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Unformatted text preview: ODE of vectors Example: Projectile motion with air drag Systems of coupled ODEs Example: Spread of an epidemic Midterm project due date extended until next Tuesday (3/30) No HW this week – finish your projects! Reading for Differential Equations in Appendix B. Often in science the quantities we are interested in are vectors which require a magnitude and direction. Typically we express a vector by it’s individual components: = (vx , vy , vz ) v
Computationally, our 1D problems now become 3 1D problems. We just perform the calculations on each of the components individually. dvx 1 = Fdx (vx ) dt m dvy 1 = Fdy (vy ) − g dt m 1 Fd = − Cd ρA  vv 2 Restrict to 2D, so only need x and y components of position (r) and velocity (v). Force of gravity is only in y direction, drag force is always opposite to velocity (x and y components constantly change). drx = vx dt dry = vy dt In many ‘systems’ we encounter in science, the evolution of a variable depends on the values of other variables, … which in turn depend on the value of the original variable. We say ODEs which describe the evolution of these variables are coupled. Examples: Predator – prey models (LotkaVolterra) Spread of infectious diseases Chemical reactions (BelousovZhabotinski) Climate models (Lorentz) Simple model of how an infectious disease (like the flu) might spread in a ‘closed’ population. Two groups of people: number of susceptibles (S) who can catch the disease and infectives (I) which have the disease. dS = −rSI dt
dI = rSI − aI dt The parameters r and a are characteristic of the disease (i.e. how infectious the disease is and rate of recovery respectively) Think about the logic behind these equations! Also need initial (t=0) conditions: S (0) = S0 I (0) = I0 Need to discretize using forward Euler method. For time = 0 to T Sk+1 − Sk in n time steps: = −rSk Ik T ∆t = n For a time step k: ∆t Ik+1 − Ik = rSk Ik − aIk ∆t So to compute the S and I values for the next time step (k+1): Ik+1 = Ik + ∆t(rSk Ik − aIk ) Sk+1 = Sk − ∆trSk Ik Data was collected during a flu outbreak in a British boarding school (closed population) in 1978. This simple ODE system was used to successfully model the data by adjusting r and a values (nonlinear fitting using a model rather than an analytic function).
Data from Mathematical Biology by J.D. Murray – excellent book! An important question to ask is “Are there values for the variables for which the system does NOT evolve?” These values define steadystate solutions and define “fixed points” for the equations. Remember, variables are not changing if their derivatives are 0. There is one fixed point for the epidemic model: I = 0 since I is in every term on the RHS of all equations. Let’s take a look at a model which has a more interesting fixed point. System of ODEs describing the change in population of predators (y) and prey (x). dx = x(a − cy − bx) dt
dy = −y (d − ex) dt Interpretation of parameters: a  prey birth rate, b – prey death rate (other than being eaten), c – rate of consumption, d – predator death rate, e – predator birth rate Find fixed points by setting both derivatives to 0 and solving for x and y: a bd d There are others, y0 = − x0 = see them? c ce e ...
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 Spring '09
 Gladden
 Projectile Motion

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