## Topic 14.7: Systems of Initial-Value Problems (HOWTO)

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# Problem

Given the vector u(t) = (u1(t), ..., un(t))T, where, for each function uk(t), we have a first-order ODE deined by:

u(1)k(t) = fk(t, u(t))

then we may define

u(1)(t) = f(t, u(t))

where

Given this, together with the initial value u(t0) = u0 = (u0,1, ..., u0,n)T, we would like to approximate the solution to this IVP.

# Assumptions

The function f(t, u) is continous.

# Tools

We will use Euler, Heun, and 4th-order Runge Kutta, though the extension to RKF45 is straight-forward.

# Process

Given a step size h, then tk = t0 + kh. We now examine how each method may be implemented.

### Euler

Given uk, let:

uk + 1 = uk + h f(tk, uk)

### Heun

Given uk, let:

K0 = f(tk, uk)
K1 = f(tk + h, uk + hK0)

Then we define:

### 4th-order Runge Kutta

Given uk, let:

K0 = f(tk, uk)
K1 = f(tk + ½h, uk + ½hK0)
K2 = f(tk + ½h, uk + ½hK1)
K3 = f(tk + h, uk + hK2)

Then we define: