GEOGG 121 Methods Differential Equations MC etc Dr
![GEOGG 121: Methods Differential Equations, MC etc Dr. Mathias (Mat) Disney UCL Geography Office: GEOGG 121: Methods Differential Equations, MC etc Dr. Mathias (Mat) Disney UCL Geography Office:](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-1.jpg)
![Lecture outline • Differential equations – Introduction & importance – Types of DE • Lecture outline • Differential equations – Introduction & importance – Types of DE •](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-2.jpg)
![Reading material • Textbooks These are good UG textbooks that have WAY more detail Reading material • Textbooks These are good UG textbooks that have WAY more detail](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-3.jpg)
![Introduction • What is a differential equation? – General 1 st order DEs – Introduction • What is a differential equation? – General 1 st order DEs –](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-4.jpg)
![Examples • Velocity – Change of distance x with time t i. e. • Examples • Velocity – Change of distance x with time t i. e. •](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-5.jpg)
![Examples • Radioactive decay of unstable nucleus – Random, independent events, so for given Examples • Radioactive decay of unstable nucleus – Random, independent events, so for given](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-6.jpg)
![Examples • Compound Interest – How does an investment S(t), change with time, given Examples • Compound Interest – How does an investment S(t), change with time, given](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-7.jpg)
![Examples • Population dynamics – Logistic equation (Malthus, Verhulst, Lotka…. ) – Rate of Examples • Population dynamics – Logistic equation (Malthus, Verhulst, Lotka…. ) – Rate of](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-8.jpg)
![Examples • Population dynamics: II – Lotka-Volterra (predator-prey) equations – Same form, but now Examples • Population dynamics: II – Lotka-Volterra (predator-prey) equations – Same form, but now](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-9.jpg)
![Examples http: //www. evolution-ofideas. com/homepage/Mathematics/mathsinner/A%20 model%20 of%20 corruption. htm Examples http: //www. evolution-ofideas. com/homepage/Mathematics/mathsinner/A%20 model%20 of%20 corruption. htm](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-10.jpg)
![Examples • Transport: momentum, heat, mass…. – Transport usually some constant (proportionality factor) x Examples • Transport: momentum, heat, mass…. – Transport usually some constant (proportionality factor) x](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-11.jpg)
![Types: analytical, non-analytical • Analytical, closed form – Exact solution e. g. in terms Types: analytical, non-analytical • Analytical, closed form – Exact solution e. g. in terms](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-12.jpg)
![Types: analytical, non-analytical • Analytical example – Exact solution e. g. – Solve by Types: analytical, non-analytical • Analytical example – Exact solution e. g. – Solve by](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-13.jpg)
![Types: analytical, non-analytical • Particular solution? – BOUNDARY conditions e. g. set t = Types: analytical, non-analytical • Particular solution? – BOUNDARY conditions e. g. set t =](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-14.jpg)
![Types: analytical, non-analytical • Analytical: population growth/decay example Log scale – obviously linear…. Types: analytical, non-analytical • Analytical: population growth/decay example Log scale – obviously linear….](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-15.jpg)
![Types: ODEs, PDEs • ODE (ordinary DE) – Contains only ordinary derivatives • PDE Types: ODEs, PDEs • ODE (ordinary DE) – Contains only ordinary derivatives • PDE](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-16.jpg)
![Types: Order • ODE (ordinary DE) – Contains only ordinary derivatives (no partials) – Types: Order • ODE (ordinary DE) – Contains only ordinary derivatives (no partials) –](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-17.jpg)
![Types: Order -> Degree • ODE (ordinary DE) – Can further subdivide into different Types: Order -> Degree • ODE (ordinary DE) – Can further subdivide into different](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-18.jpg)
![Types: Linearity • ODE (ordinary DE) – Linear or non-linear? • Linear if dependent Types: Linearity • ODE (ordinary DE) – Linear or non-linear? • Linear if dependent](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-19.jpg)
![Solving • General solution – – Often many solutions can satisfy a differential eqn Solving • General solution – – Often many solutions can satisfy a differential eqn](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-20.jpg)
![Solving • But for a particular solution – We must specify boundary conditions – Solving • But for a particular solution – We must specify boundary conditions –](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-21.jpg)
![Types: analytical, non-analytical • Analytical: Beer’s Law - attenuation – k is extinction coefficient Types: analytical, non-analytical • Analytical: Beer’s Law - attenuation – k is extinction coefficient](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-22.jpg)
![Initial & boundary conditions • One point conditions – We saw as general solution Initial & boundary conditions • One point conditions – We saw as general solution](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-23.jpg)
![Solving: examples • Verify that satisfies is a solution of – (2 nd order, Solving: examples • Verify that satisfies is a solution of – (2 nd order,](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-24.jpg)
![Initial & boundary conditions • Two point conditions – – – Again consider Solution Initial & boundary conditions • Two point conditions – – – Again consider Solution](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-25.jpg)
![Separation of variables • We have considered simple cases so far – Where and Separation of variables • We have considered simple cases so far – Where and](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-26.jpg)
![Separation of variables • Equation is now separated & if we can integ. we Separation of variables • Equation is now separated & if we can integ. we](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-27.jpg)
![2 nd order linear equations • Form – Where p(x), q(x), r(x) and f(x) 2 nd order linear equations • Form – Where p(x), q(x), r(x) and f(x)](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-28.jpg)
![Partial differential equations • DEs with two or more dependent variables – Particularly important Partial differential equations • DEs with two or more dependent variables – Particularly important](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-29.jpg)
![Partial differential equations – Now 2 nd partial derivatives of u(x, t) wrt to Partial differential equations – Now 2 nd partial derivatives of u(x, t) wrt to](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-30.jpg)
![Partial differential equations • In 3 D? – Just consider y and z also, Partial differential equations • In 3 D? – Just consider y and z also,](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-31.jpg)
![SOLVING: Numerical approaches • Euler’s Method – Consider 1 st order eqn with initial SOLVING: Numerical approaches • Euler’s Method – Consider 1 st order eqn with initial](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-32.jpg)
![SOLVING: Numerical approaches • Euler’s Method – True soln passes thru (x 0, y SOLVING: Numerical approaches • Euler’s Method – True soln passes thru (x 0, y](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-33.jpg)
![SOLVING: Numerical approaches • Euler’s Method: example – Use Euler’s method with h = SOLVING: Numerical approaches • Euler’s Method: example – Use Euler’s method with h =](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-34.jpg)
![SOLVING: Numerical approaches • Runge-Kutta methods (4 th order here…. ) – – Euler SOLVING: Numerical approaches • Runge-Kutta methods (4 th order here…. ) – – Euler](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-35.jpg)
![SOLVING: Numerical approaches • Runge-Kutta example – As before, but now use R-K with SOLVING: Numerical approaches • Runge-Kutta example – As before, but now use R-K with](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-36.jpg)
![Very brief intro to Monte Carlo • Brute force method(s) for integration / parameter Very brief intro to Monte Carlo • Brute force method(s) for integration / parameter](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-37.jpg)
![Basics: MC integration • • Pick N random points in a multidimensional volume V, Basics: MC integration • • Pick N random points in a multidimensional volume V,](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-38.jpg)
![Basics: MC integration • Why not choose a grid? Error falls as N-1 (quadrature Basics: MC integration • Why not choose a grid? Error falls as N-1 (quadrature](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-39.jpg)
![Summary • Differential equations – Describe dynamic systems – wide range of examples, particularly Summary • Differential equations – Describe dynamic systems – wide range of examples, particularly](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-40.jpg)
![Summary • Solving – Analytical methods? • Find general solution by integrating, leaves constants Summary • Solving – Analytical methods? • Find general solution by integrating, leaves constants](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-41.jpg)
![END END](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-42.jpg)
![Example • Radioactive decay – Random, independent events, so for given sample of N Example • Radioactive decay – Random, independent events, so for given sample of N](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-43.jpg)
![Example • Radioactive decay – EG: 14 C has half-life of 5730 years & Example • Radioactive decay – EG: 14 C has half-life of 5730 years &](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-44.jpg)
![Exercises • General solution of is given by • Find particular solution satisfies x Exercises • General solution of is given by • Find particular solution satisfies x](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-45.jpg)
![Exercises • Show that the analytical solution of with y(x=0)=2 is • Compare values Exercises • Show that the analytical solution of with y(x=0)=2 is • Compare values](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-46.jpg)
![Using an integrating factor • For equations of form – Where P(x) and Q(x) Using an integrating factor • For equations of form – Where P(x) and Q(x)](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-47.jpg)
![Using an integrating factor • Because it follows that • And if we can Using an integrating factor • Because it follows that • And if we can](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-48.jpg)
![Using an integrating factor • And we see that • And so (-ln. K Using an integrating factor • And we see that • And so (-ln. K](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-49.jpg)
![Using an integrating factor: example • Solve – From previous we see that – Using an integrating factor: example • Solve – From previous we see that –](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-50.jpg)
![Linear operators • When L{y} = f(x) is a linear differential equation, L is Linear operators • When L{y} = f(x) is a linear differential equation, L is](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-51.jpg)
![Linear operators • Note that L{y} = f(x) is a linear diff. eqn so Linear operators • Note that L{y} = f(x) is a linear diff. eqn so](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-52.jpg)
- Slides: 52
![GEOGG 121 Methods Differential Equations MC etc Dr Mathias Mat Disney UCL Geography Office GEOGG 121: Methods Differential Equations, MC etc Dr. Mathias (Mat) Disney UCL Geography Office:](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-1.jpg)
GEOGG 121: Methods Differential Equations, MC etc Dr. Mathias (Mat) Disney UCL Geography Office: 113, Pearson Building Tel: 7670 0592 Email: mathias. disney@ucl. ac. uk www. geog. ucl. ac. uk/~mdisney
![Lecture outline Differential equations Introduction importance Types of DE Lecture outline • Differential equations – Introduction & importance – Types of DE •](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-2.jpg)
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
![Reading material Textbooks These are good UG textbooks that have WAY more detail Reading material • Textbooks These are good UG textbooks that have WAY more detail](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-3.jpg)
Reading material • Textbooks These are good UG textbooks that have WAY more detail than we need – Boas, M. L. , 1985 (2 nd ed) Mathematical Methods in the Physical Sciences, Wiley, 793 pp. – Riley, K. F. , M. Hobson & S. Bence (2006) Mathematical Methods for Physics & Engineering, 3 rd ed. , CUP. – Croft, A. , Davison, R. & Hargreaves, M. (1996) Engineering Mathematics, 2 nd 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, 2 nd 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 1 st order DEs Introduction • What is a differential equation? – General 1 st order DEs –](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-4.jpg)
Introduction • What is a differential equation? – General 1 st order DEs – 1 st case t is independent variable, x is dependent variable – 2 nd 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 Examples • Velocity – Change of distance x with time t i. e. •](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-5.jpg)
Examples • Velocity – Change of distance x with time t i. e. • Acceleration – Change of v with t i. e. • Newton’s 2 nd 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 Examples • Radioactive decay of unstable nucleus – Random, independent events, so for given](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-6.jpg)
Examples • Radioactive decay of unstable nucleus – Random, independent events, so for given sample of N atoms, no. of decay events –d. N 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 etc
![Examples Compound Interest How does an investment St change with time given Examples • Compound Interest – How does an investment S(t), change with time, given](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-7.jpg)
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 http: //www. thecreditexaminer. com/five-things-to-know-about-compound-interest-and-saving http: //www. singaporeolevelmaths. com/tag/compound-interest-formula/`
![Examples Population dynamics Logistic equation Malthus Verhulst Lotka Rate of Examples • Population dynamics – Logistic equation (Malthus, Verhulst, Lotka…. ) – Rate of](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-8.jpg)
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, d. P/dt r. P but as P increases (asymptotically) to K, d. P/dt goes 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 LotkaVolterra predatorprey equations Same form but now Examples • Population dynamics: II – Lotka-Volterra (predator-prey) equations – Same form, but now](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-9.jpg)
Examples • Population dynamics: II – Lotka-Volterra (predator-prey) equations – Same form, but now 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 A ‘phase space’ plot – see later on logistic growth, http: //www. scholarpedia. org/article/Predator-prey_model#Lotka-Volterra_Model
![Examples http www evolutionofideas comhomepageMathematicsmathsinnerA20 model20 of20 corruption htm Examples http: //www. evolution-ofideas. com/homepage/Mathematics/mathsinner/A%20 model%20 of%20 corruption. htm](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-10.jpg)
Examples http: //www. evolution-ofideas. com/homepage/Mathematics/mathsinner/A%20 model%20 of%20 corruption. htm
![Examples Transport momentum heat mass Transport usually some constant proportionality factor x Examples • Transport: momentum, heat, mass…. – Transport usually some constant (proportionality factor) x](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-11.jpg)
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 1 D 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 nonanalytical Analytical closed form Exact solution e g in terms Types: analytical, non-analytical • Analytical, closed form – Exact solution e. g. in terms](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-12.jpg)
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 nonanalytical Analytical example Exact solution e g Solve by Types: analytical, non-analytical • Analytical example – Exact solution e. g. – Solve by](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-13.jpg)
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 nonanalytical Particular solution BOUNDARY conditions e g set t Types: analytical, non-analytical • Particular solution? – BOUNDARY conditions e. g. set t =](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-14.jpg)
Types: analytical, non-analytical • Particular solution? – BOUNDARY conditions e. g. set t = 0 to get c 1, 2 i. e. – So x 0 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 nonanalytical Analytical population growthdecay example Log scale obviously linear Types: analytical, non-analytical • Analytical: population growth/decay example Log scale – obviously linear….](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-15.jpg)
Types: analytical, non-analytical • Analytical: population growth/decay example Log scale – obviously linear….
![Types ODEs PDEs ODE ordinary DE Contains only ordinary derivatives PDE Types: ODEs, PDEs • ODE (ordinary DE) – Contains only ordinary derivatives • PDE](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-16.jpg)
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 Types: Order • ODE (ordinary DE) – Contains only ordinary derivatives (no partials) –](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-17.jpg)
Types: Order • ODE (ordinary DE) – Contains only ordinary derivatives (no partials) – Can be of different order • Order of highest derivative 2 nd 1 st
![Types Order Degree ODE ordinary DE Can further subdivide into different Types: Order -> Degree • ODE (ordinary DE) – Can further subdivide into different](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-18.jpg)
Types: Order -> Degree • ODE (ordinary DE) – Can further subdivide into different degree • Degree (power) to which highest order derivative raised 1 st order 3 rd degree 1 st order 1 st degree 2 nd order 2 nd degree
![Types Linearity ODE ordinary DE Linear or nonlinear Linear if dependent Types: Linearity • ODE (ordinary DE) – Linear or non-linear? • Linear if dependent](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-19.jpg)
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 Non-linear y dy/dx Non-linear y 2 term Non-linear sin y term
![Solving General solution Often many solutions can satisfy a differential eqn Solving • General solution – – Often many solutions can satisfy a differential eqn](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-20.jpg)
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 Solving • But for a particular solution – We must specify boundary conditions –](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-21.jpg)
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 = Ce 0 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 nonanalytical Analytical Beers Law attenuation k is extinction coefficient Types: analytical, non-analytical • Analytical: Beer’s Law - attenuation – k is extinction coefficient](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-22.jpg)
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 ϕ 0 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 Initial & boundary conditions • One point conditions – We saw as general solution](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-23.jpg)
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 secondition i. e. dy/dx = 1 when x = 0 so differentiate – Particular solution is then
![Solving examples Verify that satisfies is a solution of 2 nd order Solving: examples • Verify that satisfies is a solution of – (2 nd order,](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-24.jpg)
Solving: examples • Verify that satisfies is a solution of – (2 nd order, 1 st degree, linear)
![Initial boundary conditions Two point conditions Again consider Solution Initial & boundary conditions • Two point conditions – – – Again consider Solution](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-25.jpg)
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 secondition 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 Separation of variables • We have considered simple cases so far – Where and](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-26.jpg)
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 Separation of variables • Equation is now separated & if we can integ. we](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-27.jpg)
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 both sides wrt x – i. e. and so and then integrate and – If we define D = 2 C 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
![2 nd order linear equations Form Where px qx rx and fx 2 nd order linear equations • Form – Where p(x), q(x), r(x) and f(x)](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-28.jpg)
2 nd order linear equations • Form – Where p(x), q(x), r(x) and f(x) are fns of x only – This is inhomogeneous (dep on y) – Related homogeneous form 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
![Partial differential equations DEs with two or more dependent variables Particularly important Partial differential equations • DEs with two or more dependent variables – Particularly important](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-29.jpg)
Partial differential equations • DEs with two or more dependent variables – Particularly important for motion (in 2 or 3 D), where eg position (x, y, z) varying with time t • Key example of wave equation – Eg in 1 D 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 2 nd partial derivatives of ux t wrt to Partial differential equations – Now 2 nd partial derivatives of u(x, t) wrt to](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-30.jpg)
Partial differential equations – Now 2 nd 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 3 D Just consider y and z also Partial differential equations • In 3 D? – Just consider y and z also,](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-31.jpg)
Partial differential equations • In 3 D? – 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
![SOLVING Numerical approaches Eulers Method Consider 1 st order eqn with initial SOLVING: Numerical approaches • Euler’s Method – Consider 1 st order eqn with initial](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-32.jpg)
SOLVING: Numerical approaches • Euler’s Method – Consider 1 st order eqn with initial cond. y(x 0) = y 0 – Find an approx. solution yn at equally spaced discrete values (steps) of x, xn – Euler’s method == find gradient at x = x 0 i. e. – Tangent line approximation True solution y y(x 1) Tangent approx. y 1 y 0 0 x 1 http: //en. wikipedia. org/wiki/Numerical_ordinary_differential_equations x Croft et al. , p 495 Numerical Recipes in C ch. 16, p 710 http: //apps. nrbook. com/c/index. html
![SOLVING Numerical approaches Eulers Method True soln passes thru x 0 y SOLVING: Numerical approaches • Euler’s Method – True soln passes thru (x 0, y](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-33.jpg)
SOLVING: Numerical approaches • Euler’s Method – True soln passes thru (x 0, y 0) with gradient f(x 0, y 0) at that point – Straight line (y = mx + c) approx has eqn – This approximates true solution but only near (x 0, y 0), so only extend it short dist. h along x axis to x = x 1 – Here, y = y 1 and – Since h = x 1 -x 0 we see – Can then find y 1, and we then know (x 1, y 1)…. . rinse, repeat…. True solution y y(x 1) y 1 y 0 0 Tangent approx. x x x Generate series of values iteratively Accuracy depends on h Croft et al. , p 495 Numerical Recipes in C ch. 16, p 710 http: //apps. nrbook. com/c/index. html
![SOLVING Numerical approaches Eulers Method example Use Eulers method with h SOLVING: Numerical approaches • Euler’s Method: example – Use Euler’s method with h =](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-34.jpg)
SOLVING: 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 y 1 -4 over x 1 = 0. 25, x 2 = 0. 5, x 3 = 0. 75, x 4 = 1. 0 say, so – with x 0 = 0 y 0 = 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?
![SOLVING Numerical approaches RungeKutta methods 4 th order here Euler SOLVING: Numerical approaches • Runge-Kutta methods (4 th order here…. ) – – Euler](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-35.jpg)
SOLVING: Numerical approaches • Runge-Kutta methods (4 th order here…. ) – – Euler 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 = y 0 when x = x 0, use – where http: //en. wikipedia. org/wiki/Numerical_ordinary_differential_equations http: //en. wikipedia. org/wiki/Runge%E 2%80%93 Kutta_methods Croft et al. , p 502 Rile et al. p 1026 Numerical Recipes in C ch. 16, p 710 http: //apps. nrbook. com/c/index. html
![SOLVING Numerical approaches RungeKutta example As before but now use RK with SOLVING: Numerical approaches • Runge-Kutta example – As before, but now use R-K with](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-36.jpg)
SOLVING: 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 methods for integration parameter Very brief intro to Monte Carlo • Brute force method(s) for integration / parameter](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-37.jpg)
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/N 1/2 • 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 etc http: //en. wikipedia. org/wiki/Monte_Carlo_method http: //en. wikipedia. org/wiki/Monte_Carlo_integration Numerical Recipes in C ch. 7, p 304 http: //apps. nrbook. com/c/index. html
![Basics MC integration Pick N random points in a multidimensional volume V Basics: MC integration • • Pick N random points in a multidimensional volume V,](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-38.jpg)
Basics: MC integration • • Pick N random points in a multidimensional volume V, x 1, x 2, …. x. N MC integration approximates integral of function f over volume V as • Where • +/- term is 1 SD error – falls of as 1/N 1/2 and 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 N1 quadrature Basics: MC integration • Why not choose a grid? Error falls as N-1 (quadrature](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-39.jpg)
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” Sobol method v pseudorandom: 1000 points FROM: http: //en. wikipedia. org/wiki/Lowdiscrepancy_sequence
![Summary Differential equations Describe dynamic systems wide range of examples particularly Summary • Differential equations – Describe dynamic systems – wide range of examples, particularly](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-40.jpg)
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 Summary • Solving – Analytical methods? • Find general solution by integrating, leaves constants](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-41.jpg)
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…. .
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![Example Radioactive decay Random independent events so for given sample of N Example • Radioactive decay – Random, independent events, so for given sample of N](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-43.jpg)
Example • Radioactive decay – Random, independent events, so for given sample of N atoms, no. of decay events –d. N 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=T 1/2) = time taken to decay to half initial N i. e. N 0/2 – Express T 1/2 in terms of tau
![Example Radioactive decay EG 14 C has halflife of 5730 years Example • Radioactive decay – EG: 14 C has half-life of 5730 years &](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-44.jpg)
Example • Radioactive decay – EG: 14 C 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/N 0 = 4/14 = 0. 286 – From prev. , tau = T 1/2/ln 2 = 5730/ln 2 = 8267 yrs – So t = -tau x ln(N/N 0) = 10356 yrs
![Exercises General solution of is given by Find particular solution satisfies x Exercises • General solution of is given by • Find particular solution satisfies x](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-45.jpg)
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 (p 458, 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 yx02 is Compare values Exercises • Show that the analytical solution of with y(x=0)=2 is • Compare values](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-46.jpg)
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 •
![Using an integrating factor For equations of form Where Px and Qx Using an integrating factor • For equations of form – Where P(x) and Q(x)](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-47.jpg)
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. pdf http: //en. wikipedia. org/wiki/Integrating_factor
![Using an integrating factor Because it follows that And if we can Using an integrating factor • Because it follows that • And if we can](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-48.jpg)
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_rule http: //www. cse. salford. ac. uk/profiles/gsmcdonald/H-Tutorials/ordinary-differentialequations-integrating-factor. pdf
![Using an integrating factor And we see that And so ln K Using an integrating factor • And we see that • And so (-ln. K](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-49.jpg)
Using an integrating factor • And we see that • And so (-ln. K is const. of integ. ) • 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_rule http: //www. cse. salford. ac. uk/profiles/gsmcdonald/H-Tutorials/ordinary-differentialequations-integrating-factor. pdf
![Using an integrating factor example Solve From previous we see that Using an integrating factor: example • Solve – From previous we see that –](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-50.jpg)
Using an integrating factor: example • Solve – From previous we see that – Using the formula above – And we know the solution is given by – So , as and
![Linear operators When Ly fx is a linear differential equation L is Linear operators • When L{y} = f(x) is a linear differential equation, L is](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-51.jpg)
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 f 1 and f 2 as follows 1. 2. where a is a constant 3. where a, b are constants – Example: if – and show that
![Linear operators Note that Ly fx is a linear diff eqn so Linear operators • Note that L{y} = f(x) is a linear diff. eqn so](https://slidetodoc.com/presentation_image_h/0c283d31fc83b033bc8342580cc02ae3/image-52.jpg)
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
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