ode45 solving of diff.equation with further fitting to exp.results

Question

I am building a code to solve a diff. equation:

function dy = KIN1PARM(t,y,k)
%
% version : first order reaction
%   A  --> B
%   dA/dt = -k*A
%   integrated          


        

Answer 1

So here comes the best which I can get right now. For my way I tread ordinatus values as time and the abscissa values as measured quantity which you try to model. Also, you seem to have set alot of options for the solver, which I all omitted. First comes your proposed solution using ode45(), but with a non-zero y0 = 100, which I just "guessed" from looking at the data (in a semilogarithmic plot).

function main 

abscissa = [0; 
            240;
            480;
            720;
            960;
            1140;
            1380;
            1620;
            1800;
            2040;
            2220;
            2460;
            2700;
            2940];

ordinatus = [  0;
            19.6;
            36.7;
            49.0;
            57.1;
            64.5;
            71.4;
            75.2;
            78.7;
            81.3;
            83.3;
            85.5;
            87.0;
            87.7];

tspan = [min(ordinatus), max(ordinatus)]; % // assuming ordinatus is time

y0 = 100; % // <---- Probably the most important parameter to guess
k0 = -0.1; % // <--- second most important parameter to guess (negative for growth)

        k_opt = fminsearch(@minimize, k0) % // optimization only over k
        % nested minimization function
        function e = minimize(k)
            sol = ode45(@KIN1PARM, tspan, y0, [], k);
            y_hat = deval(sol, ordinatus); % // evaluate solution at given times
            e = sum((y_hat' - abscissa).^2); % // compute squarederror           
        end

% // plot with optimal parameter
[T,Y] = ode45(@KIN1PARM, tspan, y0, [], k_opt);
figure
plot(ordinatus, abscissa,'ko', 'markersize',10,'markerfacecolor','black')
hold on
plot(T,Y, 'r--', 'linewidth', 2)


% // Another attempt with fminsearch and the integral form
t = ordinatus;
t_fit = linspace(min(ordinatus), max(ordinatus))
y = abscissa;

% create model function with parameters A0 = p(1) and k = p(2)
model = @(p, t) p(1)*exp(-p(2)*t);
e = @(p) sum((y - model(p, t)).^2); % minimize squared errors
p0 = [100, -0.1]; % an initial guess (positive A0 and probably negative k for exp. growth)
p_fit = fminsearch(e, p0); % Optimize 

% Add to plot
plot(t_fit, model(p_fit, t_fit), 'b-', 'linewidth', 2)

legend('location', 'best', 'data', 'ode45 with fixed y0', ...
    sprintf ('integral form: %5.1f*exp(-%.4f)', p_fit))
end

function dy = KIN1PARM(t,y,k)
%
% version : first order reaction
%   A  --> B
%   dA/dt = -k*A
%   integrated form A = A0*exp(-k*t)
%
dy = -k.*y; 
end 

The result can be seen below. Quit surprisingly to me, the initial guess of y0 = 100 fits quite well with the optimal A0 found. The result can be seen below: