6.3.3 Practice set: partial differential equations

Partial differential equations

What a partial differential equation is (lesson 6.3.1)

0 of 10 correct

1.
What is the order of $u_t = u_{xx} + 5u_x$ ?
2.
For $u = e^{-4t}\cos 2x$ , find $u_{xx}$ at $(0, 0)$
3.
For $u = x^3 + 6xt$ , find the residual $u_t - u_{xx}$
4.
$e^{-at}\sin 2x$ solves $u_t = 3u_{xx}$ . Find $a$
5.
$u = \sin(x - 5t)$ solves $u_t + c\,u_x = 0$ . Find $c$
6.
$u = f(x - 3t)$ solves $u_t + 3u_x = 0$ , and $u(1, 0) = 7$ . Find $u(7, 2)$
7.
Find $b$ so that $x^3 + bxy^2$ solves $u_{xx} + u_{yy} = 0$
8.
Is $e^{-t}\sin x + 1$ a solution of $u_t = u_{xx}$ ? Type 1 for yes, 0 for no
9.
Find the residual $u_t - u_{xx}$ of $e^{-t}\sin 3x$ at $(\frac{\pi}{6}, 0)$
10.
Central differences give a residual of $2 \times 10^{-4}$ with $h = 0.1$ and $2 \times 10^{-6}$ with $h = 0.01$ . If it keeps shrinking like $h^2$ , it is $2 \times 10^{k}$ with $h = 0.001$ . Find $k$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Conditions

Initial, boundary and terminal conditions (lesson 6.3.2)

0 of 9 correct

1.
On $0 \le x \le \pi$ , with both ends held at 0, $u_t = u_{xx}$ starts from $u(x, 0) = 3\sin 2x$ . Find $u(\frac{\pi}{4}, 0.5)$
2.
A rod on $0 \le x \le 4$ has its ends held at 0 and 100. Find its steady temperature at $x = 1$
3.
$u_t + 2u_x = 0$ on $x \ge 0$ , with $u(x, 0) = 0$ and $u(0, t) = 5t$ . Find $u(2, 3)$
4.
With $\tau = T - t$ , a solution has $u_\tau = 5$ at a point. What is $u_t$ there?
5.
Find $u(\frac{\pi}{4}, 2.5)$ for $u_t + u_{xx} = 0$ with $u(x, 3) = \sin 2x$
6.
Which needs a terminal condition: $u_t = 2u_{xx}$ (type 0) or $u_t + 2u_{xx} = 0$ (type 1)?
7.
$u_t + u_x = 0$ on $x \ge 0$ , with $u(x, 0) = 2$ and $u(0, t) = 5$ . Find $u(1, 3)$
8.
Under $u_t = u_{xx}$ , a ripple $\sin 20x$ seen at time $T$ was $e^{400(T - t)}\sin 20x$ earlier. Going back 0.05 in time it grows by $e^{k}$ . Find $k$
9.
How many conditions does $u_t + 3u_x = 0$ need at the right end of $0 \le x \le 1$ ?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

Residuals by central differences

Write two functions that test a candidate solution $u(x, t)$ at one point, as in lesson 6.3.1. Here u is a function of two arguments, u(x, t).

heat_residual(u, x, t, h=1e-4) returns $u_t - u_{xx}$ at $(x, t)$ . Estimate $u_t$ by the central difference in $t$ and $u_{xx}$ by the second difference in $x$ , with step h.
transport_residual(u, c, x, t, h=1e-5) returns $u_t + c\,u_x$ at $(x, t)$ , with central differences in $t$ and in $x$ .

Raise ValueError if h is not positive.

The tests use the solutions and failures of lesson 6.3.1. They need about six correct digits, which a one-sided difference cannot reach.

See the tests

import math


def test_heat_solution():
    """e⁻ᵗ sin x solves the heat equation: its residual at (1, 0.5) is about 0"""
    r = heat_residual(lambda x, t: math.exp(-t) * math.sin(x), 1.0, 0.5)
    expect_close(r, 0.0, what="heat_residual of e^-t sin x at (1, 0.5)", abs_tol=1e-6)


def test_heat_failure():
    """e⁻ᵗ sin 2x fails: its residual at (1, 0.5) is 3e^(−0.5) sin 2, about 1.654550"""
    r = heat_residual(lambda x, t: math.exp(-t) * math.sin(2 * x), 1.0, 0.5)
    expect_close(r, 3 * math.exp(-0.5) * math.sin(2), what="heat_residual of e^-t sin 2x at (1, 0.5)", rel=1e-6)


def test_heat_polynomial():
    """x² + 2t solves the heat equation: its residual at (3, 2) is about 0"""
    r = heat_residual(lambda x, t: x**2 + 2 * t, 3.0, 2.0)
    expect_close(r, 0.0, what="heat_residual of x^2 + 2t at (3, 2)", abs_tol=1e-5)


def test_heat_frozen():
    """x² does not solve the heat equation: its residual is −2 everywhere"""
    r = heat_residual(lambda x, t: x**2, 0.7, 0.3)
    expect_close(r, -2.0, what="heat_residual of x^2 at (0.7, 0.3)", rel=1e-6)


def test_transport_solution():
    """e^(−(x − 2t)²) solves u_t + 2u_x = 0: its residual at (2.5, 1) is about 0"""
    r = transport_residual(lambda x, t: math.exp(-(x - 2 * t) ** 2), 2, 2.5, 1.0)
    expect_close(r, 0.0, what="transport_residual of the bump with c = 2", abs_tol=1e-7)


def test_transport_wrong_speed():
    """The same bump with c = −2 leaves a residual of about 3.115203 at (2.5, 1)"""
    r = transport_residual(lambda x, t: math.exp(-(x - 2 * t) ** 2), -2, 2.5, 1.0)
    expect_close(r, 4 * math.exp(-0.25), what="transport_residual of the bump with c = -2", rel=1e-6)


def test_transport_wrong_direction():
    """(x + 2t)² moves left: with c = 2 its residual at (1, 1) is 8x + 16t = 24"""
    r = transport_residual(lambda x, t: (x + 2 * t) ** 2, 2, 1.0, 1.0)
    expect_close(r, 24.0, what="transport_residual of (x + 2t)^2 with c = 2", rel=1e-6)


def test_bad_step():
    """A step that is zero or negative raises ValueError in both functions"""
    for name, call in [("heat_residual", lambda: heat_residual(lambda x, t: x, 1.0, 1.0, h=0)),
                       ("transport_residual", lambda: transport_residual(lambda x, t: x, 1, 1.0, 1.0, h=-1e-5))]:
        try:
            call()
        except ValueError:
            continue
        raise AssertionError(f"{name} should raise ValueError for a step that is not positive")

Partial differential equations

Conditions

Coding exercise

Free further reading