Difference between revisions of "MAT 5673"

From Department of Mathematics at UTSA
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==Course description==
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Partial differential equations arise in many different areas as one tries
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to describe the behavior of a system ruled by some law. Typically, this has to do with
 +
some physical process such as heat diffusion in a material, vibrations of a bridge, circulation
 +
of fluids, the behavior of microscopic particles or the evolution of the universe as a whole.
 +
Modeling by means of partial differential equations has been successful in other disciplines
 +
as well, like in the case of the Black-Scholes equation for stock options pricing and the
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Hodgkin–Huxley equations for firing patterns of a neuron. Partial differential equations are
 +
an important tool in Applied Math and Pure Math. This course gives an introduction to PDE's in the setting of two independent variables.
 
==Topics List==
 
==Topics List==
 
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Introduction and classification of PDE, Calculus review
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Multivariable Calculus, Chain Rule
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* Definition of a PDE as a relation between partial derivatives of an unknown function. Classification of PDE according to order - linear/nonlinear/quasilinear
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|-
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|Week 2
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*
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Applied examples of PDE
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Multivariable Calculus, Chain Rule
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* Origin and background of common PDE's: heat equation, wave equation, transport equation, etc.
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|-
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|Week 3
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*
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The method of characteristics for first-order quasilinear equations
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Multivariable Calculus, Chain Rule
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* Solving quasilinear first-order equations using the method of characteristics
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|-
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|Week 4
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*
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The method of characteristics for first-order fully nonlinear equations
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Multivariable Calculus, Chain Rule
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* Solving fully nonlinear first-order equations (e.g. the Eikonal equation) using the method of characteristics
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|-
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|Week 5
 
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*  
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Heat and wave equation on the whole real line
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Differentiation of integrals with respect to a parameter, integration by parts
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* Fundamental solution of the heat equation, D'Alembert's formula for the wave equation
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|-
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|Week 6
 
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*  
 
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* Learning outcome
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Initial-boundary value problem for heat and wave equation I
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Partial derivatives, chain rule
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* Separation of variables method for heat and wave equation
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|Week 7
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Initial-boundary value problem for heat and wave equation II, introduction to Fourier series
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Partial derivatives, chain rule
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* Infinite superposition of basic solutions found by separation to form more general solutions
 
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Revision as of 08:54, 12 March 2023

Course description

Partial differential equations arise in many different areas as one tries to describe the behavior of a system ruled by some law. Typically, this has to do with some physical process such as heat diffusion in a material, vibrations of a bridge, circulation of fluids, the behavior of microscopic particles or the evolution of the universe as a whole. Modeling by means of partial differential equations has been successful in other disciplines as well, like in the case of the Black-Scholes equation for stock options pricing and the Hodgkin–Huxley equations for firing patterns of a neuron. Partial differential equations are an important tool in Applied Math and Pure Math. This course gives an introduction to PDE's in the setting of two independent variables.

Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1

Introduction and classification of PDE, Calculus review

Multivariable Calculus, Chain Rule

  • Definition of a PDE as a relation between partial derivatives of an unknown function. Classification of PDE according to order - linear/nonlinear/quasilinear
Week 2

Applied examples of PDE

Multivariable Calculus, Chain Rule

  • Origin and background of common PDE's: heat equation, wave equation, transport equation, etc.
Week 3

The method of characteristics for first-order quasilinear equations

Multivariable Calculus, Chain Rule

  • Solving quasilinear first-order equations using the method of characteristics
Week 4

The method of characteristics for first-order fully nonlinear equations

Multivariable Calculus, Chain Rule

  • Solving fully nonlinear first-order equations (e.g. the Eikonal equation) using the method of characteristics
Week 5

Heat and wave equation on the whole real line

Differentiation of integrals with respect to a parameter, integration by parts

  • Fundamental solution of the heat equation, D'Alembert's formula for the wave equation
Week 6

Initial-boundary value problem for heat and wave equation I

Partial derivatives, chain rule

  • Separation of variables method for heat and wave equation
Week 7

Initial-boundary value problem for heat and wave equation II, introduction to Fourier series

Partial derivatives, chain rule

  • Infinite superposition of basic solutions found by separation to form more general solutions
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