Chapter 2.8, Problem 7E

Interpretation Introduction

Interpretation:

The function ${\text{x(t}}_{\text{1}}{\text{) = x(t}}_{\text{0}}\text{+Δt)}$ is to be expanded as a Taylor series in time step $\text{Δt}$, up to order of ${\text{(Δt)}}^{\text{2}}$. It is to be shown that the local error is $\left|{\text{x(t}}_{\text{1}}{\text{)-x}}_{\text{1}}\right|\sim {\text{C(Δt)}}^{\text{2}}$.

Concept Introduction:

The Taylor series is a series expansion of a function $\text{f(x)}$ at a point in time step $\text{Δt}$

$\text{f(x) = f(a) + Δt f ' (a) +}\frac{{\text{(Δt)}}^{\text{2}}}{\text{2!}}\text{f '' (a) +}\frac{{\text{(Δt)}}^{\text{3}}}{\text{3!}}\text{f ''' (a) + }\text{. }\text{. }\text{. }$

From the given condition ${\text{x = x}}_{\text{0}}{\text{ at t = t}}_{\text{0}}$, we can substitute $\dot{\text{x}}\text{ = f(x)}$.

Euler method is a numerical approach for solving ordinary differential equation with a given initial value.

The general rule for Euler approximation is ${\text{x}}_{\text{n+1}}{\text{ = x}}_{\text{n}}{\text{+ f (x}}_{\text{n}}\text{)Δt}$

Local Error = $\left|{\text{ x(t}}_{\text{1}}{\text{) - x}}_{\text{1}}\right|$