Question

I'm still trying to learn how to use matlab but I'm not sure how to do this at all. I am to implement three different explicit methods for approximating an IVP using Eulers, Runge-Kutta 4th order method, and the trapezoidal method using the same parameters. I'd much appreciate having just the Euler methods. but having all three done would be greatly appreciated and will give a great rating!

function y forward_euler (n, m, a, b, eta,F) % Use the forward Euler method to approximate the solution of % the initial value problem %y'(t) - F(t,y(t)) fora<t< b y (a)eta % where F is a smooth mapping R x R^n \to Ran and eta % a row vector of length n speciying the initial values for the % solution. % Input parameters: % n - the number of unknown variables in the ordinary dfferential equation; also, the length of the vector y % m - the number of steps to take % (a,b) the interval on which to solve the system F - a handle to a function specifing F and whose calling % syntax is function val F(t,y) Return the value of the function F at the point (t,y) The input t isa real number, the input y is a ROW vector of length n and the output val is a ROW vector of length n % Return value: % y - a (n,m) matrix whose jth column contains the approximation of the solution at the point t_j

Answer 1

Answer:)

Code:

function y = forwardeuler(n,m,a,b,eta,F)
    y = zeros(n,m); %pre-allocate the result with all zeros
    if(size(eta,2) ~= 1)
        y(:,1) = eta'; %set the first column to be initial condition
    else
        y(1) = eta;
    end
    h = (b-a)/m; %define the step-size
    t = a:h:b-h; %set up the time vector;
    for i = 1:m-1
        y(:,i+1) = y(:,i) + h*F(t(i),y(:,i)')'; %euler update step
                                                %note that we have
                                                %transposed the F-output so
                                                %that the dimensions match
    end
end