ASO: Integral Transforms (2024-25)
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- Lecturer: Profile: Andreas Muench
Course information
General Prerequisites:
This course is highly recommended for Differential Equations 2.
Course Term: Hilary
Course Lecture Information: 8 lectures
Course Overview:
The Laplace and Fourier Transforms aim to take a differential equation in a function and turn it into an algebraic equation involving its transform. Such an equation can then be solved by algebraic manipulation, and the original solution determined by recognizing its transform or applying various inversion methods.
The Dirac delta-function, which is handled particularly well by transforms, is a means of rigorously dealing with ideas such as instantaneous impulse and point masses, which cannot be properly modelled using functions in the normal sense of the word. \(delta\) is an example of a distribution or generalized function and the course provides something of an introduction to these generalized functions and their calculus.
The Dirac delta-function, which is handled particularly well by transforms, is a means of rigorously dealing with ideas such as instantaneous impulse and point masses, which cannot be properly modelled using functions in the normal sense of the word. \(delta\) is an example of a distribution or generalized function and the course provides something of an introduction to these generalized functions and their calculus.
Learning Outcomes:
Students will gain a range of techniques employing the Laplace and Fourier Transforms in the solution of ordinary and partial differential equations. They will also have an appreciation of generalized functions, their calculus and applications.
Course Synopsis:
Motivation for a "function'' with the properties the Dirac delta-function. Test functions. Continuous functions are determined by \(phi \rightarrow \int f \phi\). Distributions and \(delta\) as a distribution. Differentiating distributions. (3 lectures)
Theory of Fourier and Laplace transforms, inversion, convolution. Inversion of some standard Fourier and Laplace transforms via contour integration.
Use of Fourier and Laplace transforms in solving ordinary differential equations, with some examples including \(delta\).
Use of Fourier and Laplace transforms in solving partial differential equations; in particular, use of Fourier transform in solving Laplace's equation and the Heat equation. (5 lectures)
Theory of Fourier and Laplace transforms, inversion, convolution. Inversion of some standard Fourier and Laplace transforms via contour integration.
Use of Fourier and Laplace transforms in solving ordinary differential equations, with some examples including \(delta\).
Use of Fourier and Laplace transforms in solving partial differential equations; in particular, use of Fourier transform in solving Laplace's equation and the Heat equation. (5 lectures)
Section outline
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Sheets 1 and 2 as a single document, to cover two tutorials. See the sheet for what to cover when.
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Background reading on distributions, from the book Practical Applied Mathematics.
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Opened: Monday, 6 January 2025, 12:00 AM
The full lecture notes for the course, by Richard Earl/Sam Howison. You will need this to fill in details from lectures. It also has many examples different from those in lectures.
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