How to do numerical integration using trapezoidal rule?

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Nathi 2023-6-30

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https://ww2.mathworks.cn/matlabcentral/answers/1990083-how-to-do-numerical-integration-using-trapezoidal-rule

评论： Nathi 2023-7-5

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.

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Nathan Hardenberg 2023-7-4

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 2023-7-5