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| − | ==Course description==
| + | #REDIRECT[[MAT5143]] |
| − | Mathematical Physics tentative topics list. This course is aimed at physics majors who wish to deepen their understanding or mathematical methods used in physics.
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| − | ==Topics List==
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| − | {| class="wikitable sortable"
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| − | ! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes
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| − | |Week 1
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| − | *
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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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| − | |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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| − | |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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| − | |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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| − | |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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| − | |Week 6
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| − | *
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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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| − | *
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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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| − | * Forming more general solutions out of infinite superposition of basic solutions
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| − | |Week 8
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| − | *
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| − | Introduction to Fourier series
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| − | Infinite series
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| − | * Orthonormal systems of functions, spectral method for the wave and heat equation
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| − | |Week 9
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| − | *
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| − | Schroedinger equation
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| − | Complex numbers
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| − | * Basic properties of Schroedinger equation, particle in a potential well
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| − | |Week 10
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| − | Complex numbers
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| − | Qualitative properties of PDE's
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| − | Differentiation of integrals with respect to parameter
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| − | * Uniqueness of solutions, finite and infinite propagation speed for wave and heat equation
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| − | |Week 11
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| − | *
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| − | Introduction to numerical methods for PDE (optional)
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| − | Derivatives, Calculus
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| − | * Basic finite difference schemes for first-order quasilinear equations, CFL condition
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| − | |Week 12
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| − | Matrices, Linear Algebra
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| − | Introduction to the Laplace and Poisson equation
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| − | *
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| − | * Solving the Laplace equation on the whole space and on a simple bounded region (square, disc)
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| − | |Week 13
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| − | *
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| − | Introduction to the Calculus of Variations
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| − | Differentiation of an integral with respect to a parameter, parametric surfaces
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| − | * Compute the variational derivative of a functional
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| − | |Week 14
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| − | *
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| − | Review, advanced topics
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| − | *
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| − | *
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| − | |}
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