Finite Difference method solver
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I have the following code in Mathematica using the Finite difference method to solve for c1(t), 
where 
. However, I am having trouble writing the sum series in Matlab. The attatched image shows how the plot of real(c(t) should look like.
where . However, I am having trouble writing the sum series in Matlab. The attatched image shows how the plot of real(c(t) should look like.\[CapitalOmega] = 0.3;
\[Alpha][\[Tau]] := Exp[I \[CapitalOmega] \[Tau]] ;
dt = 0.1;
Ns = 1000;
ds = dt/Ns;
Ttab = Table[T, {T, 0, 10, dt}];
Stab = Table[s, {s, 0, dt - ds, ds}];
c[0] = 1;
Do[corrSum[n] = Sum[c[nn - 1]*Sum[\[Alpha][n dt - m ds]*ds, {m, Ns (nn - 1), Ns nn , 1}], {nn, 1, n}];
c[n] = c[n - 1] - dt*corrSum[n](*c[n-1]*\[Alpha][n dt]*), {n, 1, 100}]
cTtab = Table[{n*dt, c[n]}, {n, 0, 100}]
FDiff = ListPlot[Re[cTtab], PlotStyle -> Orange, PlotLegends -> {"Finite Difference"}]
采纳的回答
Alan Stevens
2020-11-7
By differentiating c' again you can solve the resulting second order ode as follows
c0 = 1;
v0 = 0;
IC = [c0 v0];
tspan = [0 10];
[t, C] = ode45(@odefn, tspan, IC);
c = C(:,1);
plot(t,real(c),'ro'),grid
xlabel('t'), ylabel('real(c)')
function dCdt = odefn(~,C)
Omega = 0.3;
c = C(1);
v = C(2);
dCdt = [v;
-1i*Omega*v - c];
end
This results in
7 个评论
Jose Aroca
2020-11-7
Alan Stevens
2020-11-7
Ok, I didn't realize you were doing a straight comparison. I went for the 2nd order ode approach because the Mathematica finite difference method looked so messy! I'll have a second look, but no promises!
Jose Aroca
2020-11-7
Alan Stevens
2020-11-7
Do you have the next page of the pdf?
Jose Aroca
2020-11-7
Alan Stevens
2020-11-8
Well, here is an explicit finite difference version that uses an outer and an inner loop, with the same values of dt and ds as in the Mathematica version. However, it is probably not what you want still, as it isn't a line by line translation of the Mathematica code. I'll leave further development to you!
Omega = 0.3;
dt = 0.1;
Ns = 1000;
ds = dt/Ns;
t = 0:dt:10;
N = numel(t);
c = zeros(1,N);
c(1) = 1;
v = 0;
for n = 1:N-1 % Outer loop; c is updated each iteration of this loop
% Inner loop: c is taken to be constant through this loop, the same
% approximation as is made in section 5.2.1 of the pdf.
for nn = 1:Ns
v = v - (1i*Omega*v + c(n))*ds;
end
c(n+1) = c(n) + v*dt;
end
plot(t,real(c),'.'),grid
xlabel('t'),ylabel('real(c)')
Jose Aroca
2020-11-9
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