%% Section 10.6 Problem 9 %% % The problem is to solve the heat equation in a 20 cm aluminum bar (so % $\alpha^2 = 0.86$), with initial value 25, u(0,t)=0, u(20,t)=60 for t>0. %% % The first step is to find a steady state solution v with v(0,t)=0, % v(20,t)=60. The solution is v=3x. Write u = v+w, so w = u-v solves the % heat equation with homogeneous boundary conditions and initial value % 25 -3x. Let b(n) be the nth coefficient in the Fourier sine series of % 25-3x. syms x t n k b = @(n) int(2*(25-3*x)*sin(n*pi*x/20),x,0,20)/20; %% % The temperature at time t is u = @(x,t,n) 3*x + symsum(b(k)*exp(-0.86*k^2*pi^2*t/400)*sin(k*pi*x/20),k,1,n); %% % with $n = \infty$. %% % I plot the initial value, the steady state solution, and the solution at % two other times, using n=3. X = 0:.04:20; U = inline(vectorize(u(x,t,3))); plot(X,3.*X,'r') hold on plot(X,U(5,X),'g') plot(X,U(50,X),'k') ezplot(25+0*x,[0,20]) hold off legend1 = legend({'t=Steady state','t=5','t=50','Initial value'},'Position',[0.1625 0.6967 0.1957 0.1869]); title('') %% % Now I see what happens if I take n=10. U = inline(vectorize(u(x,t,10))); plot(X,3.*X,'r') hold on plot(X,U(5,X),'g') plot(X,U(50,X),'k') ezplot(25+0*x,[0,20]) hold off legend1 = legend({'t=Steady state','t=5','t=50','Initial value'},'Position',[0.1625 0.6967 0.1957 0.1869]); title('') %% % Next I plot u as a function of t for certain values of x. ezplot(u(5,t,3),[0,50]) title('x=5') %% ezplot(u(10,t,3),[0,50]) title('x=10') %% ezplot(u(15,t,3),[0,50]) title('x=15') %% % Next I find where the first term in the series is within 1% of the steady % state temperature (which is 3*5=15) at x=5. The first term is u(5,t,1) %% % which is within f = 15 - u(5,t,1) % of the steady state solution. So, I want to find when $|f|\le 0.15$. %% f = 15 - u(5,t,1); double(solve(f-0.15,t))