Difference between revisions of "Math 547: Partial Differential Equations 1"

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== Chris Grant's Proposed Core Topics for Math 547/548 ==
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== Catalog Information ==
 +
 
 +
=== Title ===
 +
Partial Differential Equations 1. [Recommended change:  Applied Partial Differential Equations]
 +
 
 +
=== Credit Hours ===
 +
3
 +
 
 +
=== Offered ===
 +
On demand (contact department)
 +
 
 +
=== Prerequisite ===
 +
[[Math 334]], [[Math 342|342]]; or equivalents.
 +
 
 +
=== Recommended ===
 +
[[Math 352]] or equivalent.
 +
 
 +
=== Description ===
 +
Methods of analysis for hyperbolic, elliptic, and parabolic equations, including characteristic manifolds, distributions, Green's functions, maximum principles and Fourier analysis.
 +
 
 +
== Desired Learning Outcomes ==
 +
It is proposed that the focus of this course be '''applied''' partial differential equations.  Thus, the students ought to be less concerned with knowing existence/uniqueness proofs than with understanding the properties and representation of solutions and with using PDEs to model important phenomena.
 +
 
 +
=== Prerequisites ===
 +
Since ODEs appear in certain approaches to PDEs (in particular, the method of characteristics), [[Math 334]] is a prerequisite.  [[Math 342]] is a prerequisite to ensure that students are reasonably comfortable with analysis in several dimensions.
 +
 
 +
=== Minimal learning outcomes ===
 +
Outlined below are topics that all successful Math 547 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, and ability to make direct application of those results to related problems, including calculations.
 +
 
 
<div style="-moz-column-count:2; column-count:2;">
 
<div style="-moz-column-count:2; column-count:2;">
 
#  General Cauchy problem
 
#  General Cauchy problem
#* Cauchy-Kovalevskaya Theorem
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#* Cauchy-Kowalevski Theorem
 
#* Lewy Example
 
#* Lewy Example
 
#  Method of characteristics for first-order equations
 
#  Method of characteristics for first-order equations
Line 15: Line 43:
 
#* Rarefaction waves
 
#* Rarefaction waves
 
#  Classification of general second-order equations
 
#  Classification of general second-order equations
#  Canonical forms for semilinear second-order equations
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#  Canonical forms for semilinear second-order equations<br><br><br><br>
Classical theory for the canonical second-order linear equations on '''R'''<sup>''n''</sup>
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Hyperbolic equations
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#*  The wave equation
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#* Cauchy problem
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#* Problems with boundary data
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#* Huygens' principle
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#* Applications
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#  Elliptic equations
 
#* Laplace's equation
 
#* Laplace's equation
#** Green's first and second identities
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#* Poisson's equation
#** Mean Value Principle and its converse
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#* Green's functions
#** Weak and strong maximum principles
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#* Maximum principles
#** Uniqueness for the Dirichlet problem
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#* Applications
#** Poisson integral formula
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# Parabolic equations
#** Existence for the Dirichlet Problem on a ball
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#* The heat equation
#** Fundamental solutions
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#* Green's functions
#** Green's functions
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#* The heat kernel
#** Harnack inequality
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#* Maximum principles
#** Liouville's Theorem
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#* Applications
#** Harnack's Convergence Theorem
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#** Existence for the Dirichlet Problem on domains with regular boundaries and for continuous boundary data
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#** Interior and exterior sphere conditions
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#* Wave equation
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#** Method of spherical means
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#** Hadamard’s method of descent
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#** Huygen’s Principle
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#** Conservation of Energy
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#** Domain of Dependence
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#* Heat equation
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#** Fourier transforms
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#** The heat kernel
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#** Existence for the IVP
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#** Weak and strong maximum principles
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#** Uniqueness for the IBVP
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</div>
 
</div>
  
== Desired Learning Outcomes ==
+
=== Textbooks ===
 +
Possible textbooks for this course include (but are not limited to):
  
=== Prerequisites ===
+
* John Ockendon, Sam Howison, Andrew Lacey, and Alexander Movchan, ''Applied Partial Differential Equations (Revised Edition)'', Oxford University Press, 1999.
 
+
=== Minimal learning outcomes ===
+
 
+
<div style="-moz-column-count:2; column-count:2;">
+
 
+
</div>
+
  
 
=== Additional topics ===
 
=== Additional topics ===
  
 
=== Courses for which this course is prerequisite ===
 
=== Courses for which this course is prerequisite ===
 +
 +
Currently this course is listed as a prerequisite for [[Math 647]].  It is recommended that this connection be dropped, because the intended clientele of the two courses are quite different.
  
 
[[Category:Courses|547]]
 
[[Category:Courses|547]]

Latest revision as of 11:06, 14 November 2019

Catalog Information

Title

Partial Differential Equations 1. [Recommended change: Applied Partial Differential Equations]

Credit Hours

3

Offered

On demand (contact department)

Prerequisite

Math 334, 342; or equivalents.

Recommended

Math 352 or equivalent.

Description

Methods of analysis for hyperbolic, elliptic, and parabolic equations, including characteristic manifolds, distributions, Green's functions, maximum principles and Fourier analysis.

Desired Learning Outcomes

It is proposed that the focus of this course be applied partial differential equations. Thus, the students ought to be less concerned with knowing existence/uniqueness proofs than with understanding the properties and representation of solutions and with using PDEs to model important phenomena.

Prerequisites

Since ODEs appear in certain approaches to PDEs (in particular, the method of characteristics), Math 334 is a prerequisite. Math 342 is a prerequisite to ensure that students are reasonably comfortable with analysis in several dimensions.

Minimal learning outcomes

Outlined below are topics that all successful Math 547 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, and ability to make direct application of those results to related problems, including calculations.

  1. General Cauchy problem
    • Cauchy-Kowalevski Theorem
    • Lewy Example
  2. Method of characteristics for first-order equations
    • Semilinear case
    • Quasilinear case
    • General case
  3. Quasilinear systems of conservation laws on a line
    • Riemann problem
    • Rankine-Hugoniot jump condition
    • Entropy condition
    • Shocks
    • Rarefaction waves
  4. Classification of general second-order equations
  5. Canonical forms for semilinear second-order equations



  6. Hyperbolic equations
    • The wave equation
    • Cauchy problem
    • Problems with boundary data
    • Huygens' principle
    • Applications
  7. Elliptic equations
    • Laplace's equation
    • Poisson's equation
    • Green's functions
    • Maximum principles
    • Applications
  8. Parabolic equations
    • The heat equation
    • Green's functions
    • The heat kernel
    • Maximum principles
    • Applications

Textbooks

Possible textbooks for this course include (but are not limited to):

  • John Ockendon, Sam Howison, Andrew Lacey, and Alexander Movchan, Applied Partial Differential Equations (Revised Edition), Oxford University Press, 1999.

Additional topics

Courses for which this course is prerequisite

Currently this course is listed as a prerequisite for Math 647. It is recommended that this connection be dropped, because the intended clientele of the two courses are quite different.