21 P3: Chapter 3

Problem Class 3: Finite Differences

Published

April 6, 2026

Abstract

Exercises based on content from Lectures 12-15

using Plots, LinearAlgebra, LaTeXStrings

Exercise 21.1 So far, we have considered homogeneous boundary conditions u(0) = u(1) = 0. How would you approximate the problem

\begin{align} &-u'' + p u' + q u = f \qquad \text{on }(0,1) \nonumber\\ &u(0) = \alpha, \qquad u(1) = \beta \end{align}

with finite differences?

Exercise 21.2 So far, we have considered ADR

\begin{align} -u'' + p u' + q u = f \end{align}

with constant coefficients p,q. How would you write down a finite difference approximation to this equation if p and q are functions of x?

Exercise 21.3 (FNC example 10.4.2) Following on from the previous two questions, we try and solve the problem

\begin{align} &-u''(x) + a \cos(ax) u'(x) - a^2\sin(ax) u(x) = 0\nonumber\\ &u(0) = 1 \nonumber\\ &u(\tfrac{3\pi}2) = e^{-1}. \end{align}

Show that the exact solution is u(x) = e^{\sin ax}.

Spot and correct the error in the following code [we did this in class - instead tyr and understand the following code]:

"""
    adr_1d(n, p, q, f; a=0,b=1, α=0,β=0)

Approximate the solution to 
    -u′′ + p u′ + q u = 0 on (a,b)
    u(a) = α, u(b) = β
(linear Advection-Diffusion-Reaction equation with Dirichlet boundary conditions)
using central finite differences
"""
function adr_1d(n, p, q, f; a=0.,b=1., α=0.,β=0.)
    # mesh size
    h = (b-a)/n 
    # interior grid points 
    x = [ a + j *(b-a)/n for j ∈ 1:n-1 ] 

    # Finite Difference Coefficients
    # (using a second order central difference) 
    d = (2/h^2) * ones(n-1) + q.(x)
    d₋  = (-1/h^2)*ones(n-2) - p.(x[2:end])/(2h)
    d₊  = (-1/h^2)*ones(n-2) + p.(x[1:end-1])/(2h)
    Lh = Tridiagonal(d₋, d, d₊)
    F = f.(x)
    F[1] += (1/h^2 + p(x[1])/(2h))*α
    F[end] += (1/h^2 - p(x[end])/(2h))*β
    U = Lh \ F

    # add the boundary conditions
    U = [α, U..., β]
    x = [a, x..., b]

    return x, U
end

p(x) = cos(x)
q(x) = -sin(x)
f(x) = 0.    

x, u = adr_1d(50, p, q, f; a=0.,b=3π/2, α=1.,β=exp(-1))
P1 = scatter(x, u, title=L"$-u′′ + \cos(x) u′ - \sin(x) u = 0$", 
     label="finite difference approximation", markersize=2)
u_e(x) = exp(sin(x))
plot!( u_e, label="exact solution" )

Exercises from the end of lecture 15

Exercise 21.4 (i) Show that \hat{U}_0 = \frac{1}{\sqrt{n}} \sum_j U_j.
(ii) Write -L_{\rm per} U = \bm f in terms of \hat{U}_k and \hat{F}_k,
(iii) Hence show that, if F has zero mean \sum_{j=0}^{n-1} f(x_j) = 0, then there exists U solving -L_{\rm per} U = F.

Exercise 21.5 Read the introduction to the book

“Brigham, E. O. (1988). The Fast Fourier Transform and Its Applications.”

Trefethen (2000). Spectral Methods in MATLAB
Brigham, E. O. (1988). The Fast Fourier Transform and Its Applications.

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