Difference Equation

It defines how an elements in a sequence are related.

Consider $y : \mathbb{N} \rightarrow \mathbb{R}$ defined by the recursive relations.

$$y_1 = c$$ $$y_{n+1} = ay_n$$

$$ \implies \boxed{y_n = a^{n-1}c}$$

This is 1st order homogeneous linear difference equation.

Similarly for kth order:

  • Law of motion: $$y_{n+k} + a_{k-1}y_{n+k-1} + ... a_0 y_n = 0 $$

  • Initial Conditions for the law of motions: (k initials). $y_0 = c_0, ... y_{k-1} = c_{k-1}$ to compute the initial $y_{n+k}$

  • a's are law of motions. c's are initial conditions.

2nd Order

We have the characteristic equation $r^2 + a_1 r + a_0 = 0$

Roots can be

  • real and distict $\implies y_n = A(r_+)^n + B(r_-)^n$
  • real and same $\implies y_n = A(r)^n + Bn(r)^n$
  • complex $\implies y_n = A(r)^n cos(n\theta) + Bn(r)^n sin(n \theta)$

Ordinary Differential Equation (ODEs)

$$ y' = f(y, x) $$ $$ y(0) = c $$

  • Wehn differential eqn is solution?
  • Is it unique?
  • When depends continuously on parameters? - this is important from numerical perspective!- because $y_p$ might be very different from $y_{p+\epsilon}$

Lets compare with linear equation. $y = Ax$

  • A invertible $\implies$ unique solution
  • not invertible $\implies$ no / infinite solutions

This carries over to linear ODEs.

Picard's Theorem

Consider the first order ODE: $\frac{dy}{dt} = f(t, y(t))$, $y(t_0) = y_0$. Givem that f is Lipschitz continuous (i.e. Holder with exponent 1) in y and continuous in t. Then there exists a unique solution to the ODE in a neighbourhood of $t+0$.