Solving the function ODE45

I have to solve the following equation with the initial condition of C_{11}=1 and C_{12}=0.

Large D means material derivative.

Following is the code, and I am not sure how I could run the code.

function solve_pde()
    % Parameters
    Gamma = 4.0;  % Circulation
    nu = 0.01;  % Kinematic viscosity
    x = linspace(-2, 2, 100);  % x-coordinates
    y = linspace(-2, 2, 100);  % y-coordinates
    [X, Y] = meshgrid(x, y);
    tspan = [0, 1];  % Time span for the simulation

    % Initial conditions for the conformation tensor components
    C_11_init = ones(size(X));
    C_12_init = zeros(size(X));

    % Combine initial conditions into a single vector for the ODE solver
    C0 = [C_11_init(:); C_12_init(:)];

    % Solve the system of ODEs
    [t, C] = ode45(@(t, C) conformation_tensor_pde(t, C, X, Y, Gamma, nu), tspan, C0);

    % Extract the results for each component of the conformation tensor
    num_points = numel(X);
    C_11 = reshape(C(end, 1:num_points), size(X));
    C_12 = reshape(C(end, num_points+1:end), size(X));

    % Plot the results
    figure
    surf(X, Y, C_11)
    xlabel('X');
    ylabel('Y');
    zlabel('C_{11}');
    title('Evolution of C_{11}');
end

function dCdt = conformation_tensor_pde(t, C, X, Y, Gamma, nu)
    % Reshape C into the conformation tensor components
    num_points = numel(X);
    C_11 = reshape(C(1:num_points), size(X));
    C_12 = reshape(C(num_points+1:end), size(X));

    % Velocity field components in Cartesian coordinates
    r = sqrt(X.^2 + Y.^2);
    u = -Gamma * Y ./ (2 * pi * (X.^2 + Y.^2)) .* (1 - exp(-r.^2 / (4 * nu * t)));
    v = Gamma * X ./ (2 * pi * (X.^2 + Y.^2)) .* (1 - exp(-r.^2 / (4 * nu * t)));

    % Velocity gradients in Cartesian coordinates
    dudx = Gamma * Y .* (2 * X .* (1 - exp(-r.^2 / (4 * nu * t))) + X .* exp(-r.^2 / (4 * nu * t)) / (2 * nu * t)) ./ (2 * pi * (X.^2 + Y.^2).^2);
    dudy = -Gamma * (1 - exp(-r.^2 / (4 * nu * t))) ./ (2 * pi * (X.^2 + Y.^2)) + Gamma * Y.^2 .* (2 * (1 - exp(-r.^2 / (4 * nu * t))) - exp(-r.^2 / (4 * nu * t)) / (2 * nu * t)) ./ (2 * pi * (X.^2 + Y.^2).^2);

    % Finite differences for spatial derivatives
    [dC11dx, dC11dy] = gradient(C_11, X(1, :), Y(:, 1));

    % Compute the material derivative of the conformation tensor component C_11
    dC11dt = 2 * (C_11 .* dudx + C_12 .* dudy) - u .* dC11dx - v .* dC11dy;

    % For completeness, compute the derivative of C_12 (if necessary for other equations)
    dC12dt = zeros(size(C_12));  % Adjust based on the specific problem requirements

    % Pack the derivatives into a column vector
    dCdt = [dC11dt(:); dC12dt(:)];
end

[Torsten]

Are u and v given function ?

You only have one partial differential equation for C_11. Where is the equation for C_12 ?

Further, assuming u and v are positive, you need boundary conditions at the lines (xmin,y) and (x,ymin).

Answer 1

[Nipun]

Hi JungYeon,

I understand that you intend to solve the given ordinary differential equation using MATLAB and require help with running the given code.

Based on the given code and upon close inspection, I can verify that the given code corresponds to the solution for the given ODE.

In order to call your solution, type the following in the MATLAB command window:

>> solve_pde();

This command will call the function "solve_pde", which in turn uses "conformation_tensor_pde" to solve the equation. Additionally, a plot will be generated at the end of the function.

For more information on calling functions in MATLAB, refer to the following MathWorks documentation: https://www.mathworks.com/help/matlab/ref/function.html

Hope this helps.

Regards,

Nipun

[John D'Errico]

That is NOT an ordernary differential equation. It is a PARTIAL differential equation. There are derivatives with respect to three variables, t, x, and y.