GEOGG121: Methods Differential Equations

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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: MethodsDifferential 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
• 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
• 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.
• http://www.math.ust.hk/~machas/differential-equations.pdf
• http://www.physics.ohio-state.edu/~physedu/mapletutorial/tutorials/diff_eqs/intro.html
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
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
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
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
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

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
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
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
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
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
Types: analytical, non-analytical
• Analytical: population growth/decay example

Log scale – obviously linear….

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
Types: Order
• ODE (ordinary DE)
• Contains only ordinary derivatives (no partials)
• Can be of different order
• Order of highest derivative

2nd

1st

2nd

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

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

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
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
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!]
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
Solving: examples
• Verify that satisfies
• Verify that is a solution of
• (2nd order, 1st degree, linear)
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
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
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

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

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

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

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
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
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
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
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.
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

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

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

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

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?

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

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
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

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

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

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
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…..
Example
• 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
Example