(a) For a given ODE system, what are an initial value problem, a terminal value problem, and a…

(a) For a
given ODE system, what are an initial value problem, a terminal value problem, and
a boundary value problem?

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(b) What are
the basic differences between the numerical solution of ODEs and numerical integration?

(c)
Distinguish between the forward Euler and backward Euler methods.

(d) Define
local truncation error and order of accuracy and show that both the forward
Euler and the backward Euler methods are first order accurate.

(e) How does
the global error relate to the local truncation error?

(f) What is
an explicit RK method? Write down its general form.

(g) Define
convergence rate, or observed order.

(h) Name
three advantages that RK methods have over multistep methods and three
advantages that multistep methods have over RK methods.

(i) Why is
it difficult to apply error control and step size selection using the global
error?

(j) In what
sense are linear multistep methods linear?

(k) What are
the two Adams families of methods? What is the main distinguishing point between
them?

(l) Write
down the PECE method for the two-step formula pair consisting of the two-step Adams–Bashforth
method and the one-step, second order Adams–Moulton method.

(m) Define
region of absolute stability and explain its importance.

(n) What is
a stiff ODE problem? Why is this concept important in the numerical solution of
ODEs?

(o) Define
A-stability and L-stability and explain the difference between these concepts.