In this course, we will examine partial differential equations (PDEs) and their solutions,
including methods for solving partial differential equations.
- What is a partial differential equation?
It is exactly what it sounds like. It is an equation involving partial derivatives.
- Why would one want to study PDEs?
PDEs are used to model a vast array of physical phenomena.
PDEs are model simple everyday phenomena; the shape of a soap film,
the motion of a water drop, the flow of a fluid, the buckling of a beam,
traffic flow on a road. They are used in all fields of engineering
electrical, chemical, mechanical and civil engineering and through out the
sciences. PDEs are also the fundamental objects
in the most complex physical and social theories: gravity (Einstein's general
theory of relativity), quantum mechanics and the new theories of quantum
gravity and quantum field theory, and even economic theory, stock pricing.
Image of a Breather soliton - a solution to a nonlinear
Schrodinger equation
- What are the prerequisites?
Courses: MA 221, MA 222 Differential Equations I and II, and either MA 330 Vector Calculus or MA 336 Boundary Value Problems or
MA 366 Functions of a Real Variable or instructor's permission.
It is also useful to have an inquiring mind and some mathematical sophistication.
This of course is an introductory course in PDEs, we will start with the basic partial
differential equations:
- The transport equation, which governs the transport of materials in a
fluid or the flow of cars on a road.
- The heat equation, which is used to model the dispersion of heat
(or other substances) in a body.
- The wave equation, which is used to model the oscillation of a string
or a lamina, and water waves.
- Laplace's equation and Poisson's equation, which are used to model the
distribution of charge (or mass) in a fluid (electrical field or potential field
We will also cover extensions and variations of these basic equations along with other
equations arising from student interest. This course will cover both theory
(existence, uniqueness, and structure of solutions) and practice (solving partial
differential equations). By theory, we mean there will be proofs and deriving
properties of solutions without actually solving the PDE. By practice, we mean
applying the theory and solving PDEs. However, solving PDEs typically means
using numerical techniques either finite differences and or finite elements or other
numerical methods of numerical approximation, which will be covered briefly.
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Evaluation will be by homework sets, quizzes, class participation, and a final project.
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