A comprehensive treatment of the theory of partial differential equations (pde) from an applied mathematics perspective. Equilibrium, propagation, diffusion, and other phenomena. Initial and boundary value problems. Transform methods, eigenvalue and eigenfunction expansions, Green's functions. Theory of characteristics and shocks. Boundary layers and other singular perturbation phenomena. Elementary concepts for the numerical solution of pde's. Illustrative examples from fluid dynamics, nonlinear waves, geometrical optics, and other applications.

The module discusses the process of electrons moving across a p-n or n-p junction known as diffusion.

The module discusses the process of electrons moving across a p-n or n-p junction known as diffusion.

This course covers the following topics: conduction, diffusion, convection in electrolytes; fields in heterogeneous media; electrical double layers; Maxwell stress tensor and electrical forces in physiological systems; and fluid and solid continua: equations of motion useful for porous, hydrated biological tissues. Case studies considered include membrane transport; electrode interfaces; electrical, mechanical, and chemical transduction in tissues; electrophoretic and electroosmotic flows; diffusion/reaction; and ECG. The course also examines electromechanical and physicochemical interactions in biomaterials and cells; orthopaedic, cardiovascular, and other clinical examples.

This course covers the following topics: conduction, diffusion, convection in electrolytes; fields in heterogeneous media; electrical double layers; Maxwell stress tensor and electrical forces in physiological systems; and fluid and solid continua: equations of motion useful for porous, hydrated biological tissues. Case studies considered include membrane transport; electrode interfaces; electrical, mechanical, and chemical transduction in tissues; electrophoretic and electroosmotic flows; diffusion/reaction; and ECG. The course also examines electromechanical and physicochemical interactions in biomaterials and cells; orthopaedic, cardiovascular, and other clinical examples.

This course introduces the basic driving forces for electric current, fluid flow, and mass transport, plus their application to a variety of biological systems. Basic mathematical and engineering tools will be introduced, in the context of biology and physiology. Various electrokinetic phenomena are also considered as an example of coupled nature of chemical-electro-mechanical driving forces. Applications include transport in biological tissues and across membranes, manipulation of cells and biomolecules, and microfluidics.

Hands-on introduction to NMR presenting background in classical theory and instrumentation. Each lecture followed by lab experiments both to demonstrate ideas presented during the lecture and to familiarize students with state-of-the-art NMR instrumentation. Experiments cover topics ranging from spin dynamics to spectroscopy, and include imaging.

This course examines the process of heat transfer, or the movement of thermal energy from one place to another as the result of a temperature difference. The student will thoroughly examine each type of heat transfer (conduction, convection, and radiation), as well as combinations of these modes. Upon successful completion of this course, the student will be able to: Formulate basic equation for heat transfer problems; Apply heat transfer principles to design and to evaluate performance of thermal systems; Solve differential and algebraic equations associated with thermal systems using analytical and numerical approaches; Calculate the performance of heat exchangers; Calculate radiation heat transfer between objects with simple geometries; Calculate and evaluate the impacts of initial and boundary conditions on the solutions of a particular heat transfer problem; Evaluate the relative contributions of different modes of heat transfer. (Mechanical Engineering 204)

Partial differential equations (PDEs) describe the relationships among the derivatives of an unknown function with respect to different independent variables, such as time and position. Experiment and observation provide information about the connections between rates of change of an important quantity, such as heat, with respect to different variables. Upon successful completion of this course, the student will be able to: State the heat, wave, Laplace, and Poisson equations and explain their physical origins; Define harmonic functions; State and justify the maximum principle for harmonic functions; State the mean value property for harmonic functions; Define linear operators and identify linear operations; Identify and classify linear PDEs; Identify homogeneous PDEs and evolution equations; Relate solving homogeneous linear PDEs to finding kernels of linear operators; Define boundary value problem and identify boundary conditions as periodic, Dirichlet, Neumann, or Robin (mixed); Explain physical significance of boundary conditions; Show uniqueness of solutions to the heat, wave, Laplace and Poisson equations with various boundary conditions; Define well-posedness; Define, characterize, and use inner products; Define the space of L2 functions, state its key properties, and identify L2 functions; Define orthogonality and orthonormal basis and show the orthogonality of certain trigonometric functions; Distinguish between pointwise, uniform, and L2 convergence and show convergence of Fourier series; Define Fourier series on [0,pi] and [0,L] and identify sufficient conditions for their convergence and uniqueness; Compute Fourier coefficients and construct Fourier series; Use the method of characteristics to solve linear and nonlinear first-order wave equations; Solve the one-dimensional wave equation using d'Alembert's formula; Use similarity methods to solve PDEs; Solve the heat, wave, Laplace, and Poisson equations using separation of variables and apply boundary conditions; Define the delta function and apply ideas from calculus and Fourier series to generalized functions; Derive Green's representation formula; Use Green's functions to solve the Poisson equation on the unit disk; Define the Fourier transform; Derive basic properties of the Fourier transform of a function, such as its relationship to the Fourier transform of the derivative; Show that the inverse Fourier transform of a product is a convolution; Compute Fourier transforms of functions; Use the Fourier transform to solve the heat and wave equations on unbounded domains. (Mathematics 222)

Unified treatment of phenomenological and atomistic kinetic processes in materials. Provides the foundation for the advanced understanding of processing, microstructural evolution, and behavior for a broad spectrum of materials. Emphasis on analysis and development of rigorous comprehension of fundamentals. Topics include: irreversible thermodynamics; diffusion; nucleation; phase transformations; fluid and heat transport; morphological instabilities; gas-solid, liquid-solid, and solid-solid reactions.

This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems.

The classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. Methods of solution, such as separation of variables, Fourier series and transforms, eigenvalue problems. Green's function methods are emphasized. 18.04 or 18.112 are useful, as well as previous acquaintance with the equations as they arise in scientific applications.

The classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. Methods of solution, such as separation of variables, Fourier series and transforms, eigenvalue problems. Green's function methods are emphasized. 18.04 or 18.112 are useful, as well as previous acquaintance with the equations as they arise in scientific applications.

The classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. Methods of solution, such as separation of variables, Fourier series and transforms, eigenvalue problems. Green's function methods are emphasized. 18.04 or 18.112 are useful, as well as previous acquaintance with the equations as they arise in scientific applications.

This course discusses the basics every manager needs to organize successful technology-driven innovation in both entrepreneurial and established firms. We start by examining innovation-based strategies as a source of competitive advantage and then examine how to build organizations that excel at identifying, building and commercializing technological innovations. Major topics include how the innovation process works; creating an organizational environment that rewards innovation and entrepreneurship; designing appropriate innovation processes (e.g. stage-gate, portfolio management); organizing to take advantage of internal and external sources of innovation; and structuring entrepreneurial and established organizations for effective innovation. The course examines how entrepreneurs can shape their firms so that they continuously build and commercialize valuable innovations. Many of the examples also focus on how established firms can become more entrepreneurial in their approach to innovation.

The goal of 3.044 is to teach cost-effective and sustainable production of solid material with a desired geometry, structure or distribution of structures, and production volume. Toward this end, it is organized around different types of phase transformations which determine the structure in various processes for making materials, in roughly increasing order of entropy change during those transformations: solid heat treatment, liquid-solid processing, fluid behavior, deformation processing, and vapor-solid processing. The course ends with several lectures that place the subject in the context of society at large.

Insert channels in a membrane and see what happens. See how different types of channels allow particles to move through the membrane.

Introduces the theory and technology of integrated-circuit fabrication. Lectures and laboratory sessions on basic processing techniques such as diffusion, oxidation, epitaxy, photolithography, chemical vapor deposition, and plasma etching. Emphasis on the interrelationships between material properties, device structure, and the electrical behavior of devices. Provides background for thesis work in microelectronics or for 6.151.

Introduces the theory and technology of integrated-circuit fabrication. Lectures and laboratory sessions on basic processing techniques such as diffusion, oxidation, epitaxy, photolithography, chemical vapor deposition, and plasma etching. Emphasis on the interrelationships between material properties, device structure, and the electrical behavior of devices. Provides background for thesis work in microelectronics or for 6.151. This course introduces the theory and technology of micro/nano fabrication. Lectures and laboratory sessions focus on basic processing techniques such as diffusion, oxidation, photolithography, chemical vapor deposition, and more. Through team lab assignments, students are expected to gain an understanding of these processing techniques, and how they are applied in concert to device fabrication. Students enrolled in this course have a unique opportunity to fashion and test micro/nano-devices, using modern techniques and technology.