Chapter 2.8, Problem 4E

Interpretation Introduction

Interpretation:

Using the improved Euler method, the analytical solution to $\dot{\text{x}}\text{ = - x}$ for an initial condition $\text{x(0) = 1}$, the exact value of $\text{x(1)}$ as well as $\widehat{\text{x}}{\text{(1) with step size Δt = 1 and Δt = 10}}^{\text{-n}}\text{, for n = 1,2,3,4}$ is to be obtained. Also, the error $\text{E = |}\widehat{\text{x}}\text{(1) - x(1)|}$ as a function of $\text{Δt}$ and $\text{ln E vs ln t}$ graph should be plotted, and corresponding results should be explained.

Concept Introduction:

The improved error method is a first-order numerical approach to solve the ordinary differential equations with some given initial value.

The improved Euler method is

${\text{x}}_{\text{n+1}}{\text{= x}}_{\text{n}}\text{+ }\frac{\text{1}}{\text{2}}\left[{\text{x}}_{\text{n}}{\text{+ f(}\tilde{\text{x}}}_{\text{n+1}}\text{)}\right]\text{(Δt)}$