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Introduction to Symmetry Analysis

Brian CantwellDepartment of Aeronautics and Astronautics

Stanford University

Chapter 9 - Partial Differential Equations

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Finite transformation of partial derivatives

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The p-th extended finite group is

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Variable count

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Infinitesimal transformation of first partial derivatives

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Substitute

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The (p-1)th order extended infinitesimal transformation is

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The number of terms in the infinitesimal versus the derivative order

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Isolating the determining equations of the group - the Lie algorithm

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The classical point group of the heat equation

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Invariance condition

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The fully expanded invariance condition is

Apply the constraint that the solution must satisfy theheat equation and gather terms

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The determining equations of the point group of the heat equation

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Series solution of the determining equations

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The classical six-parameter group of the heat equation

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Impulsive source solutions of the heat equation

Boundary conditions

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Is the integral conserved?

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This problem is invariant under the three parameter group of dilationsin the dependent and independent variables and translation in time.

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Similarity variables are the invariants of the infinitesimal transformation.

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Now substitute the similarity form of the solution into the heat equation.The result is a second order ODE of Sturm-Liouville type.

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Stanford University Department of Aeronautics and Astronautics

A modified problem of an instantaneous heat source

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Stanford University Department of Aeronautics and Astronautics

Invariant group