Plotting a multivariable function, that also has a summation

Good Afternoon everyone,

I am attempting to plot the funtion above in matlab, I have been trying with symsum, for loops & 3D plots but can't get it to work. Hopefully someone can help me!

Thank you

[Torsten]

Show us what you have tried.

[Ali Mosa]

My initial attempt was to do the summation using a for loop similar to how you'd solve a fourier transform using the inital value, but it would only plot the final value then after some googling I found the symsum function, so I have been trying to use that, but I can't find examples of it used with two variables. Ive also got the issue of the resultant arrays needing to be the same size so they can be plotted which is difficult sonce x is defined and t is infinite.

Any clarification would be greatly appreciated, sorry for not including this in my question.

Answer 1

[Torsten]

xstart = 0.0;
xend = 1.0;
nx = 100;

tstart = 0.0;
tend = 5.0;
nt = 100;

nsum = 31;

X = linspace(xstart,xend,nx);
T = linspace(tstart,tend,nt);
U = zeros(nx,nt);

for ix = 1:nx
    x = X(ix);
    for it = 1:nt
        t = T(it);
        for isum = 1:2:nsum
            U(ix,it) = U(ix,it) + ?
        end
    end
end

U = U * 400/pi^3;

surf(X,T,U)

[Ali Mosa]

Thanks so much for the help, I just wanted to ask about the use of nx and nt. Why there is two separate variables since they appear to do the same thing, so which would take the place of n in the equation for U(ix,it)?

Also the value chosen for nsum, is that random?

Sorry for all the questions, I want to make sure I understand this so I can apply these techniques in other problems I come across!

[Torsten]

You have two dimensions in your equation: x in space and t in time.

Both dimensions are independent from each other.

So you need two loops to calculate values for U depending on a given value of x (this is the loop with ix) and a given value of t (this is the loop with it) simultaneously, thus for a pair (X(ix),T(it)).

The value U(ix,it) at X(ix) and T(it) is given by an infinite series. Thus you need a third loop over the number of elements of the series you want to account for. This is the loop with isum. If you feel better then, you can replace isum by n. For simplicity, I chose a fixed value for the number of elements of the series that are summed (31) (independent of x and t). Usually, this is done by evaluating the size of the series elements and a stopping criterion for each pair (X(ix),T(it)) separately.

[Ali Mosa]

Thanks so much for taking the time to explain!

Answer 2

[Paul]

Hi Ali,

Another approach that you might be interested in, which is less code but requires more memory, is to use an anonymous function

u = @(x,t,N) 400/pi^3*sum(sin(pi*reshape(1:2:N,1,1,[]).*x).*exp(-reshape(1:2:N,1,1,[]).^2.*pi^2.*0.003.*t)./reshape(1:2:N,1,1,[]).^3,3);
[X,T] = meshgrid(0:.1:1);
Nsum = 31; % uppler limit on the n-summation
surf(X,T,u(X,T,Nsum))

Memory could be reduced by loop-summing u over scalar values of n for a fixed set of values of X and T

U = 0*X;
for n = 1:2:31
    U = U + u(X,T,n);
end
surf(X,T,U)