Euler Method - Formulation of The Method

Formulation of The Method

Suppose that we want to approximate the solution of the initial value problem

Choose a value for the size of every step and set . Now, one step of the Euler method from to is

The value of is an approximation of the solution to the ODE at time : . The Euler method is explicit, i.e. the solution is an explicit function of for .

While the Euler method integrates a first-order ODE, any ODE of order N can be represented as a first-order ODE: to treat the equation

,

we introduce auxiliary variables and obtain the equivalent equation

\mathbf{z}'(t) = \begin{pmatrix} z_1'(t)\\ \vdots\\ z_{N-1}'(t)\\ z_N'(t) \end{pmatrix} = \begin{pmatrix} y'(t)\\ \vdots\\ y^{(N-1)}(t)\\ y^{(N)}(t) \end{pmatrix} = \begin{pmatrix} z_2(t)\\ \vdots\\ z_N(t)\\ f(t,z_1(t),\ldots,z_N(t)) \end{pmatrix}

This is a first-order system in the variable and can be handled by Euler's method or, in fact, by any other scheme for first-order systems.

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