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GEOGG121: Methods Differential Equations. Dr. Mathias (Mat) Disney UCL Geography Office: 113, Pearson Building Tel: 7670 0592 Email: [email protected] www.geog.ucl.ac.uk /~ mdisney. Lecture outline. Differential equations Introduction & importance Types of DE Examples

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geogg121 methods differential equations

GEOGG121: MethodsDifferential Equations

Dr. Mathias (Mat) Disney

UCL Geography

Office: 113, Pearson Building

Tel: 7670 0592

Email: [email protected]

www.geog.ucl.ac.uk/~mdisney

slide2
Lecture outline
  • Differential equations
    • Introduction & importance
    • Types of DE
  • Examples
  • Solving ODEs
    • Analytical methods
      • General solution, particular solutions
      • Separation of variables, integrating factors,linear operators
    • Numerical methods
      • Euler, Runge-Kutta
  • V. short intro to Monte Carlo (MC) methods
slide3
Reading material
  • Textbooks

These are good UG textbooks that have WAY more detail than we need

    • Boas, M. L., 1985 (2nd ed) Mathematical Methods in the Physical Sciences, Wiley, 793pp.
    • Riley, K. F., M. Hobson & S. Bence (2006) Mathematical Methods for Physics & Engineering, 3rd ed., CUP.
    • Croft, A., Davison, R. & Hargreaves, M. (1996) Engineering Mathematics, 2nd ed., Addison Wesley.
  • Methods, applications
    • Wainwright, J. and M. Mulligan (eds, 2004) Environmental Modelling: Finding Simplicity in Complexity, J. Wiley and Sons, Chichester. Lots of examples particularly hydrology, soils, veg, climate. Useful intro. ch 1 on models and methods
    • Campbell, G. S. and J. Norman (1998) An Introduction to Environmental Biophysics, Springer NY, 2nd ed. Excellent on applications eg Beer’s Law, heat transport etc.
    • Monteith, J. L. and M. H. Unsworth(1990) Principles of Environmental Physics, Edward Arnold. Small, but wide-ranging and superbly written.
  • Links
    • http://www.math.ust.hk/~machas/differential-equations.pdf
    • http://www.physics.ohio-state.edu/~physedu/mapletutorial/tutorials/diff_eqs/intro.html
slide4
Introduction
  • What is a differential equation?
    • General 1st order DEs
    • 1st case t is independent variable, x is dependent variable
    • 2nd case, x is independent variable, y dependent
  • Extremely important
    • Equation relating rate of change of something (y) wrt to something else (x)
    • Any dynamic system (undergoing change) may be amenable to description by differential equations
    • Being able to formulate & solve is incredibly powerful
slide5
Examples
  • Velocity
    • Change of distance x with time t i.e.
  • Acceleration
    • Change of v with t i.e.
  • Newton’s 2nd law
    • Net force on a particle = rate of change of linear momentum (m constant so…
  • Harmonic oscillator
    • Restoring force F on a system  displacement (-x) i.e.
    • So taking these two eqns we have
slide6
Examples
  • Radioactive decay of unstable nucleus
    • Random, independent events, so for given sample of N atoms, no. of decay events –dN in time dt N
    • So N(t) depends on No (initial N) and rate of decay
  • Beer’s Law – attenuation of radiation
    • For absorption only (no scattering), decreases in intensity (flux density) of radiation at some distance x into medium, Φ(x) is proportional to x
    • Same form as above – will see leads to exponential decay
    • Radiation in vegetation, clouds etcetc
slide7
Examples
  • Compound Interest
    • How does an investment S(t), change with time, given an annual interest rate r compounded every time interval Δt, and annual deposit amount k?
    • Assuming deposit made after every time interval Δt
    • So as Δt0
slide8
Examples
  • Population dynamics
    • Logistic equation (Malthus, Verhulst, Lotka….)
    • Rate of change of population P with t depends on Po, growth rate r (birth rate – death rate) & max available population or ‘carrying capacity’ K
    • P << K, dP/dt rP but as P increases (asymptotically) to K, dP/dtgoes to 0 (competition for resources – one in one out!)
    • For constant K, if we set x = P/K then

http://www.scholarpedia.org/article/Predator-prey_model#Lotka-Volterra_Model

slide9
Examples
  • Population dynamics: II
    • Lotka-Volterra (predator-prey) equations
    • Same form, butnow two populations x and y, with time –
    • y is predator and yt+1 depends on yt AND prey population (x)
    • x is prey, and xt+1 depends on xt AND y
    • a, b, c, d – parameters describing relationship of y to x
  • More generally can describe
    • Competition – eg economic modelling
    • Resources – reaction-diffusion equations
slide10
Examples
  • Transport: momentum, heat, mass….
    • Transport usually some constant (proportionality factor) x driving force
    • Newton’s Law of viscosity for momentum transport
      • Shear stress, τ, between fluid layers moving at different speeds - velocity gradient perpendicular to flow, μ = coeff. of viscosity
    • Fourier’s Law of heat transport
      • Heat flux density H in a material is proportional to (-) T gradient and area perpendicular to gradient through which heat flowing, k = conductivity. In 1D case…
    • Fick’s Law of diffusive transport
      • Flux density F’j of a diffusing substance with molecular diffusivity Dj across density gradient dρj/dz (j is for different substances that diffuse through air)
  • See Campbell and Norman chapter 6
slide11
Types: analytical, non-analytical
  • Analytical, closed form
    • Exact solution e.g. in terms of elementary functions such as ex, log x, sin x
  • Non-analytical
    • No simple solution in terms of basic functions
    • Solution requires numerical methods (iterative) to solve
    • Provide an approximate solution, usually as infinite series
slide12
Types: analytical, non-analytical
  • Analytical example
    • Exact solution e.g.
    • Solve by integrating both sides
    • This is a GENERAL solution
      • Contains unknown constants
    • We usually want a PARTICULAR solution
      • Constants known
      • Requires BOUNDARY conditions to be specified
slide13
Types: analytical, non-analytical
  • Particular solution?
    • BOUNDARY conditions e.g.set t = 0 to get c1, 2 i.e.
    • So x0 is the initial value and we have
    • Exponential model ALWAYS when dx/dt x
      • If a>0 == growth; if a < 0 == decay
      • Population: a = growth rate i.e. (births-deaths)
      • Beer’s Law: a = attenuation coeff. (amount x absorp. per unit mass)
      • Radioactive decay: a = decay rate
slide14
Types: analytical, non-analytical
  • Analytical: population growth/decay example

Log scale – obviously linear….

slide15
Types: ODEs, PDEs
  • ODE (ordinary DE)
    • Contains only ordinary derivatives
  • PDE (partial DE)
    • Contains partial derivatives – usually case when depends on 2 or more independent variables
    • E.g. wave equation: displacement u, as function of time, t and position x
slide16
Types: Order
  • ODE (ordinary DE)
    • Contains only ordinary derivatives (no partials)
    • Can be of different order
      • Order of highest derivative

2nd

1st

2nd

slide17
Types: Order -> Degree
  • ODE (ordinary DE)
    • Can further subdivide into different degree
      • Degree (power) to which highest order derivative raised

1st order

3rd degree

1st order

1st degree

2nd order

2nd degree

slide18
Types: Linearity
  • ODE (ordinary DE)
    • Linear or non-linear?
      • Linear if dependent variable and all its derivatives occur only to the first power, otherwise, non-linear
      • Product of terms with dependent variable == non-linear
      • Functions sin, cos, exp, ln also non-linear

Non-linear

ydy/dx

Non-linear

y2 term

Non-linear

sin y term

Linear

slide19
Solving
  • General solution
    • Often many solutions can satisfy a differential eqn
    • General solution includes all these e.g.
    • Verify that y = Cex is a solution of dy/dx = y, C is any constant
    • So
    • And for all values of x, and eqn is satisfied for any C
    • C is arbitrary constant, vary it and get all possible solutions
    • So in fact y = Cex is the general solution of dy/dx = y
slide20
Solving
  • But for a particular solution
    • We must specify boundary conditions
    • Eg if at x = 0, we know y = 4 then from general solution
    • 4 = Ce0 so C = 4 and
    • is the particular solution of dy/dx = y that satisfies the condition that y(0) = 4
    • Can be more than one constant in general solution
    • For particular solution number of given independent conditions MUST be same as number of constants
slide21
Types: analytical, non-analytical
  • Analytical: Beer’s Law - attenuation
    • k is extinction coefficient – absorptivity per unit depth, z (m-1)
    • E.g. attenuation through atmosphere, where path length (z) 1/cos(θsun), θsun is the solar zenith angle
    • Take logs:
    • Plot z against ln(ϕ), slope is k, intercept is ϕ0 i.e. solar radiation with no attenuation (top of atmos. – solar constant)
    • [NB taking logs v powerful – always linearise if you can!]
slide22
Initial & boundary conditions
  • One point conditions
    • We saw as general solution of
    • Need 2 conditions to get particular solution
      • May be at a single point e.g. x = 0, y = 0 and dy/dx = 1
      • So and solution becomes
      • Now apply second condition i.e. dy/dx = 1 when x = 0 so differentiate
    • Particular solution is then
slide23
Solving: examples
  • Verify that satisfies
  • Verify that is a solution of
    • (2nd order, 1st degree, linear)
slide24
Initial & boundary conditions
  • Two point conditions
    • Again consider
    • Solution satisfying y = 0 when x = 0 AND y = 1 when x = 3π/2
    • So apply first condition to general solution
    • i.e. and solution is
    • Applying second condition we see
    • And B = -1, so the particular solution is
    • If solution required over interval a ≤ x ≤ b and conditions given at both ends, these are boundary conditions (boundary value problem)
    • Solution subject to initial conditions = initial value problem
slide25
Separation of variables
  • We have considered simple cases so far
    • Where and so
  • What about cases with ind. & dep. variables on RHS?
    • E.g.
  • Important class of separable equations. Div by g(y) to solve
    • And then integrate both sides wrt x
slide26
Separation of variables
  • Equation is now separated & if we can integ. we have y in terms of x
    • Eg where and
    • So multiply both sides by y to give and then integrate both sides wrt x
    • i.e. and so and
    • If we define D = 2C then

Eg See Croft, Davison, Hargreaves section 18, or

http://www.cse.salford.ac.uk/profiles/gsmcdonald/H-Tutorials/ordinary-differential-equations-separation-variables.pdf

http://en.wikipedia.org/wiki/Separation_of_variables

slide27
Using an integrating factor
  • For equations of form
    • Where P(x) and Q(x) are first order linear functions of x, we can multiply by some (as yet unknown) function of x, μ(x)
    • But in such a way that LHS can be written as
    • And then
    • Which is said to be exact, with μ(x) as the integrating factor
    • Why is this useful?

Eg See Croft, Davison, Hargreaves section 18, or

http://www.cse.salford.ac.uk/profiles/gsmcdonald/H-Tutorials/ordinary-differential-equations-integrating-factor.pdfhttp://en.wikipedia.org/wiki/Integrating_factor

slide28
Using an integrating factor
  • Because it follows that
  • And if we can evaluate the integral, we can determine y
  • So as above, we want
  • Use product rule i.e. and so, from above
  • and by inspection we can see that
  • This is separable (hurrah!) i.e.

http://en.wikipedia.org/wiki/Product_rulehttp://www.cse.salford.ac.uk/profiles/gsmcdonald/H-Tutorials/ordinary-differential-equations-integrating-factor.pdf

slide29
Using an integrating factor
  • And we see that (-lnK is const. of integ.)
  • And so
  • We can choose K = 1 (as we are multiplying all terms in equation by integ. factor it is irrelevant), so
    • Integrating factor for is given by
    • And solution is given by

http://en.wikipedia.org/wiki/Product_rulehttp://www.cse.salford.ac.uk/profiles/gsmcdonald/H-Tutorials/ordinary-differential-equations-integrating-factor.pdf

slide30
Using an integrating factor: example
  • Solve
    • From previous we see that and
    • Using the formula above
    • And we know the solution is given by
    • So , as
slide31
2nd order linear equations
  • Form
    • Where p(x), q(x), r(x) and f(x) are fns of x only
    • This is inhomogeneous(depon y)
    • Related homogeneousform ignoring term independent of y
    • Use shorthand L{y} when referring to general linear diff. eqn to stand for all terms involving y or its derivatives. From above
    • for inhomogeneous general case
    • And for general homogenous case
    • Eg if then where
slide32
Linear operators
  • When L{y} = f(x) is a linear differential equation, L is a linear differential operator
    • Any linear operator L carries out an operation on functions f1 and f2 as follows
    • where a is a constant
    • where a, b are constants
    • Example: if show that
    • and
slide33
Linear operators
  • Note that L{y} = f(x) is a linear diff. eqn so L is a linear diff operator
  • So
    • we see
    • And rearrange:
    • & because differentiation is a linear operator we can now see
  • For the second case
  • So
slide34
Partial differential equations
  • DEs with two or more dependent variables
    • Particularly important for motion (in 2 or 3D), where eg position (x, y, z) varying with time t
  • Key example of wave equation
    • Eg in 1D where displacement u depends on time and position
    • For speed c, satisfies
    • Show is a solution of
    • Calculate partial derivatives of u(x, t) wrt to x, then t i.e.
slide35
Partial differential equations
  • Now 2nd partial derivatives of u(x, t) wrt to x, then t i.e.
  • So now
  • More generally we can express the periodic solutions as (remembering trig identities)
  • and
  • Where k is the wave vector (2π/λ); ω is the angular frequency (rads s-1) = 2π/T for period T;

http://en.wikipedia.org/wiki/List_of_trigonometric_identities

http://www.physics.usu.edu/riffe/3750/Lecture%2018.pdf

http://en.wikipedia.org/wiki/Wave_vector

slide36
Partial differential equations
  • In 3D?
    • Just consider y and z also, so for q(x, y, z, t)
  • Some v. important linear differential operators
    • Del (gradient operator)
    • Del squared (Laplacian)
  • Lead to eg Maxwell’s equations

http://www.physics.usu.edu/riffe/3750/Lecture%2018.pdf

slide37
Numerical approaches
  • Euler’s Method
    • Consider 1st order eqn with initial cond. y(x0) = y0
    • Find an approx. solution yn at equally spaced discrete values (steps) of x, xn
    • Euler’s method == find gradient at x = x0 i.e.
    • Tangent line approximation

True solution

y

y(x1)

Tangent approx.

y1

y0

Croft et al., p495

Numerical Recipes in C ch. 16, p710

http://apps.nrbook.com/c/index.html

x0

x1

0

x

http://en.wikipedia.org/wiki/Numerical_ordinary_differential_equations

slide38
Numerical approaches
  • Euler’s Method
    • True soln passes thru (x0, y0) with gradient f(x0, y0) at that point
    • Straight line (y = mx + c) approx has eqn
    • This approximates true solution but only near (x0, y0), so only extend it short dist. h along x axis to x = x1
    • Here, y = y1 and
    • Since h = x1-x0 we see
    • Can then find y1, and we then know (x1, y1)…..rinse, repeat….

True solution

y

y(x1)

Tangent approx.

y1

y0

Generate series of values iteratively

Accuracy depends on h

Croft et al., p495

Numerical Recipes in C ch. 16, p710

http://apps.nrbook.com/c/index.html

x0

x1

0

x

slide39
Numerical approaches
  • Euler’s Method: example
    • Use Euler’s method with h = 0.25 to obtain numerical soln. of

with y(0) = 2, giving approx. values of y for 0 ≤ x ≤ 1

    • Need y1-4 over x1 = 0.25, x2 = 0.5, x3 = 0.75, x4 = 1.0 say, so
    • with x0 = 0 y0 = 2
    • And

NB There are more accurate variants of Euler’s method..

Exercise: this can be solved ANALYTICALLY via separation of variables. What is the difference to the approx. solution?

slide40
Numerical approaches
  • Runge-Kutta methods (4th order here….)
    • Family of methods for solving DEs (Euler methods are subset)
    • Iterative, starting from yi, no functions other than f(x,y) needed
    • No extra differentiation or additional starting values needed
    • BUT f(x, y) is evaluated several times for each step
    • Solve subject to y = y0 when x = x0, use
    • where

Euler

Croft et al., p502

Rile et al. p1026

Numerical Recipes in C ch. 16, p710

http://apps.nrbook.com/c/index.html

http://en.wikipedia.org/wiki/Numerical_ordinary_differential_equations

http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods

slide41
Numerical approaches
  • Runge-Kutta example
    • As before, but now use R-K with h = 0.25 to obtain numerical soln. of with y(0) = 2, giving approx. values of y for 0 ≤ x ≤ 1
    • So for i = 0, first iteration requires
    • And finally
    • Repeat! c.f. 2 from Euler, and 1.8824 from analytical
slide42
Very brief intro to Monte Carlo
  • Brute force method(s) for integration / parameter estimation / sampling
    • Powerful BUT essentially last resort as involves random sampling of parameter space
    • Time consuming – more samples gives better approximation
    • Errors tend to reduce as 1/N1/2
      • N = 100 -> error down by 10; N = 1000000 -> error down by 1000
    • Fast computers can solve complex problems
  • Applications:
    • Numerical integration (eg radiative transfer eqn), Bayesian inference, computational physics, sensitivity analysis etcetc

Numerical Recipes in C ch. 7, p304

http://apps.nrbook.com/c/index.html

http://en.wikipedia.org/wiki/Monte_Carlo_method

http://en.wikipedia.org/wiki/Monte_Carlo_integration

slide43
Basics: MC integration
  • Pick N random points in a multidimensional volume V, x1, x2, …. xN
  • MC integration approximates integral of function f over volume V as
  • Where and
  • +/- term is 1SD error – falls of as 1/N1/2

Choose random points in A

Integral is fraction of points under curve x A

From

http://apps.nrbook.com/c/index.html

slide44
Basics: MC integration
  • Why not choose a grid? Error falls as N-1 (quadrature approach)
  • BUT we need to choose grid spacing. For random we sample until we have ‘good enough’ approximation
  • Is there a middle ground? Pick points sort of at random BUT in such a way as to fill space more quickly (avoid local clustering)?
  • Yes – quasi-random sampling:
    • Space filling: i.e. “maximally avoiding of each other”

FROM: http://en.wikipedia.org/wiki/Low-discrepancy_sequence

Sobol method v pseudorandom: 1000 points

slide45
Summary
  • Differential equations
    • Describe dynamic systems – wide range of examples, particularly motion, population, decay (radiation – Beer’s Law, mass – radioactivity)
  • Types
    • Analytical, closed form solution, simple functions
    • Non-analytical: no simple solution, approximations?
    • ODEs, PDEs
    • Order: highest power of derivative
      • Degree: power to which highest order derivative is raised
    • Linear/non:
      • Linear if dependent variable and all its derivatives occur only to the first power, otherwise, non-linear
slide46
Summary
  • Solving
    • Analytical methods?
      • Find general solution by integrating, leaves constants of integration
      • To find a particular solution: need boundary conditions (initial, ….)
      • Integrating factors, linear operators
    • Numerical methods?
      • Euler, Runge-Kutta – find approx. solution for discrete points
  • Monte Carlo methods
    • Very useful brute force numerical approach to integration, parameter estimation, sampling
    • If all else fails, guess…..
slide48
Example
  • Radioactive decay
    • Random, independent events, so for given sample of N atoms, no. of decay events –dN in time dt N so
    • Where λ is decay constant (analogous to Beer’s Law k) units 1/t
    • Solve as for Beer’s Law case so
    • i.e. N(t) depends on No (initial N) and rate of decay
    • λ often represented as 1/tau, where tau is time constant – mean lifetime of decaying atoms
    • Half life (t=T1/2) = time taken to decay to half initial N i.e. N0/2
    • Express T1/2 in terms of tau
slide49
Example
  • Radioactive decay
    • EG: 14C has half-life of 5730 years & decay rate = 14 per minute per gram of natural C
    • How old is a sample with a decay rate of 4 per minute per gram?
    • A: N/N0 = 4/14 = 0.286
    • From prev., tau = T1/2/ln2 = 5730/ln2 = 8267 yrs
    • So t = -tau x ln(N/N0) = 10356 yrs
slide50
Exercises
  • General solution of is given by
  • Find particular solution satisfies x = 3 and dx/dt = 5 when t =0
  • Resistor (R) capacitor (L) circuit (p458, Croft et al), with current flow i(t) described by
  • Use integrating factor to find i(t)….approach: re-write as
slide51
Exercises
  • Show that the analytical solution of with y(x=0)=2 is
  • Compare values from x = 0 to 1 with approx. solution obtained by Euler’s method
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