How to integrate this function numerically?

2019 12 14
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更新时间：2019 12 15
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Hello everyone,
I'm stuck at this equation and I want to integrate it numerically
where
and r = 695510, a = -0.0021, b = 1.34, Wo = 438.1
I need to integrate the first equation numerically to get V as a function of R
Numerical integration from R = 10, where it is assumed V = U, to 215 should give V(R).
I appreciate your help.
Thank you,

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Walter Roberson
Walter Roberson 2019-12-14
Is e the base of the natural logarithms? And is it acting as a constant multiplier or is e being raised to the part after it?
Mohamed Nedal
Mohamed Nedal 2019-12-14
Yes that's right, it's the base of the natural logarithm and it's being raised to the part after it.

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Chuguang Pan
Chuguang Pan 2019-12-14

1 个投票

You can use Euler formula. V(n+1)=V(n)+h*f(R,V(n)), which f(R,V) is the right side of differential equation.
But you need to know the initial value V(10).
r=695510;a=-0.0021;b=1.34;Wo=438.1;
h=0.1;%Integral step size, you can change this value
N=100;%It means that you want to integral from 10 to 10+h*N
V=zeros(1,N+1);%Initialization V array
V(1)=?;%Need to know the initial value V(10)
for n=1:N
R=10+(n-1)*h;
W=Wo*sqrt(1-exp(1)*(2.8-R)/8.1);
V(n+1)=V(n)+h*r*a*R^(-b)*(1-W/V(n));
end
X=10+(0:N)*h;
Y=V;
plot(X,Y);

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Mohamed Nedal
Mohamed Nedal 2019-12-14
编辑：Mohamed Nedal 2019-12-14
Thanks for your response. The initial value of V(10) is known (693).
I think the 3rd line of the for loop should be like this
W=Wo*sqrt(1-exp((2.8-R)/8.1));
Could you please elaporate on h and N?
What are the consequences of changing the integral step size (h) and N?
Chuguang Pan
Chuguang Pan 2019-12-14
r=695510;a=-0.0021;b=1.34;Wo=438.1;
h=0.1;%Integral step size, you can change this value
N=(215-10)/h;%It means that you want to integral from 10 to 215
V=zeros(1,N+1);%Initialization V array
V(1)=693;%Need to know the initial value V(10)
for n=1:N
R=10+(n-1)*h;
W=Wo*sqrt(1-exp((2.8-R)/8.1));
V(n+1)=V(n)+h*r*a*R^(-b)*(1-W/V(n));
end
X=10+(0:N)*h;
Y=V;
plot(X,Y);
you can decrease h for increasing presision
Mohamed Nedal
Mohamed Nedal 2019-12-14
Thank you so much, that's very helpful.
Mohamed Nedal
Mohamed Nedal 2019-12-14
Please I have another question. Knowing the above parameters, How do we get t from the folowing equation?
Chuguang Pan
Chuguang Pan 2019-12-14
If t is independent variable, it is similar to R. You need to decide for youself.
Such as t=t0+(0:N)*h.
Mohamed Nedal
Mohamed Nedal 2019-12-14
编辑：Mohamed Nedal 2019-12-14
t is the time (in seconds) the object takes to cross the distance R at speed V.
to = 0, I need to find t(R=215).
Mohamed Nedal
Mohamed Nedal 2019-12-14
Could you check this please.
r = 695510; a = -0.0021; b = 1.34; Wo = 438.1;
h = 1; % Integral step size, you can change this value
N = (215 - 10)/h; % It means that you want to integral from 10 to 215
V = zeros(1,N+1); % Initialization V array
t = zeros(1,N+1);
t(1) = 0;
V(1) = 693; % Need to know the initial value V(10)
for n = 1:N
R = 10 + (n-1)*h;
W = Wo * sqrt(1-exp((2.8-R)/8.1));
V(n+1) = V(n) + h*r*a*R^(-b)*(1-W/V(n));
t(n+1) = t(n) + h*a*R^(-b)*(V(n)-W);
end
X = 10 + (0:N)*h;
t = (0:N)*h; % to = 0
plot(X,V);
t here is monotonically increasing (t=0,1,2,...) and when I removed this line
t = (0:N)*h;
it gives negative fractions and it's not correct.
Chuguang Pan
Chuguang Pan 2019-12-14
V=dR/dt. dt=dR/V.
R=10+(0:N)*h;
dR=diff(R);
V=(V(2:end)+V(1:end-1))*0.5;%average speed
dt=dR./V;
t=[0 cumsum(dt)];
Mohamed Nedal
Mohamed Nedal 2019-12-15
Excuse me another question..
I would like to apply this equation on several objects and I tried to replace V(1), which's the initial value V(10), with a vector column instead of a single value like this one for instance
Vtest = [136
490
359
361
693
1374
585
542
1118
910
1188];
so that each value is assigned for one object. But I got this error
In an assignment A(:) = B, the number of elements
in A and B must be the same.
Error in numeric_integ (line 22)
V(1) = Vtest; % Need to know the initial value V(10)
Could you help me out?
and if I want to find only the value of V(215) and t(215), which are the last values of V and t at distance 215, how can I do that?
Chuguang Pan
Chuguang Pan 2019-12-15
result_V=zeros(1,length(Vtest));% save last V for every Vtest
result_t=zeros(1,length(Vtest));% save last t for every Vtest
for i=1:length(Vtest) %for every Vtest
V=zeros(1,N+1);
V(1)=Vtest(i);
%do the computing
result_V=V(end); %using end to index last value of array V
result_t=t(end); %using end to index last value of array t
end

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