Initial value problems | Métodos Numéricos

What an initial value problem is

Differential equations model dynamical systems: the trajectory of a particle, the evolution of a temperature, an electrical circuit or the growth of a population. An initial value problem (IVP) combines the differential equation with the state of the system at the initial time:

y'(t)=f(t,y(t)),\qquad t\in[a,b],\qquad y(a)=y_a

A classic example is the Malthus population model, $y'(t)=k\,y(t)$ : the growth rate is proportional to the number of individuals (with $k<0$ it describes decay). The logistic or Verhulst model adds a nonlinear term that slows growth when resources are limited: $y'(t)=k\,y(t)-p\,y(t)^2$ . The latter appears as a recurring example in the methods of this section.

Existence and uniqueness of the solution

Before approximating a solution it helps to know that one exists and is unique. The condition controls how fast $f$ changes with respect to $y$ :

From the continuous to the discrete solution

Analytic techniques produce the continuous solution $y(t)$ , but they only work for specific families of equations. Numerical methods approximate the solution when the analytic computation is impossible or too costly: they produce a discrete solution $y_k\approx y(t_k)$ on the nodes of a partition of the interval.

t_k=a+kh,\qquad k=0,1,\dots,N,\qquad h=\frac{b-a}{N}

Equally spaced discretization of

[a,b]

into

N

subintervals of step

h

.

Every method in this area (Euler, Heun, Runge-Kutta and the Adams multistep methods) follows this scheme: start from $y_0=y_a$ and advance node by node, building $y_{k+1}$ from the available information.

Systems of first-order equations

When several quantities evolve together (for instance interacting populations), the IVP is stated with $m$ unknown functions $y_1,\dots,y_m$ and one equation for each. With vector notation $Y(t)=[y_1(t),\dots,y_m(t)]^T$ the system mirrors the scalar case:

Y'(t)=F\bigl(t,Y(t)\bigr),\qquad Y(a)=\bigl[y_{1,a},\dots,y_{m,a}\bigr]^T

This notation carries over to computation: one-step methods apply to the system component by component with no conceptual change, as the SIR epidemic model exercise shows.

Higher-order equations

An order- $m$ equation, $y^{(m)}(t)=f\bigl(t,y,y',\dots,y^{(m-1)}\bigr)$ , defines an IVP when $y$ and its derivatives up to order $m-1$ are known at the initial time. To solve it numerically, convert it into a first-order system:

Reduction to a first-order system

Introduce new variables: the function and its successive derivatives.
$y_1(t)=y(t),\quad y_2(t)=y'(t),\quad \dots,\quad y_m(t)=y^{(m-1)}(t)$
Differentiating each new variable yields the next one, and the last one carries the original equation.
$y_1'=y_2,\quad y_2'=y_3,\quad \dots,\quad y_m'=f\bigl(t,y_1,\dots,y_m\bigr)$
The initial conditions of the equation translate directly into those of the system: $y_1(a)=y(a)$ , $y_2(a)=y'(a)$ , and so on. The sought solution is the first component, $y(t)=y_1(t)$ .

ExampleThe pendulum as a system

Write as a first-order system the equation of a pendulum of length $L$ , $\theta''(t)-\frac{g}{L}\sin\theta(t)=0$ , with initial angle $\theta(0)=\pi/6$ and initial angular velocity $\theta'(0)=0$ .

With $y_1=\theta$ and $y_2=\theta'$ , the second-order equation becomes two first-order equations:
$\begin{bmatrix} y_1' \\ y_2' \end{bmatrix}=\begin{bmatrix} y_2 \\ \tfrac{g}{L}\sin y_1 \end{bmatrix}$

The initial conditions of the system are $y_1(0)=\pi/6$ , $y_2(0)=0$ , and any one-step method can integrate it in vector form.