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: Methods Differential Equations

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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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential 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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

Types: analytical, non-analytical

  • Analytical: population growth/decay example

Log scale – obviously linear….


Geogg121 methods differential equations

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


Geogg121 methods differential equations

Types: Order

  • ODE (ordinary DE)

    • Contains only ordinary derivatives (no partials)

    • Can be of different order

      • Order of highest derivative

2nd

1st

2nd


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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!]


Geogg121 methods differential equations

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


Geogg121 methods differential equations

Solving: examples

  • Verify that satisfies

  • Verify that is a solution of

    • (2nd order, 1st degree, linear)


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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.


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods 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


Geogg121 methods differential equations

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?


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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…..


Geogg121 methods differential equations

END


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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


Geogg121 methods differential equations

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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