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GEOGG121: Methods Differential Equations

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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

- Analytical methods
- V. short intro to Monte Carlo (MC) methods

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

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 Δt0

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

- Can further subdivide into different degree

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

- Linear or 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

- Use Euler’s method with h = 0.25 to obtain numerical soln. of

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

- Analytical methods?
- Monte Carlo methods
- Very useful brute force numerical approach to integration, parameter estimation, sampling
- If all else fails, guess…..

END

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

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

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

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