Chapter 2.8, Problem 5E

Interpretation Introduction

Interpretation:

Using the Runge-Kutta 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 Runge-Kutta method is used for finding the approximate values of a solution of a non-linear initial value problem.

It is preferred over the Euler method since it is a more accurate method than the Euler method.

The error which is obtained by the Runge-Kutta method is relatively smaller.

According to the ${\text{4}}^{\text{th}}\text{order}$ Runge-Kutta method, the solution of a differential equation of the form $\dot{\text{x}}\text{ = -f(x)}$ is

${\text{x}}_{\text{n+1}}{\text{= x}}_{\text{n}}\text{+}\frac{\text{1}}{\text{6}}{\text{(k}}_{\text{1}}{\text{+2k}}_{\text{2}}{\text{+2k}}_{\text{3}}{\text{+k}}_{\text{4}}\text{)}$