ENV 264 - APPLIED DIFFERENTIAL EQUATIONS IN ENVIRONMENTAL SCIENCES

General Information

Instructor: Gabriel Katul
Number of Credits: 2 Units
Prerequisits: Mathematics 31 or equivalent
Work load: 6 Homework, an in-class (open book) midterm and a term project

Objectives and Scope

It is well recognized that many environmental problems require the construction and solution to equations (or system of equations) that involve relations between rates and states of environmental variables. The objective of this course is to illustrate the use of differential equations and analytical tools from calculus to solve such problems. The course covers basic analytical and numerical solutions to ordinary differential equations (O.D.E.) with an introduction to partial differential equations commonly encountered in environmental studies (mainly diffusion and reaction-diffusion equations). Example applications include atomic waste disposal in oceans, refined population forecasting, hydrologic transport problems in xylems, predator-prey systems, heat transport in soils, and spatial models of biomass-water interactions. It is envisaged that this special topics course will also serve as an applied mathematics review for students who have not been in contact with calculus in the last two years.

Benefit to Graduate and Professional Students

This course offers unique opportunities for professional and graduate students to be exposed to differential equations in a less formal mathematical setting. Emphasis will be placed on a dynamical systems interpretation of differerntial equations. Concepts such as stability, resilience, and equilibrium are routinely used in environmental sciences (climate, ecosystem, conservation efforts, etc...), yet the 'genesis' of these concepts remains embedded in differential equations. Finally, ENV264 will make use of a computer-aided software ( MATHEMATICA and Matlab) capable of solving analytically and numerically much of the ODEs encountered in environmental problems.

Topics

The course is divided into four parts:

Part - 1: General Calculus Review and Introduction to Ordinary Differential Equations:
- Review of differential and integral calculus.
- Riemann sequences, and integral.
- Introduction to differential equations and general definitions.
- Classification of differential equations.
- The role of differential equations in environmental transport problems.
Part - 2: First Order Ordinary Differential Equations
- Elementary concepts regarding homogeneous and non-homogeneous functions.
- First order O.D.E. - Method of Variable Separation
- Introduction to exact differentials and the Integrating Factor Methods
- Bernoulli's equation
- Example applications: (1) Flow of water from dams, (2) Atomic waste disposal in oceans, (3) Refined logistic equations for population forecasting, (4) Water movement in the soil-plant-atmosphere continuum using the electric circuit analogies.
Part - 3 : Extension to Higher Order Differential Equations
- Linear Differential Equations of the second order
- Superposition principle
- General solutions to homogenous equations with constant coefficients of order n.
- Example application: ground water hydraulics.
- Systems of Differential Equations
- Example applications: Predator-Prey models for population forecasting with emphasis on stability, equilibrium, and resilience of systems.
Part - 4 : Numerical Methods
- Introduction to forward, backward, and central differencing.
- Euler's method, Predictor-Corrector approaches, and Runge-Kutta approximations.
- Limitations and stability problems.
- Examples: Heat movement in soils with time-dependent soil heat flux, nonlinear reaction-diffusion equations with applications to pattern formation - a case study in biomass-water interactions.
- Project: A requirement of the course is a term project presented in-class as an oral report (15 minutes). A starting point can be a manuscript from the list below.

Examples:
R. F. Costantino, R. A. Desharnais, J. M. Cushing, B. Dennis, 1997, Chaotic Dynamics in an Insect Population, Science, 275, 389-391.
M. Scheffer, S. Carpenter, J. A. Foley, and B. Walker, 2001, Catastrophic shifts in ecosystems, Nature, 413, 591-596.
D. J. D. Earn, P. Rohani,B. M. Bolker, B. T. Grenfell, 2000, A Simple Model for Complex Dynamical Transitions in Epidemics, Science, 287, 667-670.
P. Turchin, A.D. Taylor, and J.D. Reeve, Dynamical Role of Predators in Population Cycles of a Forest Insect: An Experimental Test, Science, 285, 1068-1071.
M. Rietkerk,, M. C. Boerlijst, F. van Langevelde, R. HilleRisLambers, J. van de Koppel, L. Kumar, H. H. T. Prins, and A. M. de Roos, 2002, Self-Organization of Vegetation in Arid Ecosystems, American Naturalist, 160, 524-530.
M. Rietkerk, S. C. Dekker, P. C. de Ruiter, J. van de Koppel, 2004, Self-Organized Patchiness and Catastrophic Shifts in Ecosystems, Science, 305, 1926-1929.
References

Abromowitz, M., and I.A. Stegun, 1964, Handbook of Mathematical Functions, Dover, 1046 pp.
Coddington, E.A., 1961, An Introduction to Ordinary Differential Equations, Dover, 292 pp.
DuChateau, P.C., 1992, Advanced Math for Engineers and Scientists, Harper Collins College Outline, 389 pp.
Edwards, C.H., and D.E. Penney, 1982, Calculus and Analytic Geometry, Prentice- Hall, 895 pp.
Farlow, S.J., 1982, Partial Differential Equations for Scientists and Engineers, Dover, 414 pp.
Greenberg, M.D., 1978, Foundations of Applied Mathematics, Prentice Hall, 636 pp.
Kreyzig, E., 1988, Advanced Engineering Mathematics, John Wiley and Sons, 1294 pp.
Lebedev, N.N., 1965, Special Functions and Their Applications, Dover, 308 pp.
Moursund, D.G., and C.S. Duris, 1967, Elementary Theory and Applications of Numerical Analysis, Dover, 297 pp.
Schey, H.M., 1992, Div., Grad., Curl., and All That, W.W. Norton and Company, 163 pp.
Spiegel, M.R., 1993, Mathematical Handbook, Shaum's Outline Series, 271 pp.
Spiegel, M.R., 1963, Advanced Calculus, Shaum's Outline Series, 384 pp.
Spiegel, M.R., 1963, Advanced Mathematics, Shaum's Outline Series, 407 pp.

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