AMATH – Introduction to Differential Equations. David Harmsworth. Spring Course Notes by John Wainwright. AMATH is an advanced-level version of AMATH Compared to AMATH , AMATH offers a more theoretical treatment of differential equations and . Is this really as easy as people say? Is the difficulty prof dependant? How is the course otherwise (proof vs computation)?.

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Vector integral calculus-line integrals, surface integrals and vector fields, Green’s theorem, the Divergence theorem, and Stokes’ theorem. Applications include conservation laws, fluid flow and electromagnetic fields. An introduction a,ath Fourier analysis.

Fourier series and the Fourier transform. Frequency analysis of signals. Discrete and continuous spectra. A rigorous introduction to the field of computational mathematics. The focus is on the interplay between continuous models and their solution via discrete processes. Applications are used as motivation. This course may be substituted for CS in any degree plan or for prerequisite purposes; lab is not scheduled and students are expected to find time in open hours to complete their work.

Physical systems which lead to differential equations examples include mechanical vibrations, population dynamics, and mixing processes. Dimensional analysis and dimensionless variables. Solving linear differential equations: Laplace transform methods of solving differential equations.

In addition, emphasis will be placed on computational analysis of differential equations and on applications in science and engineering. Topology of Euclidean spaces, continuity, norms, completeness. Various applications, for example, to ordinary differential equations, optimization and numerical approximation.

Complex numbers, Cauchy-Riemann equations, analytic functions, conformal maps and applications to the solution of Laplace’s equation, contour integrals, Cauchy integral formula, Taylor and Laurent expansions, residue calculus and applications.

An introduction to numerical methods for ordinary and partial differential equations. The course focuses on introducing widely used methods and highlights applications in the natural sciences, the health sciences, engineering and finance.

Difference equations, Laplace and z transforms applied to discrete and continuous mathematical models taken from ecology, biology, economics and other fields. First order ordinary differential equations.

Applications to continuous compounding and the dynamics of supply and demand. Higher order linear ordinary differential equations. Systems of linear ordinary differential equations. Introduction to linear partial differential equations. The Fourier Transform and the diffusion equation.

Discussion of the Black-Scholes partial differential equations, and solutions thereof. Second order linear differential equations with non-constant coefficients, Sturm comparison, oscillation and separation theorems, series solutions and special functions. Linear vector differential equations in Rn, an introduction to dynamical systems. Laplace transforms applied to linear vector differential equations, transfer functions, the convolution theorem.

Perturbation methods for differential equations.

Numerical methods for differential equations. Applications are qmath throughout. Second order linear partial differential equations – the diffusion equation, wave equation, and Laplace’s equation. Methods of solution – separation of variables and eigenfunction expansions, the Fourier transform. Physical interpretation of solutions in terms of diffusion, waves and steady states. First order non-linear partial differential equations and the method of characteristics. Applications are ajath throughout.

Stress and strain tensors; analysis of stress and strain. Lagrangian and Eulerian methods for describing flow. Equations of continuity, motion and energy, constitutive equations. Basic equations of elasticity. Critical experiments and old quantum theory.

Basic concepts of quantum mechanics: Uncertainty, correspondence and superposition principles. Simple applications to finite and extended one-dimensional systems, harmonic oscillator, rigid rotor and hydrogen atom.

## MODERATORS

An introduction to dynamic mathematical modeling of cellular processes. The emphasis is on using computational tools to investigate differential equation-based models. A variety of cellular phenomena are discussed, including ion pumps, membrane potentials, intercellular communication, genetic networks, regulation of metabolic pathways, and signal transduction. An introduction to the mathematical modelling of biological processes, with emphasis on population biology.

Topics include ecology, epidemiology, microbiology, and physiology. Techniques include difference equations, ordinary differential equations, partial differential equations, stability analysis, phase plane analysis, travelling wave solutions, mathematical software.

Includes collaborative projects and computer labs. An introduction to some of the deep connections between mathematics and music. An introduction to contemporary mathematical concepts in signal analysis. Fourier series and Fourier transforms FFTthe classical sampling theorem and the time-frequency uncertainty principle.

Wavelets and multiresolution analysis. Applications include oversampling, denoising of audio, data compression and singularity detection. This course studies several classes of methods for the numerical solution of partial differential equations in multiple dimensions on structured and unstructured grids.

Finite volume methods for hyperbolic conservation laws: Finite element methods for elliptic and parabolic equations: Examples from fluid and solid mechanics.

Additional topics as time permits. A unified view of linear and nonlinear systems of ordinary differential equations in Rn.

Flow operators and their classification: Stable and unstable manifolds. Nonlinear systems, stability of equilibria and Lyapunov functions. The special case of flows in the plane, Poincare-Bendixson theorem and limit cycles. Applications to physical problems will be a motivating influence.

### AMATH : uwaterloo

Amat thorough discussion of the class of second-order linear partial differential equations with constant coefficients, in two independent variables.

Laplace’s equation, the wave equation and the heat equation in higher dimensions. Feedback control with applications. System theory in both time and frequency domain, state-space computations, stability, system uncertainty, loopshaping, linear quadratic regulators and estimation.

## AMATH 250 – Introduction to Differential Equations

Concept of functional and its variations. The solution of problems using variational methods – the Euler-Lagrange equations. Applications include an introduction to Hamilton’s Principle and optimal control. Introduction to wave motion and geophysical fluid amatj. Elements of compressible flow. The Hilbert space of states, observables and time evolution. Feynman path integral and Greens functions.

Coordinate transformations, angular momentum and spin. The relation between symmetries and conservation laws. Density matrix, Ehrenfest theorem and decoherence. Bell inequality and basics of quantum computing. Curved space-time and the Einstein field equations. The Schwarzschild solution and applications. The Friedmann-Robertson-Walker cosmological models.