How to do numerical integration using trapezoidal rule?

2023 6 30
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更新时间：2023 7 5
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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.

6 个评论
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Nathan Hardenberg
Nathan Hardenberg 2023-6-30
Do you know about the trapz()-function in MATLAB? This does "trapezoidal numerical integration" and might help you
Nathi
Nathi 2023-7-1
@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.
Ran in:
Ran in:
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
Nathi 2023-7-3
编辑：Nathi 2023-7-4
@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
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

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Nathi
Nathi
2023-6-30

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Nathi
2023-7-5

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