Hello! I am having trouble with an academic problem; my professor would like me to 'replace the inner most for loop with two lines of code using .* as an operator and the sum function. I am utterly lost, because I think in for loops and this curveball feels outside the strike zone. Any insight would be awesome!
Question:Solutions to partial differential equations are often in the form of "Fourier series", which we will learn more about later in the course. Consider heat conduction in a one-dimensional rod of length L (0≤x≤L), with an initial temperature q(x, 0) = 0, the end at x = L kept insulated at all times (i.e. temperature gradient = 0) and the end at x = 0 increased to a temperature q0at time = 0 and maintained at q0thereafter.€∂θ∂t−α∂2θ∂x2=0, 0≤x≤L;θ(x,0)=0θ(0,t)=θ0 for t >0 and ∂θ∂x(L,t)=0 for all tHere, ais the thermal diffusivity of the rod, as an example, takehypothetical valuesof a= 1 and L = 1, and you can also considerq0= 1.The Fourier-series solution to the heat equation, which describes the evolution of the temperature distribution with time, is given by(note that the summation over n starts from n=0, not n=1):€θ(x,t)=θ01−4sin(2n+1)πx2Lexp−(2n+1)2π2αt4L2(2n+1)πn=0∞∑(a) We wish to plot the solution at several times (t = 0.01, 0.1, 0.2, 0.5, 1, 2, 5). For this we will compute the solution at each of these time values, at discrete points from x=0 to x=1, say using a spacing of 0.05, by summing the series at each of these values of x. Write a MATLAB script that will begin with an array of x values, and compute a matrix theta (7x21) corresponding to the 7 different time values for which we need the solution at 21 discrete x values. I want you to use 3 nested for loops, in which the outer-most loop changes the time value, the next loop goes over the discrete x values, and the innermost loop accomplishes the summationThe series sum cannot be carried to infinity, but truncated at some value of n. Note that this is not so bad, because as n increases, the exponential terminside the summation decreases rapidly. Let us simply use 50 terms in the series.I want you to plot (on a single graph) the temperature distributions versus x at the 7 different times and comment on the behavior you see in the plot.(b) Rewrite the script from Part(a) by replacing the inner-most loop that does the summation with two lines of MATLAB code that take advantage of MATLAB's vector-matrix capabilities (hint: think .* and search MATLAB help for "sum").
Code:
theta0=1;
L=1;
alpha=1;
n=0;
a=zeros(7,21);
b=[];
c=1;
d=1;
for t=[0.01, 0.1, 0.2, 0.5, 1, 2, 5];
for x=0:0.05:length(L);
for n=0:50;
y=(((sin(2*n+1)*pi()*x)/(2*L))*exp((-(2*n+1)^2*pi()^2*...
alpha*t)/(4*L^2)))/((2*n+1)*pi());
theta=theta0*(1-4*(y));
b(n+1)=theta;
z=sum(b);
end
a(c,d)=z;
d=d+1;
end
d=1;
c=c+1;
end
x=0:0.05:1;
t=[0.01, 0.1, 0.2, 0.5, 1, 2, 5];
figure
plot(x,a)
title('Rod temperature distribution')
xlabel('Length of Rod, x')
ylabel('Temperature, T')
