Partial Differential Equations and Integral Transforms

Course #MA3132

Est.imated Completion Time: 3 months

Overview

Solution of boundary value problems by separation of variables; Sturm-Liouville problems; Fourier and Bessel series solutions, Fourier transforms; classification of second-order equations; applications, method of characteristics. Applications to engineering and physical science. Satisfies the ESR in differential equations for the Applied Mathematics program.

Prerequisites

  • MA2121
  • MA1116

Learning Outcomes

  • Understand linearity and homogeneity in the context of PDEs. Be able to distinguish linear and nonlinear problems and homogeneous and inhomogeneousproblems.
  • Be able to apply the method of separation of variables to develop Fourier series solutions to linear initial boundary value problems in one and several spatial dimensions.
  • Understand Fourier series in the context of PDEs. Be able to expand functions in a Fourier series, including sine and cosine series, by calculating their Fourier coefficients and/or exploitingsymmetries.
  • Develop both qualitative and quantitative understanding of the three fundamental second-order linear PDEs: the heat equation, the wave equation, and Laplace’s equa- tion.  Understand the basic physics that each describes.  Be able to solve each of   these equations with constant coefficients for common boundary conditions in simple geometries using separation ofvariables.
  • Understand the difference between Dirichlet, Neumann, and periodic boundary con- ditions physically and mathematically. Understand the correspondence between these boundary conditions and the various forms of Fourier series solutions to PDEs.
  • Develop a basic understanding of Sturm–Liouville theory as it applies to PDEs. Recognize Sturm–Liouville eigenvalue problems. Be able to identify regular, singular,  and self-adjoint Sturm–Liouville problems. Recognize these problems as they occur in separation-of-variables solutions toPDEs.
  • Be able to apply techniques for inhomogeneous problems: solution for displacement from an equilibrium/reference solution, eigenfunction expansion.
  • Be able to apply the method of characteristics to solve first-order linear and quasi  -linear PDEs. Understand and apply d’Alembert’s solution to thewave equation.
  • Be able to classify second-order linear PDEs as parabolic, hyperbolic, or elliptic. Understand qualitatively how the classification of a PDE determines features of its solution.
  • Be able to apply Fourier transform techniques for solving linear initial boundary value problems posed on unbounded domains. Understand the Fourier, sine, and cosine transforms as  “continuous analogues” of their series counterparts.
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Application Deadlines

  •  08 Jul 2024

    Fall Quarter applications due

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