Download Engineering Mathematics | CHEN30101 problem sheet 6 1

Engineering Mathematics | CHEN30101
problem sheet 6
1. Suppose that we are using the FTCS approximation method to solve the
heat equation. Denote the spatial grid resolution by h and suppose that
the grid points are x0 , x1 , . . . , x10 at time level tn = nk.
Determine the discrete equation that un9 , un10 and un11 will satisfy if the
condition at the right-hand boundary is given by (i) u = 1, (ii) ∂u
∂x = 0,
∂u
(iii) ∂u
=
2.3,
(iv)
+
4.7u
=
2.1.
∂x
∂x
2. Consider the initial value problem: find u(x, t) such that
ut − uxx = 0
u(0, t) = 0;
in (0, 1) × (0, τ )
u(1, t) = 0,
u(x, 0) = x2 (1 − x),
t>0
x ∈ (0, 1).
Using a grid resolution of h = 0.2 and k = 0.01, carry out two steps of
the FTCS approximation method.
What is the maximum time step possible if a stable FTCS approximation
is to be computed for a spatial grid resolution of (i) h = 10, (ii) h = 1,
(iii) h = 0.1, (iv) h = 0.01?
3. Consider the initial value problem: find u(x, t) such that
ut − uxx = 0
u(0, t) = 0;
in (0, 1) × (0, τ )
u(1, t) = 1,
u(x, 0) = x2 ,
t>0
x ∈ (0, 1).
Using a grid resolution of h = 0.25 and k = 0.1, carry out two steps of
the Crank–Nicolson approximation method by working out the discrete
equations one at a time, and then solving a 3 × 3 equation system at each
time step. Work to an accuracy of four decimal places.
4. Consider the initial value problem: find u(x, t) such that
ut − uxx = 0
u(0, t) = 0;
in (0, 1) × (0, τ )
u(1, t) = 0,
u(x, 0) = sin(πx),
t>0
x ∈ (0, 1).
Using a grid resolution of h = 0.2 and k = 0.01, carry out the first step of
(a) the FTCS approximation method
(b) the BTCS approximation method∗
(c) the Crank–Nicolson method.∗
∗
An efficient way of doing this is to use the function trisolve.m that is associated with
Computational Exercise IV.
1
Work to an accuracy of four decimal places and compare your numerical
2
results with the exact values that are given by u(x, t) = e−π t sin(πx).
5. Consider the initial value problem: find u(x, t) such that
utt − uxx = 0
u(0, t) = 0;
in (0, 1) × (0, τ )
u(1, t) = 0,
u(x, 0) = sin(πx),
t>0
ut (x, 0) = 0,
x ∈ (0, 1).
(a) By defining a new dependent variable v = ut , convert the problem
to a system of two linear PDEs that are first order in time.
(b) Hence, using a grid resolution of h = 0.25 and k = 0.01, carry out two
steps of the FTCS approximation method. Record all your answers
to an accuracy of four decimal places.
6. Consider the boundary value problem: find u(x, y) such that
uxx + uyy = 0
in (0, 3) × (0, 3)
u(0, y) = 0,
u(3, y) = 0,
y ∈ (0, 3)
u(x, 0) = x(3 − x),
u(x, 3) = 0,
x ∈ (0, 3).
Consider the grid with h = 1. Compute an approximation to the solution
using the 5-point difference scheme by working out the discrete equations
one at a time, and then solving a 4 × 4 equation system by hand.†
7. Consider the boundary value problem: find u(x, y) such that
uxx + uyy = 0
in (0, 1) × (0, 1)
u(0, y) = y,
u(1, y) = 1 + 2y,
y ∈ (0, 1)
u(x, 0) = x,
u(x, 1) = 1 + 2x,
x ∈ (0, 1).
Consider the grid with h = 0.25. Construct an approximation to the
solution using the 5-point difference scheme by working out the discrete
equations one at a time, and forming a 9 × 9 equation system. (You do
not need to solve the system.† )
†
One way to check your answers is solve the same problem using the function laplacefd.m
presented in Computational Exercise V.
2