Thanks @vv.

What about  L-2 error norm ?

Thanks, dear @Carl Love, it's a brilliant solution. 

I want to change the question a little bit.

Let's select as follows:

 =

where

is a real number.

In fact; it is a generalization of the question in the post. If we select , it transforms to original post.

How to get an explicit formula?

I wrote a code as follows by adapting your code above:

Although I waited a lot, Maple did not give any results. (I think we need more assumptions ) 

restart:
interface(showassumed= 0):


assume(n::posint, j::nonnegint, j <= n, T > 0, alpha > 0,rho::realcons);
h:= T/n:



psi:= t->
    local i:= op(procname);
    convert(
        if i=0 then piecewise(t<0, 0, t<h, 1-t/h)
        elif i=n then piecewise(t < T-h, 0, 1-(T-t)/h)
        else 
            piecewise(
                t < (i-1)*h, 0, 
                t < i*h,     t/h-(i-1), 
                t < (i+1)*h, (i+1)-t/h
            )
        fi,
        Heaviside
    ):   

 

g:= proc()
local i,j,rho,t; 
    (i,j):= op(procname);
    simplify(
        int(t^(rho)*((j*h)^(rho+1)-t^(rho+1))^(alpha-1)*psi[i](t), t= 0..j*h)*(rho+1)^(1-alpha)/GAMMA(alpha)
    )
end proc
:
g[0,j]() assuming additionally, j > 0 ;

 
g[j,j]() assuming additionally, j > 0; 
 
g[i,j]() assuming i::posint, additionally, i < j; 
 

download_the_codeabove.mw

Thanks, dear @Kitonum,

I use `Standard Worksheet Interface, Maple 2020.2, Windows 10, November 11 2020 Build ID 1502365`, everything is the same as yours.

Yes, your solution way is an alternative, but in fact, everybody wants/hopes to accomplish this by using the plot, plots:-textplot, plottools:-rectangle  commands etc. 

Thanks for your interest @dharr,

if alpha=1;

fracdiff(t, t,alpha)

gives 1. But it doesn't appear in the figure. 

if alpha=0, 

fracdiff(t, t,alpha)

gives t. But it doesn't appear in the figure, also. 

You are right dear @acer,

I will try to achieve it.

Thanks.

Thanks, @acer,

In fact, I don't mean the location of the legends in my last question. 

In your code, since the lines in the figure include the solid circles, the legends must include also them. I marked one of them on the following figure. How can we achieve this?

Dear @acer,

Your solution is good.

But, while the lines includes solid circles, but the circles doesn't appear in the legends. Can we solve this?

On the other hand, ı know Maple has some options instead of circles. But is it possible as follows?

@Thomas Richard,

Oh yes, you are right. I edited the post.

@vv 

You can find the Maple_code.mw for , P(1), P(2). 

For N=1, you are right may be  int(Psi(s), s=0..t) <>P(1)*Psi(t).

But this is a numerical approximation. When N increases, we get more exact results. I tested it in the Maple_code.mw

Thanks for your interest Dear @vv 

But for N=1,  Psi(t)  is not equal to zero. Why do you think it's zero?  (I don't understand exactly)

Meanwhile, this method is a kind of wavelet method, and it's commonly used.

Dear Prof. @Carl Love,

I explained some more below.

The following is the thing which I want to write a procedure in my question post.

IN FACT;

I want to check the below solution algorithm for Differential Equations by Maple. This is a simple example. But I have more complicated examples. So I want to use MAPLE for the symbolic operations below.

This is one part of the mey method. If we write a Maple code for this part, I hope so I can handle the rest.

Thanks for your interest dear @Carl Love.

P(1), P(2) are NxN matrix functions. 
For N:=2, they are as follows, respectively.

P(1):=
Matrix(2, 2, [[0.5, 0.2886751345229591], [-0.216506351, 0.]]);

P(2):=Matrix(2, 2, [[0.16666666666666669, 0.14433756726147953], [-0.1082531755, -0.08333333333333333]]);

Dear @Mariusz Iwaniuk,

Thanks for your valuable response.

I have a question about the fractional differential equation.

I download the code from the following link:
https://data.mendeley.com/datasets/dn82gkvf4f/1

I write the following code, but I get an error when I run the code.
Could you help me, please?
Best regards.

QUESTION:
for 0<x<1, and alpha=1,
exact Solution of the following equation is u(x,t)= 3*exp(t)/(1-3+3*exp(t)):
-------
CODE:
eq:=diff(u(x,t),t$alpha)-diff(u(x,t),x,x)-u(x,t)*(1-u(x,t))=0;
IC:=[u(x,0)=3];
Asolve([eq],IC,alpha=1,index=11,x=0..1,u=0..1,output=plot,pade=[20,20]); 
-------
But the error is  "Error, (in SeparateOperator) improper op or subscript selector |InitialInterface.mpl:257| "

And I want to also run the code for alpha=0.5 with suitable initial/boundary condition.

Dear @acer , thanks for your interest. But I think I was misunderstood. 

I want to mean that

2d graphics:
for alpha=0.3,  when we run the code, we get a graphics of function UU(0.5,t) 

and after restarting the code

for alpha=1 when we run the code again, we get a graphic of function UU(0.5,t).

So, I want to display these two graphs together in one graph (2d).

And also for 3d graphics:

for alpha=0.3,  when we run the code, we get a graphics of function UU(x,t) 

and after restarting the code

for alpha=1 when we run the code again, we get a graphic of function UU(x,t).

So, I want to display these two graphs together in one graph (3d).

Dear @tomleslie, thanks for your interest. But I think I was misunderstood. 

I want to mean that

2d graphics:
for alpha=0.3,  when we run the code, we get a graphics of function UU(0.5,t) 

and after restarting the code

for alpha=1 when we run the code again, we get a graphic of function UU(0.5,t).

So, I want to display these two graphs together in one graph (2d).

3d graphics:

for alpha=0.3,  when we run the code, we get a graphics of function UU(x,t) 

and after restarting the code

for alpha=1 when we run the code again, we get a graphic of function UU(x,t).

So, I want to display these two graphs together in one graph (3d).