PARTIAL DIFFERENTIAL EQUATIONS

A graduate-level book that introduces the modern theory of PDEs, with an emphasis on elliptic PDEs. Reviews note that the book is useful for a PhD course and provides a good introduction to the basic techniques of elliptic PDEs. A concise and clear textbook that covers the foundations of PDE theory. It’s suitable for a variety of courses, including undergraduate engineering, sciences, and mathematics courses, and first-year graduate courses. A book that covers a variety of important advanced topics and has enough material for a year-long course. Partial Differential Equations (PDEs) are equations that involve unknown functions of several variables and their partial derivatives. They arise in various fields such as physics, engineering, economics, and biology because many natural phenomena depend on multiple independent variables. For instance, PDEs model the behavior of temperature, sound, fluid dynamics, electromagnetism, and financial markets.
• Order of a PDE: The order is determined by the highest derivative that appears in the equation.
• Linear vs. Nonlinear PDEs: A PDE is linear if the unknown function and its derivatives appear to the first power and are not multiplied together. If the function or its derivatives appear in nonlinear forms (e.g., squares, products), the equation is nonlinear.
• Boundary and Initial Conditions: To solve a PDE, you typically need both boundary conditions (values on the domain boundary) and initial conditions (values at the start time).
Solution Methods
• Separation of Variables: This technique is used when the PDE can be written as a product of functions, each depending on a single variable. It transforms the PDE into simpler ordinary differential equations.