???

Unfortunately, I could not solve it analytically, so it was not possible to me to compare the numerical and Exact solutions too!!!

@Carl Love 

Hi dear,

Thank you. Now I have another question:

Download SolvePDE.mwSolvePDE.mw

Thanks a lot. I got it.

@vv 

I got it...

good job dear

@vv

Excuse me dear,

Could you explain what do the boundary conditions mean??? for example one can tell the function is finite at zero, so we have: f ' (0)=0!!!

boundary conditions:

f(0)= 1;   f ' (0)= 0...  

HeunT.mwThanks

Is there anything related between the HeunT function and Hypergeometric function that I shud consider?? The Hypergeometric function is complicated for me to

HeunT(alpha, beta, gamma, z) could be expanded in the series solution form,

I have a question: does the HeunT function have singular point??? at infinity?!

I calculated the   HeunT(alpha, beta, gamma, z)  in different values of the 4 parameters... I did not understand what are the singularities points for the HeunT function???

@tomleslie 

Hi, I thank you dear,

The equation which must be solved is similar to this:

b* x^3 + x -1= 0,  in which x is equal to (Un/N) and  ( ( 8* (3+j)* K^2)/ (5+j) )*e* De^2 is defined by b for convenient.

Now we must solve that equation so that we can reach to the solution which is wrote in the third line. The fourth line introduces the parameter  delta in which two other parameters are involved, namely beta and alpha...

Please note that our desire solution is the third line.

@tomleslie 

I thank you, but I did not understand your notifications... I just want to recover the solution given by the picture for the U_n/ U  expression.

@vv 

Ok,  I got it

God bless you

Hi,

 Thanks a lot. But I coudn't do that.

What is P in

sort( collect(p,Nu,expand))

@vv 

Thank you, but it is not the same that I want. I want to have the results in this form:

()*Nu^7+ ()*Nu^6+... +()*Nu+1=0

)-:

I got it :-)

Thanks a lot dears..

God bless you

CorrectedHeunTPrime.mw   

@Carl Love 

Sorry, I noticed what's your mention dear,

I correct the error, but I cannot plot it...

I attached the corrected one again