Partial Differential Equations

Prerequisites:

Fundamental Algebra and Analysis II

Course description:

Partial differential equations (PDEs) play a pivotal role in modeling a vast array of complex phenomena across numerous scientific and engineering disciplines. These equations are essential for understanding and predicting behavior in fields such as fluid dynamics, where they describe the motion of fluids like air or water; optics, where they model the propagation of light; atomic and plasma physics, where they help in understanding the interactions between charged particles and electromagnetic fields; and elasticity, where they describe how materials deform under stress. Additionally, PDEs are indispensable in studying chemical reactions, where they model the diffusion and reaction processes, as well as in climate modeling, where they are used to predict changes in the atmosphere and oceans. PDEs have extensive applications in finance, particularly in the modeling of stock markets and options pricing, where they are applied to forecast fluctuations and trends over time.

This course is designed for advanced undergraduate and graduate students and students who have a strong interest in mathematics, science, and engineering. It will be especially beneficial for those pursuing research or careers in areas where PDEs play a crucial role, such as aerospace engineering, environmental science, mechanical engineering, physics, computational biology, and quantitative finance. Students will gain a deep understanding of the mathematical structure of PDEs, as well as a toolkit of analytical methods that can be applied across a broad range of applications.

Learning objectives:

The study of PDEs has become a cornerstone of applied mathematics, as it provides a framework for translating physical, biological, and economic systems into mathematical language, allowing for rigorous analysis and precise predictions. This course delves deeply into both the theoretical underpinnings and practical applications of PDEs, making it an essential component of the mathematical toolkit for anyone engaged in scientific research or engineering. By focusing on the derivation of a set of core PDEs from relatively simple physical models, the course will highlight how these equations often serve as the foundation for more complex systems found in real-world applications. For example, the heat equation, derived from basic principles of thermodynamics, can be extended to model heat transfer in complex materials, while the wave equation can be adapted to describe vibrations in sophisticated mechanical systems or even electromagnetic waves in optical fibers.

In addition to deriving key examples of PDEs, the course will provide a thorough exploration of the powerful analytical methods and tools used to solve them. These methods include separation of variables, a technique that reduces complex PDEs into simpler, more manageable ordinary differential equations (ODEs); Fourier series and Fourier transforms, which allow for the decomposition of functions into simpler sinusoidal components, enabling efficient solutions for many problems involving periodic or infinite domains; Sturm-Liouville theory, which provides a framework for solving eigenvalue problems that arise in a wide variety of applications; and Green's functions, which are instrumental in solving inhomogeneous PDEs by representing the solution as a superposition of responses to point sources.

Throughout the course, students will be introduced to both classical techniques and modern developments in the theory of PDEs, ensuring they are well-equipped to tackle challenges in their respective fields. The course will emphasize a balanced approach, combining theoretical rigor with practical problem-solving skills, helping students develop an intuition for how to apply these methods in real-world situations. While the examples provided will often stem from relatively simple physical systems, the techniques covered in the course are highly versatile and can be extended to more complex scenarios, such as turbulent fluid flows, nonlinear wave propagation, or multiphase chemical reactions.

Detailed topics covered:

This course will begin with a derivation of classical partial differential equations, such as heat, wave, and Laplace equations. It will then introduce the method of separations of variables and Fourier series. The Sturm-Liouville problem, higher dimensional PDE's, and Green's Functions will be studied. A brief introduction to the Fourier transform will be provided, followed by infinite domain problems and non-homogeneous problems. The method of characteristics will be applied to the wave equation. Time permitting, there will be an introduction to nonlinear equations, starting with Burgers' equations, the method of characteristics and shock waves. The Euler and Navier-Stokes equations will be derived, and questions of existence, uniqueness, regularity, and energy balance will be discussed. The course will conclude with an overview of modern directions on Kolmogorov and Onsager's conjectures in relation to turbulence.

In addition to lectures, there will be discussion sessions designed to develop strong problem-solving skills.