Introduction to Partial Differential Equations

Partial Differential Equations (PDEs) are fundamental tools in mathematics, providing powerful models for continuous change in space and time. These equations describe a vast range of natural phenomena, from the Earth's temperature and magnetic fields to fluid dynamics, electrostatic potentials, tumor growth, brain activity, stock market fluctuations, and population dynamics. With advancements in software and the rise of Artificial Intelligence (AI) and Deep Learning, the applications of PDEs are rapidly expanding across science and technology.

This course offers a solid foundation in Partial Differential Equations, beginning with their physical origins and motivations. You'll explore the classical equations of mathematical physics, including the heat equation, wave equation, and Laplace equation.

Key concepts covered include:

• Separation of Variables: A core technique for solving many PDEs.
• Integral Transforms: Powerful tools for simplifying and solving equations.
• Special Functions of Mathematical Physics: Understanding functions crucial to PDE solutions.
• Fourier Series: Decomposing functions into simpler components for analysis.

While emphasizing the underlying mathematical principles, the course maintains strong connections to the physics from which these equations arise. A significant portion of the course is dedicated to applying these methods to solve real-world physical problems, providing practical experience in solving PDEs and understanding their applications. Learn how to model and analyze complex systems using the power of PDEs and modern computational techniques.