Any mathematical mistake in my script ?
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Hi, I am trying to plot the self-derived analytical solution using matlab. However, I get different answer from what I obtained using excel sheet.
I suspect the mistake is in somewhere within this script, where I could have miss out.
Could anyone spare a help here ?
Self-derived Analytical Solution :
syms n y real
assume([n y] >= 0);
Y = 0:0.02:1;
b =1; t=1; br=1; Pr=1;
%t = t*
K1=(1-exp(-((2.*n)+1).^2.*pi.^2.*t./(4.*Pr)))./(((2.*n)+1).^2.*pi.^2);
K2=((((2.*n)+1).^2.*pi.^2.*exp(-((2.*n)+1).^2.*pi.^2.*t./(4.*Pr)))-(((2.*n)+1).^2.*pi.^2.*cos(2.*b.*t))-(8.*b.*Pr.*sin(2.*b.*t)))./((64.*b.^2.*Pr.^2)+(((2.*n)+1).^4.*pi.^4));
T = subs( sum( subs( 8.*br./(((2.*n)+1).*pi).*(K1+K2).*sin((((2.*n)+1)./2).*pi.*y), n, 1:100 )), y, Y );
Tn = double(T);
disp(Tn);
plot(Tn,Y);
Thanks,
Andy
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Alan Stevens
2021년 8월 10일
Easier to check like this
b=1; t=1; br=1; Pr=1;
d = @(n) (2*n + 1)*pi;
c = @(n) d(n).^2;
k1 = @(n) ( 1 - exp( -c(n)*t/(4*Pr) ) )./c(n);
k2 = @(n) c(n).*exp( -c(n)*t/(4*Pr) ) - c(n)*cos(2*n*t) - 8*b*Pr*sin(2*b*t);
k3 = @(n) c(n).^2 + 64*b^2*Pr^2;
K = @(n) k1(n) + k2(n)./k3(n);
term = @(y,n) 8*br./d(n).*K(n).*sin(d(n)*y/2);
y = 0:0.02:1;
N = numel(y);
T = zeros(1,N);
for i=1:N
for n = 0:100
T(i) = term(y(i),n) + T(i);
end
end
figure
plot(y,T),grid
xlabel('y'), ylabel('T')
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Alan Stevens
2021년 8월 11일
"1. Shouldn't it be all using .* and ./ since we are summing up the "n" term by term (consider as element wise sum up right) ? I tried using your current script, it gives same results with I added all with .* and ./. I could not imagine how they sum up using mixed of *; .*; / and ./."
Yes, you can replace the .* etc by just . I put .* initially, as I hadn't decided if I would use a loop or vector indexing.
"2. In your case, you first sum up the T for a particular of y location (here is up to 100 n terms, increment 1 each time). Then do the same for rest of the discretized y which defined by array of zeros N elements (from 0 to 1, increment 0.02 each time), am i right ?"
Yes, that's right.
"3. If i have a similar analytical solution that consists of triple summation terms, it involved piecewise function, using for loop method should work also right ?"
Yes, it should.
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