How to include average of a variable in the equation setup of PDEPE

Hi everyone!

I am using PDEPE to solve a set of partial differential equations over space (x) and time. My equation is , where is the average of all the values of G from 0 to x. G is also a variable in this set of partial differential equations. How can I add this G_avg function?

Edit: I wont be able to do it this way - One way I know is to run the solver from [t,t+Δt], obtain G(x) for that value of time, feed back the G_avg and then run the solver again. Can I do it without this method?

Thanks!

Answer 1

[Nipun]

Hi Jairaj,

I understand that you want to solve a PDE using "pdepe" in MATLAB where the equation involves an average function "G_avg". Here's a possible approach to include "G_avg" without iterating through time steps manually:

function sol = solvePDE()
    % Define the PDE coefficients
    m = 0; % Symmetry parameter for pdepe (0 for slab, 1 for cylindrical, 2 for spherical)

    % Time and space discretization
    x = linspace(0, 1, 100); % Spatial domain
    t = linspace(0, 10, 50); % Time domain

    % Solve the PDE
    sol = pdepe(m, @pdefun, @icfun, @bcfun, x, t);
    
    % Plot results
    surf(x, t, sol(:,:,1));
    title('Solution of the PDE');
    xlabel('Space x');
    ylabel('Time t');
    zlabel('F(x,t)');
end

function [c, f, s] = pdefun(x, t, u, dudx)
    % Calculate G_avg(x)
    global G_values G_avg C;
    G_avg = trapz(G_values(1:find(x==G_values,1)))/x; % Approximate integral
    
    % Define PDE
    c = 1;
    f = dudx;
    s = G_avg - C;
end

function u0 = icfun(x)
    % Initial condition
    u0 = 0;
end

function [pl, ql, pr, qr] = bcfun(xl, ul, xr, ur, t)
    % Boundary conditions
    pl = ul;
    ql = 0;
    pr = ur - 1;
    qr = 0;
end

In this approach, "G_avg" is calculated within the "pdefun" function. This allows you to avoid iterating through time steps manually. Ensure you define "G_values" and "C" in your workspace before running the solver.

Hope this helps.

Regards,

Nipun

[Torsten]

This won't work since G is also a variable in the system of partial differential equations, thus a component of the u vector. And the solution u is input to "pdefun" separately for each grid point so that no average can be computed there over all grid points.