Problem Sheet 6 - Systems of Equations and Boundary Value Problems#

Question 1#

  1. Apply the Euler method to estimate the solution of the system of equations

(534)#\[\begin{align} \frac{dx}{dt}= x-0.2xy; &\ \ x(1) = 10;\\ \frac{dy}{dt}= y + 0.1xy; &\ \ y(1) = 2; \end{align}\]

using a stepsize h = 0.25, giving the approximate value of the solution at t = 2.0.

Question 2#

  1. Use the Euler method to estimate the solution of the Lorenz system of first order initial value problems,

(535)#\[\begin{align} \frac{dx}{dt}=\sigma(y-x),& \ \ \ x(0)=1,\\ \frac{dy}{dt}=x(\rho-z)-y, &\ \ \ y(0)=1,\\ \frac{dz}{dt}=xy-\beta z,& \ \ \ z(0)=1,\end{align}\]

where \(\sigma=8\), \(\rho=2\) and \(\beta=\frac{7}{3}\), and using \(h=0.25\), estimate the value of the solution at \(t=1.0\).

Question 3#

  1. Consider the boundary value problem

(536)#\[\begin{equation} y''=3xy'-4y+x^2, \ \ 1\leq x \leq 2, \ \ y(1)=2,\ \ \ y(2)=-1.\end{equation}\]

Apply the linear shooting method to transform this equation into two second order initial value problems and approximate the solution using the Euler method with stepsize \(h=\frac{1}{3}.\)

Question 4#

  1. Describe in your own words how the linear shooting method is used to numerically approximate a boundary value problem as a system of initial value problems.

Question 5#

  1. The Van der Pol oscillator is a non-conservative oscillator with non-linear damping is described by the differential non-linear second order differential equation,

(537)#\[\begin{equation} u^{''}=-\epsilon(u^2-1)u^{'}-u.\end{equation}\]

Let \(\epsilon=1.5\), and given the initial condition

(538)#\[\begin{equation} u(0)=0.0, \text{ and, } u(1)=0. \end{equation}\]

Apply the one iteration of the non-linear shooting method to approximate the solution from, \(0\leq t \leq 1,\) using a stepsize of \(h=0.2.\)

Question 6#

  1. a. Describe in your own words how the non-linear shooting method is used to numerically approximate a boundary value problem as a system of initial value problems.

Question 7#

  1. Consider the boundary value problem

(539)#\[\begin{equation} y''=3xy'-4y+x^2, \ \ 1\leq x \leq 2, \ \ y(1)=2,\ \ \ y(2)=-1.\end{equation}\]

Apply the finite difference method to approximate the solution using the Euler method with stepsize \(h=\frac{1}{3}.\)