A journey through DEsMAST 20030
A Tordesillas & J Collis
A bit of history…. Napoleon asked Fourier to accompany him in his expedition to Egypt in
1798. It was during this travel that Fourier was said to have become obsessed with heat.
Well the Middle East is hot!
By late 1807, Fourier offered the first solution to the heat equation in a metal plate, and
therein introduced his much-celebrated series.
He then presented his work, entitled “Mémoire sur la propagation de la chaleur dans les
corps solides (Treatise on the propagation of heat in solid bodies)”, to the French Academy of
Science. It received mixed reviews. The panel, a distinguished line up of mathematical
giants including Laplace, Lagrange, Lacroix, Monge and Poisson dismissed the concept as
unsound
“… the manner in which the author arrives at these equations is not exempt of difficulties
and... his analysis to integrate them still leaves something to be desired on the score of
generality and even rigour.”
Their reservations were partly due to a lack of proof which did not come until the late 19th
century because it needed more advanced mathematics not yet developed in Fourier’s time.
Sound familiar?! This is echoed in our DEs class! We don’t get to see the proofs of many of
the theorems we are using in DEs. Still this should not detract from their importance in
determining whether a given formal solution from SoV is genuine or not. In the words of
Heaviside:
Why should I refuse a good dinner simply because I don't understand the digestive processes
involved. [Oliver Heavide’s reply when criticised for his daring use of operators before
they could be justified formally.]
As described by Bressoud in A Radical Approach to Real Analysis,
"The crisis struck four days before Christmas 1807. The edifice of calculus was shaken to its
foundations. In retrospect, the difficulties had been building for decades. Yet while most scientists
realized that something had happened, it would take fifty years before the full impact of the event
was understood... Here was the heart of the crisis.
Infinite sums of trigonometric functions had appeared before. Daniel Bernoulli (1700-1782)
proposed such sums in 1753 as solutions to the problem of modeling the vibrating string. They
had been dismissed by the greatest mathematician of the time, Leonhard Euler (1707-1783).
Perhaps Euler scented the danger they presented to his understanding of calculus... Well into the
1820s, Fourier series would remain suspect
because they contradicted the established
wisdom about the nature of functions.
Fourier did more than suggest that the
solution to the heat equation lay in his
trigonometric series. He gave a simple and
practical means of finding those coefficients,
the ai, for any function. In so doing, he
produced a vast array of verifiable solutions
to specific problems."
TOPIC 4: Fourier series
FS and FT combined are by far one of -if
not the- biggest deal in DEs.
Its report card is second to none. Indeed
the bulk majority of Nobel prizes in
Physics, Medicine, and Chemistry would
have some connection to, if not
underpinned by, the inventions of Joseph
Fourier.
In fact the noted physicist Lord Kelvin wrote in 1867: "Fourier's theorem is not only one of
the most beautiful results of modern analysis, but it may be said to furnish an indispensable
instrument in the treatment of nearly every recondite question in modern physics."
o Fourier series trumps power series because it can represent discontinuities even
though the component functions are smooth, continuous and infinitely differentiable!
o Note in practice that we approximate the infinite FS by truncating the sum to the first
few N terms. This has an interesting connection to the rate of convergence of a
series.
§ The faster the rate, the fewer the N terms you need to achieve a certain
accuracy in your representation of a given function by a FS.
§ The smoothness of the function is tied to the rate of convergence.
Continuous periodic extension ⇒ fast (and uniform) convergence.
Discontinuous periodic extension ⇒slow (and non-uniform but still
pointwise) convergence.
§ Differentiation retards rate of convergence
§ Integration promotes rate of convergence
o FS is the power behind why SoV can work. It allows us to propagate the information
from the initial conditions throughout space (domain of problem) and time (future).
TOPIC 5: BVP and SoV
Go see the LECTURE SUMMARIES in the relevant module as this a loaded topic!
TOPIC 6: Fourier Transform
Up until this point in our journey, we dealt only with bounded spatial domains. But many
applications call for unbounded domains or at least very large domains where we need ways
to deal with the limit 𝐿 → ∞. One example is if you want to build a cellar, or you are a
geologist hunting for precious minerals, and you need to start from zero (the surface of the
Earth) and then drill down deep!
FT extrapolates the ideas of FS as you go deeper and deeper into “the limit 𝐿 → ∞”.
Because FT is a (linear) integral transform it has many things in common with LT (shift
theorems, convolutions, calculus → algebra). One big advantage over LT, is that the inverse
formula is just as simple as the original transform.
o FS = wave number is discrete variable + infinite sum over these wave numbers.
o FT = wave number is continuous variable + integration over (infinite) domain of these
wave numbers.
o When solving DEs with FT, we don’t have BCs as we’ve seen in Topic 5, but instead
have conditions like ‘FT of solution exists’ or 𝑢 → 0 as 𝑥 → ±∞.
o For many problems our final step will usually require numerical integration.
o FT can be used with other methods like LT to turn PDE into fully algebraic equations!
o FT also vital to signal processing. The FT of a signal reveals the frequencies and
amplitudes of the trig functions that makeup a signal
o Solving a PDE using FT takes 3 steps (just like in LT)
1. Take the FT of the PDE and initial conditions. This leads to an IVP.
2. Solve the IVP using whatever method you can (integrating factor/LT/trial
solutions/power series)
3. Take the inverse FT of the solution to the IVP to find the solution to the PDE
And that’s a wrap for 2023 MAST20030!