@Axel Vogt  The basic ideas for writing this procedure, I found here 

@Markiyan Hirnyk   Оbviously  999999999999999999990 = 10^21-10  rather than  10^20+10^20-10

With Maple:

is(999999999999999999990 = 10^21-10);

is(20*9^20 > 10^21-10);

                          true

                          false

@Carl Love  Thanks. Yes you are right. I did not notice that   _Z6  and  _Z7  can be nonpositive. Therefore, solutions will be more than 106141495. I will think how to fix this situation.

@ramakrishnan   The latter calculation  5!/(5-1-4)! * 4!/(1 * 4)! * 1/5^4 *1/2 ^2  is wrong.

Must be  

5!/(5-0-2)! * 4!/(1 * 2)! * 1/5^4 *1/2^2;

@Markiyan Hirnyk   All right. Make restart  before substitution:

restart;

eval(30*a+75*b+110*c+85*d+255*e+160*f+15*g+12*h+120*i=8000, [a = 2, b = 4, c = 6, d = 5, e = 16, f = 9, g = 1, h = 5, i = 8]);

                                                                    8000 = 8000

@Carl Love 

M:=(N::nonnegint, i::nonnegint, alpha) -> Matrix(N+1, (k,j)->`if`(j=k+i,1/GAMMA((k-1)*alpha+1), 0)):

@Rouben Rostamian   I did in the classic interface Maple 12.

@acer  with rotation. I just do not understand this line

A:=plots:-display(plottools:-rotate(plottools:-transform((x,y,z)->[y,z,x])(A),0,0,Pi/2));

We can simply write

A:=plottools:-rotate(A,0,-Pi/2,0);

The result is the same.

@mostibra  For example:

plot(eval(U(x,t),t=2), x=0..10, color=red, thickness=2);

@Markiyan Hirnyk  Instead of commenting on the merits, you are asking very strange questions. If you look closely at the analyzed examples, you can see that in most cases the results of my procedure and Rouben's formula coincide. Why do not you ask such questions before?

@Markiyan Hirnyk   According to your picture positive value of  H2  is measured from the vertical axis to the left. Therefore, for values  H2=2, H1=0.5, R=1   the body  does not exist. I put a condition  H1^2+H2^2<=R^2  to exclude trivial cases (the whole ball or the ball without one or two segments). Of course, the procedure can be adapted for all cases, but it is not interesting.

@jbail  First, execute restart command in your worksheet . Then just copy the code in your worksheet and then press ENTER key.

@jbail  I removed the extra commas from  the code. Try again.

@sunit 

combine(sqrt(a)*sqrt(b))  assuming positive;

                        sqrt(a*b)

Compare

sqrt(-1)*sqrt(-1)=I^2=-1<>sqrt((-1)*(-1))=sqrt(1)=1

@Markiyan Hirnyk  I wrote "to achieve complete clarity..."  For example, your approach does not answer the question how many real solutions has the equation for different values of  a .