@Kitonum 

Thank you, Kitonum, for taking the time to provide a solution to my question! I really appreciate your effort and creativity in using the tools from the plots and plottools packages to address the problem.

However, I found that the suggested approach has some limitations in my case. In the actual plot I am working on, there are four different curves. Additionally, the axis scales are different, which makes implementing the solution challenging and less effective for my needs.

The example code I shared earlier was just a simplified representation of my problem, and applying your approach to my actual working code would require extensive adjustments.

Thank you again for your help! If you have any further suggestions or alternatives, I would be glad to hear them.

@acer 
Thank you very much, it's excellent

@dharr 
Thank you for your comment
Variables a1 have an unknown range, but k and re are positive numbers that are later substituted in the general solution for the variables a1

Thank you for your comment
I know that the number of unknowns in the above system is ten and not eight, but in my studies, k and re are arbitrary constants and not unknowns, so first all solutions of a1 must contain k and re, and then I discuss some special cases of k and re values
As you mentioned in your comment when the values of k and re are determined then fsolve easily gives a solution but that's not what I want to get.

Is there a way to get solutions of a1 with k and re

@Carl Love 

Thank you very much it's great and working properly

@Carl Love 

thank you for replying
Yes, these three assumptions are acceptable

@Carl Love 

I'm trying to doing this rule in which the multiplication of any two functions T(x, a)*T(x, b) must convert  into an addition

1/2*(T(x, a+b)+T(x, abs(a-b)))

on this series for example

p := 8*T(x, 7)*T(x, 2)+4*T(x, 5)*T(x, 1)+6*T(x, 3)*T(x, 3)+7*T(x, 1)*T(x, 4)

After applying this rule T(x,a)*T(x,b) =1/2*(T(x,a+b)+T(x,abs(a-b))) i must get

pp := 4*(T(x, 9)+T(x, 5))+2*(T(x, 6)+T(x, 4))+3*(T(x, 6)+T(x, 0))+7/2*(T(x, 5)+T(x, 3))

Please help. 

@Kitonum 

Thank you very much, i appreciate it

How to convert  this
 T(x,a)*T(x,b) into

( T(x,a+b) +T(x,abs(a-b)) )*1/2

@Kitonum 

   Thank you very much
Can I add 1 for each T(x,i) to become T(x,i+1)

8T(x, 8)+8*T(x, 3)+4*T(x, 6)+11*T(x, 2)+12*T(x, 4)+7*T(x,5) 

or minus one to become

8*T(x, 6)+8*T(x, 1)+4*T(x, 4)+11*T(x, 0)+12*T(x, 2)+7*T(x, 3)