Lecture

Modeling Heat Equation

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In course

MATH-115(b): Advanced linear algebra II

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et de démontrer rigoureusement les résultats principaux du sujet. ...

Description

This lecture covers the modeling of the heat equation using Laplacian operator, Riemann varieties, and physical constants. It discusses the Laplacian operator, partial derivatives, Laplacian operator, and heat equation modeling. The lecture also explores Riemann varieties, physical constants, and the importance of Laplacian operator in heat equation modeling.

Instructor

Marc Troyanov

Expertise Differential Geometry Geometric Analysis Teaching & PhD Past EPFL PhD Students Olivier Mermoud, Khaled Gafaiti, Serguei Timochine, Stephen Ducret, Ratiba Djelid, Zahra Sinaei, Luc Genton, Adrien Giuliano Marcone, Guillaume Buro, Bruno Luiz Santos Correia Courses Advanced linear algebra II MATH-115(b) The purpose of this course is to introduce some basic notions of linear algebra and to rigorously prove the main results in the field.

Official source

Related concepts (32)

Heat equation

In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. As the prototypical parabolic partial differential equation, the heat equation is among the most widely studied topics in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations.

Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function with respect to the variable is variously denoted by It can be thought of as the rate of change of the function in the -direction.

Heat kernel

In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum of the Laplace operator, and is thus of some auxiliary importance throughout mathematical physics. The heat kernel represents the evolution of temperature in a region whose boundary is held fixed at a particular temperature (typically zero), such that an initial unit of heat energy is placed at a point at time t = 0.

Differential equation

In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

Derivative

In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.

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