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vpde_lecture24 [2020/04/04 21:55]
trinh [Example 15.4: Solution of the inhomogeneous Dirichlet problem] 		vpde_lecture24 [2020/04/04 22:00] (current)
trinh
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 $$ $$
  
 +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: 2020/04/04 21:55