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Partial Differential Equations

Revised: March 2024

Course Description

Solution techniques, modeling and analysis of partial differential equations (PDEs) and boundary value problems. Techniques include separation of variables methods for second-order linear PDEs, Fourier series, Transform methods, Bessel functions, and Sturm-Liouville Theory. Additional topics may include Green’s functions, numerical methods, and method of characteristics.  Prerequisites: Math 256 and Math 320.

Student Learning Objectives

By the end of the course students will be able to:

• formulate partial differential equations (PDEs) and boundary value problems (BVPs) via modeling;
• apply analytic techniques to solve and examine both PDEs and BVPs;
• discuss the importance of eigenvalues and eigenfunctions, both mathematically and physically;
• construct the decomposition of a function via eigenfunction expansion;
• discuss the importance of integral operators and how they are utilized to simplify linear PDEs;
• interpret the solutions to PDEs and BVPs in a physically relevant fashion.

Text

Boundary Value Problems and Partial Differential Equations, 6th edition, Powers.

Grading Procedure

Grading procedures and factors influencing course grade are left to the discretion of individual instructors, subject to general university policy.

Attendance Policy

Attendance policy is left to the discretion of individual instructors, subject to general university policy.

Course Outline

• Chapter 0: Ordinary Differential Equations (2 weeks)
  Review, Boundary Value Problems, Eigenvalues, Eigenfunctions, Sturm-Liouville Theory
• Chapter 1: Fourier Series and Integrals (2 weeks)
  Fourier Series, Fourier Integrals, Fourier Transform
• Chapter 2: The Heat Equation (4 weeks)
  Derivation and Analysis, Steady States, Separation of Variables, Variants, Integral Solutions for Semi-Infinite/Infinite Domains
• Chapter 3: The Wave Equation (1 week)
  Derivation and Analysis, Separation of Variables, Vibrating Beam, d’Alembert’s Solution
• Chapter 4: The Potential Equation (2.5 weeks)
  Derivation and Analysis, Separation of Variables, Splitting BCs, Potential Equation on a disk, Helmholtz Equation, Poisson Equation in rectangular and circular domains
• Chapter 5: Higher Dimensions and Other Coordinates (1.5 weeks)
  Heat and Wave Equations in higher dimensions (rectangular coordinates), Polar Coordinates, Bessel Functions, Vibrations on a Circular Membrane, Spherical Coordinates and Legendre Polynomials
• Chapter 6: Laplace Transform (0.5 week)
  Integral Transforms, Convolution, Application to PDEs
• Additional Topics as time allows, based on student and instructor interest.