Vedat Erturk I'll try to help: give you some ideas to try. Using an array for: α1=1;α2=1;α3=1;α4=1;α5=1; α={1,1,1,1,1} and referencing as: α[[I]] could help. Maybe better as a[i_].Other variables could also be assigned to an array/ given a number instead of a variable name. In most model Intel based machines :ParallelTable[] and ParallelSum[] speed up calculations by the number of processes your machine allows. Integrations might be better as NIntegrate for time saving. Do Loops have ever been slow in Mathematica. Replacing the loops with Block[] might help as sometimes used in fractal programs. Compiling the Do loop section might also speed up the calculation process. I know the Parallel idea works for at least 10 places in your program. In all: buy yourself the fastest computer you can get (with as many processors as you can) with the most built in memory (16 gb or more): then the slowness of Mathematica will fade, LOL. Good luck in your real world programming. Roger Lee Bagula

For anyone to consider taking their time to study your program and suggest how to speed it up I expect they are almost certainly going to want to quickly and easily understand exactly which of all your many subscripts are connected to which of all your many variables.

Consider the first example

x[t_]:=10;
  0,0

Is it the x that is double subscripted? Or the t double subscripted, or do x and t each have one subscript? Hopefully it is not the 10 that is subscripted.

Your displayed program is filled with vast numbers of examples exactly like this. Even clicking on "Make your own copy" gives exactly the same subscript issue. And downloading the pdf file gives exactly the same subscript issue.

Is there any way that you might be able to display your entire program in InputForm ? That would remove all the two-dimensional formatting and all the positioning of all your subscripts. That might make it more feasible for someone to understand exactly what your program is doing and even be able to paste that directly into their Mathematica notebook and perform small experiments to see if a change results in a substantial increase in speed.

You write "For large K's and the small values of T/M, my code runs slowly." Can you give us some idea what "large K" is? 10^3? 10^6? 10^12? And exactly the same question for M. Are you doing thousands or millions of Simplify and thousands or millions or even billions of Integrate? Approximately how many of each of those, potentially complicated and slow, operations are you doing?

Thank you

Hello Bill, Thank you for your reply.only x is double subscripted. K is 5,10,20,30,etc. that is,it is not so big.some times 10 is even enough. Now, T/M is just 0.1. I should be able to choose T/M as small as I want. For example T/M=0.01,0.001,etc. Hello again Bill, I hope this helps you.

I should be able to get the K at least 10.

Thank you for the additional information and files. That was very helpful.

You might add a couple of Print statements to your code and look at the form of the expression that you are taking the derivative of and that result and the form of the expression you are taking the integral of and that result.

If I have not made any mistake then it looks like you are in effect taking the derivative of a constant times lambda with respect to lambda and that just gives you that constant. Then you are taking the integral of an expression containing that constant times t giving you the constant times t.

If that is the case then it might be possible that you could derive your constant times t without needing to spent all the time needed to construct the derivative and then the time to integrate that.

Since both those operations can be expensive and since you are doing this thousands of times inside a loop then this might show you a path to speed up your code significantly.

I have tried this for other small values of K and I think I see the same behavior.

You must be very careful to verify whether what I am suggesting is correct or not. You are using subscripts in a way that I have not done and I have made a variety of changes to try to get the results. Any of my changes might have introduced errors. I have seen some of the numbers becoming very large and negative and I am not confident that what I have done is correct.

This is the current state of my modifications

ClearAll["Global`*"];
(*modified code to eliminate subscripts and underscores*)
(*modified code changing z[] to z2f[] and z3f[]*)
α1=1;α2=1;α3=1;α4=1;α5=1;
g0=0.9;b10=0.03;b11=0.001;Topt=24;gama0=150;ds0=4;A0=0.2;
w=2.10;z1=3.5;Oc=1.10;d1=0.66;d=0.4;gama1=2;d2=0.2;z=0.0019;
β=1.024;γ=4;z0=10.10;T10=14.50;delta1=0.1;T0=20;
b=1.30;Tmaks=35;K=2;M=2;T=60.05;
x[t_,0,0]:=10;h1[t_,0,0]:=28;h2[t_,0,0]:=1;y[t_,0,0]:=1.2;z3f[t_,0,0]:=0.25;
x[ λ_,m_]:=Sum[ x[t,m,i]*λ^i,{i,0,K}]; 
h1[λ_,m_]:=Sum[h1[t,m,i]*λ^i,{i,0,K}]; 
h2[λ_,m_]:=Sum[h2[t,m,i]*λ^i,{i,0,K}]; 
y[ λ_,m_]:=Sum[ y[t,m,i]*λ^i,{i,0,K}];
z2f[λ_,m_]:=Sum[z3f[t,m,i]*λ^i,{i,0,K}];
N1[λ_,m_]:=g0*(Exp[-b*(h1[λ,m]-Topt)/(Tmaks-Topt)]+(z2f[λ,m]-(b10+b11*
(h1[λ,m]-Topt)))/(z2f[λ,m]+1))*x[λ,m]-g0*x[λ,m]^2/gama0;
N2[λ_,m_]:=w*h2[λ,m]-z1*(h1[λ,m]-T10)+γ*(z0-y[λ,m]);
N3[λ_,m_]:=A0-delta1*h2[λ,m];
N4[λ_,m_]:=Oc-d1*y[λ,m]-d*y[λ,m]*h2[λ,m];
N5[λ_,m_]:=gama1*β^(h1[λ,m]-T0)*(ds0/(1+h1[λ,m]-Topt)-z2f[λ,m])-
d2*z2f[λ,m]*x[λ,m]-z*(h1[λ,m]-Topt);
A1[m_,0]:=N1[0,m];A2[m_,0]:=N2[0,m];A3[m_,0]:=N3[0,m];A4[m_,0]:=N4[0,m];A5[m_,0]:=N5[0,m];
Do[
  For[i=0,i<K,i++,
    A1[m,i]=Simplify[1/i!*D[N1[λ,m],{λ,i}]]/.λ->0;
    Print["{i,m,derivative}=",{i,m,A1[m,i]}];
    A2[m,i]=Simplify[1/i!*D[N2[λ,m],{λ,i}]]/.λ->0;
    A3[m,i]=Simplify[1/i!*D[N3[λ,m],{λ,i}]]/.λ->0;
    A4[m,i]=Simplify[1/i!*D[N4[λ,m],{λ,i}]]/.λ->0;
    A5[m,i]=Simplify[1/i!*D[N5[λ,m],{λ,i}]]/.λ->0
  ];
  Do[
    x[ t_,m,k]=1/Gamma[α1]*Integrate[(t-τ)^(α1-1)*A1[m,k-1],{τ,0,t}];
    Print["{k,m,integral}=",{k,m,Simplify[1/Gamma[\[Alpha]1]*Integrate[(t-\[Tau])^(\[Alpha]1-1)*A1[m,k-1],{\[Tau],0,t}]]}];
    h1[t_,m,k]=1/Gamma[α2]*Integrate[(t-τ)^(α2-1)*A2[m,k-1],{τ,0,t}];
    h2[t_,m,k]=1/Gamma[α3]*Integrate[(t-τ)^(α3-1)*A3[m,k-1],{τ,0,t}];
    y[ t_,m,k]=1/Gamma[α4]*Integrate[(t-τ)^(α4-1)*A4[m,k-1],{τ,0,t}];
    z3f[t_,m,k]=1/Gamma[α5]*Integrate[(t-τ)^(α5-1)*A5[m,k-1],{τ,0,t}],
  {k,1,K}];
  f1[t_,m_]:=Sum[x[ t-(T/M)*m,m,i],{i,0,K}];
  f2[t_,m_]:=Sum[h1[t-(T/M)*m,m,i],{i,0,K}];
  f3[t_,m_]:=Sum[h2[t-(T/M)*m,m,i],{i,0,K}];
  f4[t_,m_]:=Sum[y[ t-(T/M)*m,m,i],{i,0,K}];
  f5[t_,m_]:=Sum[z3f[t-(T/M)*m,m,i],{i,0,K}];
  x[ t_,m+1,0]=f1[(T/M)*(m+1),m];
  h1[t_,m+1,0]=f2[(T/M)*(m+1),m];
  h2[t_,m+1,0]=f3[(T/M)*(m+1),m];
  y[ t_,m+1,0]=f4[(T/M)*(m+1),m];
  z3f[t_,m+1,0]=f5[(T/M)*(m+1),m],
{m,0,M-1}];
(*X[ t_]=X[ t]=Piecewise[Table[{f1[t,m],((T/M)*(m))<=t<((T/M)*(m+1))},{m,0,M-1}]];
  H1[t_]=H1[t]=Piecewise[Table[{f2[t,m],((T/M)*(m))<=t<((T/M)*(m+1))},{m,0,M-1}]];
  H2[t_]=H2[t]=Piecewise[Table[{f3[t,m],((T/M)*(m))<=t<((T/M)*(m+1))},{m,0,M-1}]];
  Y[ t_]=Y[ t]=Piecewise[Table[{f4[t,m],((T/M)*(m))<=t<((T/M)*(m+1))},{m,0,M-1}]];
  Z[ t_]=Z[ t]=Piecewise[Table[{f5[t,m],((T/M)*(m))<=t<((T/M)*(m+1))},{m,0,M-1}]];
  s1=Table[{t,X[t]},{t,0,T}]
  s1MSADM=ListLinePlot[s1,PlotRange->All,PlotStyle->Black,AxesLabel->{"t","N(t)"}]*)

And the output of that is

{i,m,derivative}={0,0,6.5649}
{i,m,derivative}={1,0,0.59649 x[t,0,1]+9.(-0.0736627 h1[t,0,1]-0.13824([0.+z3f[t,0,1])+0.8(0.-0.001 h1[t,0,1]+z3f[t,0,1]))}
{i,m,integral}={2,0,6.56490*t}
{i,m,integral}={2,0,0+15.2130*t^2}
{i,m,derivative}={0,1,0}
{i,m,derivative}={1,1,0}
{i,m,integral}={2,1,0}
{i,m,integral}={2,1,0}
{i,m,derivative}={0,2,0}
{i,m,derivative}={1,2,0}
{i,m,integral}={2,2,0}
{i,m,integral}={2,2,0}

I believe what that is showing is that you might not need all the machinery and horsepower to be doing derivatives and integrals to construct your Piecewise function. If that is true then this may result in substantial increases in the speed of your code, which was your stated goal. If you look at the code with the idea of finding further simplifications in the way it calculates the final result then I think it might be possible to simplify and speed this up even further.

This still needs more work and testing and verification. Please check this carefully to try to make certain that I have not introduced any errors.

Note: There are tiny changes between the code you first posted and the notebook you later posted. I believe my code is based on the original post. I also think I have probably made mistakes translating your code to mine and that my calculations probably do not match yours. If you can point out specific differences for i,m,k and the derivative or integral, what you get and I should get then I will try to track down my errors and see if I can fix those. It is also likely that the diagnostic prints that I inserted could be improved to provide better information. Please make any suggestions.

Thanks