# ASO: Integral Transforms - Material for the year 2019-2020

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This course is highly recommended for Differential Equations 2.

8 lectures

The Laplace and Fourier Transforms aim to take a differential equation in a function $f$ and turn it into an algebraic equation involving its transform $\bar{f}$ or $\hat{f}$. 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.

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.

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)

S. Howison, *Practical Applied Mathematics* (CUP 2005), Chapters 9 & 10 (for distributions).

P. J. Collins, *Differential and Integral Equations* (OUP, 2006), Chapter 14.

W. E. Boyce & R. C. DiPrima, *Elementary Differential Equations and Boundary Value Problems*, there are many editions, most recently in 2017; in all of them Chapter 6 covers Laplace Transforms.

K. F. Riley & M. P. Hobson, *Essential Mathematical Methods for the Physical Sciences* (CUP 2011) Chapter 5.

H. A. Priestley, *Introduction to Complex Analysis* (2nd edition, OUP, 2003) Chapters 21 and 22.

L. Debnath & P. Mikusinski, *Introduction to Hilbert Spaces with Applications*, (3rd Edition, Academic Press. 2005) Chapter 6