For this we will be combining while and if control structures together. Using MA

Question

For this we will be combining while and if control structures together. Using MATLAB.

• Write a pseudocode (pen and paper) algorithm that solves the ODE y 0 = f(t, y) using Euler’s method.

• Convert this algorithm into MATLAB syntax, as the function euler method .

• Your inputs should be the start time t0, the end time tend, the function f, an initial condition y0 and an initial step size h.

• If h × f(t, y) at any point exceeds 1, your function should halve the step size h.

• Equally, if h×f(t, y) at any point drops below 0.01, your function should double the step size h.

• You should also make sure that the final solution step doesn’t ‘overshoot’ tend. (i.e. change h during the final step to exactly reach tend).

• Your MATLAB code should use the function header below, and be well-commented with a sensible layout.

• Include your pseudocode algorithm, as well as the MATLAB code. Also include a discussion of when the could could break, or give incorrect outputs (you DO NOT need to design your code to avoid these).

Explanation / Answer

%
function [ts, ys] = odemeul (rhsfun, tspan, y0, options, varargin)
    try,        h=options.(matchfieldname(options, 'InitialStep'));
    catch,      error(['Option InitialStep must be supplied']);    end
    nsteps = ceil ((tspan(2) - tspan(1)) / h);
    ts = zeros(nsteps+1,1);
    ys = zeros(nsteps+1,length(y0));
    t = tspan(1);
    ts(1) = t;
    ys(1,:) = conj (y0');
    for step = 2:nsteps+1
        if (ts(step-1) + h >tspan(2)), h=tspan(2)-ts(step-1); end
        ts(step) = ts(step-1) + h;
        ys(step,:) = (feval ('MEuler', rhsfun, h, ts(step-1), ys(step-1,:)', varargin{:}))';
    end
    ys=conj(ys);
    return;
end

%
% Do one Modified Euler step for the equation dy/dt = f(t,y).
%
% y_n+1 = y_n + h/2 * (f(t_n+1, y_n+1) + f(t_n, y_n))
%
% which is approximated as
%
% y_n+1 = y_n + h/2 * (f(t_n+1, y_p) + f(t_n, y_n)) ,
%
% where y_p = y_n + h * f(t_n, y_n)
%
% Arguments:
%   rhsfun = rhs function f(t,y)
%   h      = step length
%   tn     = time t_n
%   yn     = function value y_n
%
function [yn1] = MEuler (rhsfun, h, tn, yn, varargin)
    tp = tn + h;
    fn = feval (rhsfun, tn, yn, varargin{:});
    yp = yn + h * fn;
    yn1 = yn + h/2 * (fn + feval (rhsfun, tp, yp, varargin{:}));
end

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