Initial Value Problem Review Questions#
a) Derive the Euler approximation show it has a local truncation error of \(O(h)\) of the Ordinary Differential Equation
with initial condition
[8 marks]
b) Suppose \(f\) is a continuous and satisfies a Lipschitz condition with constant L on \(D=\{(t,y)|a\leq t \leq b, -\infty < y < \infty \}\) and that a constant M exists with the property that
Let \(y(t)\) denote the unique solution of the IVP
and \(w_0,w_1,...,w_N\) be the approx generated by the Euler method for some positive integer N. Then show for \(i=0,1,...,N\)
You may assume the two lemmas:\ If s and t are positive real numbers \(\{a_i\}_{i=0}^{N}\) is a sequence satisfying \(a_0 \geq \frac{-t}{s}\) and \(a_{i+1} \leq (1+s)a_i +t \) then
For all \( x \geq 0.1\) and any positive m we have
[17 marks]
c) Use Euler’s method to estimate the solution of
at x=1, using \(h=0.25\).
[8 marks]
a) Derive the difference equation for the Midpoint Runge-Kutta method\
for solving the ordinary differential equation
by using a formula of the form
where \(k_1\) is defined as above,
and \(a\), \(b\), \(\alpha\) and \(\beta\) are constants are deteremined. Prove that \(a+b=1\) and \(b\alpha=b\beta=\frac{1}{2}\) and choose appropriate values to give the Midpoint Runge-Kutta method.
[18 marks]
b) Show that the midpoint Runge-Kutta method is stable.
[5 marks]
c)Use the Runge-Kutta method to approximate the solutions to the following initial value problem
with \(h=0.2\) with the exact solution \(y(t)=t +\frac{1}{1-t}\).
[10 marks]
a) Derive the two step Adams-Bashforth method:
and the local truncation error
[18 marks]
b) Apply the two step Adams-Bashforth method to approximate the soluion of the initial value problem:
Using \(N=4\) steps, given that \(y_1=0.6872\).
[15 marks]
a) Derive the Adams-Moulton two step method and its truncation error which is of the form
and the local truncation error
[23 marks]
b) Define the terms strongly stable, weakly stable and unstable with respect to the characteristic equation.
[5 marks]
c) Show that the Adams-Bashforth two step method is stongly stable.
[5 marks]
a) Given the initial value problem:
and a numerical method which generates a numerical solution \((w_n)_{n=0}^{N}\), explain what it means for the method to be convergent.
[5 marks]
b) Using the 2-step Adams-Bashforth method:
as a predictor, and the 2-step Adams-Moulton method:
as a corrector, apply the 2-step Adams predicitor-corrector method to approximate the solution of the initial value problem
using N=4 steps, given \(y_1=0.5\).
[18 marks]
c) Using the predictor corrector define a bound for the error by controlling the step size.
[10 marks]
6 a) Given the Midpoint point (Runge-Kutta) method
Assume that the Runge-Kutta method satisfies the Lipschitz condition. Then for the initial value problems
Show that the numerical solution \(\{ w_n\}\) satisfies
where
If the consistency condition
where
is satisfied then the numerical solution \(w_n\) converges to \(Y(x_n)\).
[18 marks]
b) Consider the differential equation
Apply the midpoint method to approximate the solution at \(y(0.4)\) using \(h=0.2.\)\
[11 marks]
c) How would you improve on this result.
[4 marks]