Advanced Analytic Methods in Science and Engineering is a comprehensive treatment of the advanced methods of applied mathematics. It was designed to strengthen the mathematical abilities of graduate students and train them to think on their own.

This course analyzes the functions of a complex variable and the calculus of residues. It also covers subjects such as ordinary differential equations, partial differential equations, Bessel and Legendre functions, and the Sturm-Liouville theory.

This resource is a video abstract of a research paper created by Research Square on behalf of its authors. It provides a synopsis that's easy to understand, and can be used to introduce the topics it covers to students, researchers, and the general public. The video's transcript is also provided in full, with a portion provided below for preview:

"Angiogenesis, the formation of new blood vessels, is a popular target of various therapies. Some therapies, like those used in tissue engineering, are designed to promote angiogenesis and new tissue growth, while other therapies, such as those designed to fight cancer, aim to suppress angiogenesis— a lifeline for tumor cells. Unfortunately, these therapies aren’t always effective. Now, a new mathematical model could help researchers understand what molecular levers to pull to effectively modulate angiogenesis. Trained on published experimental data, the model predicted the effects of activating two common targets of angiogenesis-based therapies: vascular endothelial growth factor, or VEGF, and fibroblast growth factor, or FGF. Computational experiments showed that the two factors modify both the ERK signaling pathway, which is linked to cell proliferation, and the Akt signaling pathway, which is associated with cell survival and migration..."

The rest of the transcript, along with a link to the research itself, is available on the resource itself.

Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE’s) deal with functions of one variable, which can often be thought of as time.

The laws of nature are expressed as differential equations. Scientists and engineers must know how to model the world in terms of differential equations, and how to solve those equations and interpret the solutions. This course focuses on the equations and techniques most useful in science and engineering.
Course Format
This course has been designed for independent study. It provides everything you will need to understand the concepts covered in the course. The materials include:

Lecture Videos by Professor Arthur Mattuck.
Course Notes on every topic.
Practice Problems with Solutions.
Problem Solving Videos taught by experienced MIT Recitation Instructors.
Problem Sets to do on your own with Solutions to check your answers against when you’re done.
A selection of Interactive Java® Demonstrations called Mathlets to illustrate key concepts.
A full set of Exams with Solutions, including practice exams to help you prepare.

Content Development
Haynes Miller 
Jeremy Orloff 
Dr. John Lewis 
Arthur Mattuck

This course provides an introduction to nonlinear deterministic dynamical systems. Topics covered include: nonlinear ordinary differential equations; planar autonomous systems; fundamental theory: Picard iteration, contraction mapping theorem, and Bellman-Gronwall lemma; stability of equilibria by Lyapunov’s first and second methods; feedback linearization; and application to nonlinear circuits and control systems.

This book consists of ten weeks of material given as a course on ordinary differential equations (ODEs) for second year mathematics majors at the University of Bristol. It is the first course devoted solely to differential equations that these students will take.

This course provides an introduction to linear systems, transfer functions, and Laplace transforms. It covers stability and feedback, and provides basic design tools for specifications of transient response. It also briefly covers frequency-domain techniques.