Midpoint method

Improved Euler's method is another one-stepped method which gives the approximation of the solution of the differential equation

$y'=f(x,y),\qquad y(a)=\alpha$

by turning it into a system called a discrete begining problem:

$u(a)=\alpha _{h}$

$u(x+h)=u(x)+hf_{h}(x,u(x))$

where u(x) is an approximation of the function f, and h is called a discretisation step. The points {x₀, x₁, ..._n} are usually equidistant, i.e. x_i = x₀ + ih, i=0,1,...n

With the Improved Euler method, function f is approximated in the following way:

$y'(x+{\frac {h}{2}})=y(x)+y'(x){\frac {h}{2}}$

Finally, we get the formula for the Improved Euler's method:

$u(a)=\alpha _{h}$

$u(x+h)=u(x)+hf(x+{\frac {h}{2}},u(x)+{\frac {h}{2}}f(x,u(x)))$