How to do numerical integration using trapezoidal rule?

I want a graph for velocity profile. In my equation have one of the term is integro_differential. for that purpose i have to solve the integral part in my equation using newtons cotes closed integration. so, I'm using one of the rule is trapezoidal form. I want to use trapezoidal rule for integration. i.e . Here, we have to notice the 'T' is another equation output value which I already solved and i got the matrix form. i.e

%===========================================%
ymax=20; m=80; dy=ymax/m; %y=dy:dy:ymax;       
xmax=1; n=20; dx=xmax/n;                        
tmax=100; nt=500; dt=tmax/nt; t=0:dt:tmax;      
tol=1e-2;
max_difference(1)=1; max_Iteration=1;
pr=6.2;
UOLD=zeros(m,nt);VOLD=zeros(m,nt); 
TNEW=0; TOLD=TNEW*ones(m,nt); TWALL=ones(1,length(t));
A=zeros([1,m]); 
B=A;
C=A;
D=A;
T=TOLD;
while max_Iteration>tol
    for j=1:nt
        for i=1:m
            if j>i
                C(i)=dt*VOLD(i,j)/4*dy-dt/(2*dy^2);
            elseif i>j
                A(i)=-dt*VOLD(i,j)/4*dy-dt/(2*dy^2);
            elseif i==j
                B(i)=1+dt*UOLD(i,j)/2*dx+dt/dy^2;
            end
        end
    end
    for j=2:nt
        for i=2:m-1
            if i==2
                D(i)=-(dt*UOLD(i,j)*(-T(i,j-1)+TOLD(i,j)-TOLD(i,j-1)/2*dx))+(dt*(TWALL(j)-2*TOLD(i,j)+TOLD(i-1,j)/(2*dy^2)))-(dt*VOLD(i,j)*(TOLD(i-1,j)-TWALL(j)+TOLD(i,j)/4*dy))-((-dt)*(VOLD(i,j)/4*dy))-(dt*(TWALL(j)/2*dy^2));
            elseif i==m-1
                D(i)=-(dt*UOLD(i,j)*(-T(i,j-1)+TOLD(i,j)-TOLD(i,j-1)/2*dx))+(dt*(TOLD(i+1,j)-2*TOLD(i,j)+TNEW/(2*dy^2)))-(dt*VOLD(i,j)*(TNEW-TOLD(i+1,j)+TOLD(i,j)/4*dy))-(dt*(VOLD(i,j)/4*dy))-(dt*(TNEW/2*dy^2));
            else
                D(i)=-(dt*UOLD(i,j)*(-T(i,j-1)+TOLD(i,j)-TOLD(i,j-1)/2*dx))+(dt*(TOLD(i+1,j)-2*TOLD(i,j)+TOLD(i-1,j)/2*dy^2))-(dt*VOLD(i,j)*(TOLD(i-1,j)-TOLD(i+1,j)+TOLD(i,j)/4*dy));
            end
        end
        T(:,j)=TriDiag(A,B,C,D);   %This 'T' is a function of integral w.r.to y/ how to solve this case usign this 'T'.
        dt=0.2+dt;
        TOLD=T;
        max_difference(j) = max(abs(T(:,j)-T(:,j-1))./max(1,abs(T(:,j-1))));
        if max_difference(j)<tol
            break
        end
        max_Iteration=max_difference;
    end
end

I want to integrate this 'T' w.r.to Y

T(:,j)=TriDiag(A,B,C,D);   

kindly give advice. thank you.

[Nathan Hardenberg]

Do you know about the trapz()-function in MATLAB? This does "trapezoidal numerical integration" and might help you

[Nathi]

@Nathan Hardenberg Yes. I tried that rule and try to include this term in my main equation. After integrate w.r.to.Y, I have to differentiate that integral term w.r.to. X.

F=@(y) T;
a=0; b=20;
h=(b-a)/m;
i=1:1:m-1;
S=F(a+i*h);
I=(h./2)*(F(a)+2*sum(S)+F(b));

I integrated this 'I' afterthat i couldn't differentiate this integral term.

R=diff(I)/h;   %dono this is the proper way for partial derivative 

This R not satisfying with my equation. or If possible, we can use it in numerical differentiation method. Give me some idea to proceed this. Thnak you.

[Nathan Hardenberg]

I can't really follow your steps. One reason might be that I can not execute your original code, since I do not have the TriDiag()-Function. Also your other code is not executable as well. I assumed:

syms F(y)  % defined as symbolic function
%F=@(y) T
a=0; b=20; m=80;  % added m=80;
h=(b-a)/m;
i=1:1:m-1;
S=F(a+i*h);
I=(h./2)*(F(a)+2*sum(S)+F(b))

I = 

R=diff(I)/h

R = 

0

Which gives a vaild result, but probably not what you want!?

I still do not know what T is! You write: "'T' is a function [...]". But in your Code it looks like T is a 2D-matrix, not a function. Maybe I'm missing something.

If possible please provide T (whatever it is)

[Nathi]

@Nathan Hardenberg Really thanks for your response.

I will explained it clearly in the following link. please find this link and give your valuable solution. https://in.mathworks.com/matlabcentral/answers/1991053-how-to-include-partial-derivative-term-of-matrix-in-finite-difference-equation

[Nathan Hardenberg]

Sorry I'll not answer the question, since I still do not understand it and your code in the new question was still not executable. Maybe try to reduce your problem. Then other users might be able to help

[Nathi]

@Nathan Hardenberg once again see this link

https://in.mathworks.com/matlabcentral/answers/1991053-how-to-include-partial-derivative-term-of-matrix-in-finite-difference-equation#comment_2805163

I explained it separately.