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vpde_lecture24 [2020/03/28 12:43]
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-====== Computation of the heat equation III ======+====== Lecture 24: Computation of the heat equation III ======
  
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-//Need to write this.//+====== Example 15.4: Solution of the inhomogeneous Dirichlet problem ====== 
 + 
 +This lecture starts off with Example 15.4 from the notes, where we look to solve 
 +$$ 
 +\begin{gathered} 
 +u_t = u_{xx} \\  
 +u(0, t) = 2, \qquad u(\pi, t) = 1 \\  
 +u(x, 0) = 0. 
 +\end{gathered} 
 +$$ 
 + 
 +We show that the solution is given by  
 +$$ 
 +u(x, t) = U(x) + \sum_{n=1}^\infty B_n \sin(nx) \mathrm{e}^{-n^2 t}  
 +$$ 
 +where we have found the steady-state solution  
 +$$ 
 +U(x) = 2 - \frac{x}{\pi}, 
 +$$ 
 +as well as the coefficients 
 +$$ 
 +B_n = \frac{2}{n\pi}[(-1)^n - 2]. 
 +$$ 
 + 
 +We also illustrated the solution using the code: 
 + 
 +<Code:Matlab linenums:1 |Inhomogeneous heat equation> 
 +%% Plot the Fourier series for Example 15.4 
 +% Written for MA20223 Vectors & PDEs 2019-20 
 + 
 +clear           % Clear all variables 
 +close all       % Close all windows 
 +N = 100;        % How many Fourier modes to include? 
 + 
 +R = @(n, t, x) 2/(n*pi)*((-1)^n - 2)*exp(-n^2*t)*sin(n*x); 
 + 
 +% Create a mesh of points between two limits 
 +x0 = pi; x = linspace(0, x0, 1000); 
 + 
 +% Create a mesh of points in time 
 +t = linspace(0, 5, 200); 
 + 
 +figure(1);                              % Open the figure 
 +plot(x, 2 - x/pi, 'b', 'LineWidth', 2); % Plot the base function 
 +ylim([-0.2,2]);                         % Set the y limits 
 +xlim([0, x0]);                          % Set the x limits 
 +xlabel('x'); ylabel('u(x,t)'); 
 +hold on 
 + 
 +for j = 1:length(t) 
 +    tj = t(j); 
 +     
 +    u = (2 - x/pi); 
 +    for n = 1:N 
 +        u = u + R(n, tj, x); 
 +    end 
 +    if j == 1 
 +        p = plot(x, u, 'r'); 
 +    else 
 +        set(p, 'YData', u); 
 +    end 
 +    drawnow 
 +    title(['t = ', num2str(tj)]); 
 +    pause(0.1); 
 +    
 +    if j == 1 
 +        pause 
 +    end 
 +end 
 +</Code> 
 + 
 +===== Problem set 8 Q1 ===== 
 + 
 +The next thing we studied was PS8, Q1, which is the solution of the homogeneous Neumann problem for the heat equation.  
 + 
 +<Code:Matlab linenums:1 |Modification of PS8 Q1> 
 +%% Plot the Fourier series for a made-up modification of PS8 Q1.  
 +% Written for MA20223 Vectors & PDEs 
 + 
 +clear           % Clear all variables 
 +close all       % Close all windows 
 +N = 20;          % How many Fourier modes to include? 
 + 
 +% Define an in-line function that takes in three inputs:  
 +R = @(n, t, x) 4*(-1)^n/n^2*cos(n*x)*exp(-n^2*t); 
 + 
 +% Create a mesh of points between two limits 
 +x0 = pi; 
 +x = linspace(0, x0, 1000); 
 + 
 +% Create a mesh of points in time 
 +t = linspace(0, 5, 200); 
 + 
 +figure(1);                              % Open the figure 
 +plot(x, x.^2, 'b', 'LineWidth', 2);     % Plot the base function 
 +ylim([-0.2,pi^2]);                      % Set the y limits 
 +xlim([0, x0]);                          % Set the x limits 
 +xlabel('x'); ylabel('u(x,t)'); 
 +hold on 
 + 
 +for j = 1:length(t) 
 +    tj = t(j); 
 +     
 +    u = 1/2*(2*pi^2/3); 
 +    for n = 1:N         
 +        u = u + R(n, tj, x); 
 +    end 
 +    if j == 1 
 +        p = plot(x, x.^2, 'r'); 
 +    else 
 +        set(p, 'YData', u); 
 +    end 
 +    drawnow 
 +    title(['t = ', num2str(tj)]); 
 +    pause(0.1); 
 +    
 +    if j == 1 
 +        pause 
 +    end 
 +end 
 +</Code>
Last modified: 2020/03/28 12:43