Confessions of an applied mathematician Chris Budd

What is applied maths? Using maths to understand an aspect of the real world … usually through a simplified model and to predict or create new things It is crazy that this works at all Learning NEW mathematics in the process Using this new mathematics to change the world

Some ways that maths has changed the modern world Maxwell: Electromagnetism … radio, TV, radar, mobile phones Linear algebra, graph theory, SVD... Google Error correcting codes

We live in interesting times with applied mathematics in a process of great transition! 20 th century.. Great drivers of applied maths are physics, engineering and more recently biology Expertise in …. Fluids Solids Reaction-diffusion problems Dynamical systems Signal processing

Usually deterministic Continuum problems, modelled by Differential Equations Solutions methods Simple analytical methods eg. Separation of variables Approximate/asymptotic approaches Phase plane analysis Numerical methods eg. finite element methods PDE techniques eg. Calculus of variations Transforms: Fourier, Laplace, Radon

What are the drivers of 21 st century applied mathematics? Information/Bio-informatics/Genetics? Commerce/retail sector? Complexity? What new techniques do we need to consider? Discrete maths? Stochastic methods? Very large scale computations? Complex systems? Optimisation (discrete and continuous)?

Example 1: What happens when we eat? Stomach Small intestine: 7m x 1.25cm Intestinal wall: Villi and Microvilli

Process: Food enters stomach and leaves as Chyme Nutrients are absorbed through the intestinal wall Chyme passes through small intestine in 4.5hrs Stomach Intestinal wall Colon, illeocecal sphincter Peristaltic wave Mixing process

Objectives Model the process of food moving through the intestine Model the process of nutrient mixing and absorption

Basic flow model: axisymmetric Stokes flow pumped by a peristaltic wave and a pressure gradient Chyne moves at slow velocity: u(x,r,t) Nutrient concentration: c(x,r,t) Peristaltic wave: r = f(x,t) x r=f(x,t) r Wavelength:8cm h = 1.25cm

Navier Stokes Slow viscous Axisymmetric flow Velocity & Stokes Streamfunction

FIXED FRAME WAVE FRAME No slip on boundary Change from Impose periodicity

Axisymmetry Amplitude: Wave Number: Small parameters

Flow depends on: Flow rate Proportional to pressure drop Amplitude Wave number gives Poiseuille flow Develop asymptotic series in powers of

Reflux Pressure Rise Particles undergo net retrograde motion Trapping Regions of Pressure Rise & Pressure Drop Streamlines encompass a bolus of fluid particles Trapped Fluid recirculates Distinct flow types

A B C D E F G Flow regions Poiseuille A: Copumping, Detached Trapping B: Copumping, Centreline Trapping C: Copumping, No Trapping Illeocecal sphincter open D: Pumping, No Trapping E: Pumping, Centreline Trapping Illeocecal sphincter closed Illeocecal sphincter closed

Case A: Copumping, Detached Trapping Recirculation Particle paths

x Case C: Copumping, No Trapping Poiseuille Flow Particle paths

x Case E: Pumping, Centreline Trapping Recirculation Reflux Particle paths

Calculate the concentration c(x,r,t) 1. Substitute asymptotic solution for u into 2. Solve for c(x,r,t) numerically using an upwind scheme on a domain transformed into a computational rectangle. 3. Calculate rate of absorption

Poiseuille flow Peristaltic flow Type C flow: no trapping

Poiseuille flow Peristaltic flow Type E flow: trapping and reflux

x t Nutrient absorbed Location of absorbed mass at final time Peristaltic flow Conclusions Peristalsis helps both pumping and mixing Significantly greater absorption with Peristaltic flow than with Poiseuille flow

Example 2: Mathematics can look inside you Modern CAT scanner CAT scanners work by casting many shadows with X-rays and using maths to assemble these into a picture

X-Ray Object Density f(x,y) ρ : Distance from the object centre θ : Angle of the X-Ray Measure attenuation of X-Ray R(ρ, θ) X-ray Source Detector

Object AttenuationR(ρ, θ) Edge

If we can measure R(ρ, θ) accurately we can calculate The density f(x,y) of the object at any point Also used to X-ray mummies Radon 1917

Example 3: Finding land mines Land mines are hidden in foliage and triggered by trip wires Land mines are well hidden.. we can use maths to find them

Find the trip wires in this picture

Digital picture of foliage is taken by camera on a long pole Effect : Image intensity f Cause : Trip wires.. These are like X-Rays Radon transfor m x y f(x,y ) R(ρ,θ) Points of high intensity in R correspond to trip wires θ ρ Isolate points and transform back to find the wires

Mathematics finds the land mines! Who says that maths isn’t relevant to real life?!?