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Concept# Heat equation

Summary

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. The heat equation can also be considered on Riemannian manifolds, leading to many geometric applications. Following work of Subbaramiah Minakshisundaram and Åke Pleijel, the heat equation is closely related with spectral geometry. A seminal nonlinear variant of the heat equation was introduced to differential geometry by James Eells and Joseph Sampson in 1964, inspiring the introduction of the Ricci flow by Richard Hamilton

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MATH-305: Introduction to partial differential equations

This is an introductory course on Elliptic Partial Differential Equations. The course will cover the theory of both classical and generalized (weak) solutions of elliptic PDEs.

ME-341: Heat and mass transfer

This course covers fundamentals of heat transfer and applications to practical problems. Emphasis will be on developing a physical and analytical understanding of conductive, convective, and radiative heat transfer.

PHYS-324: Classical electrodynamics

The goal of this course is the study of the physical and conceptual consequences of Maxwell equations.

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Partial differential equation

In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function.
The function is often thought of as

Dirac delta function

In mathematical physics, the Dirac delta distribution (δ distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is

Fourier transform

In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transfo

This work is concerned with a numerical simulation of the thermal behaviour of an electrolysis cell for the production of the aluminium. Aluminium is produced by an electrolytic reduction of alumina dissolved in a bath of molten cryolite. In this reduction process, called Hall-Héroult process, the metal is produced at about 965 °C. A frozen bath layer, called ledge, arises in the boundary region and protects the side walls of the cell from corrosive electrolyte. This ledge may change the magnetohydrodynamical equilibrium of the cell and reduce the heat loss through the walls. The ledge is thus playing a significant role in both the thermal and magnetohydrodynamical behaviour of the cell. A precise knowledge of the ledge is thus an imporant ingredient in the optimization process of the cell. The temperature field and the ledge shape in a whole smelter are obtained by simultaneously solving the system of equations formed by: a non-linear convection-diffusion heat equation, which can be considered as a Stephan problem in enthalpy and temperature in the domain of the cell occupied by fluids and ledge, Navier-Stokes equations with a free interface in the fluid domains and Maxwell equations in the whole space. The source term of the heat equation results from the Joule effect due to the electrical current crossing the cell. A Chernoff scheme is used to numerically solve Stephan problem. Three dimensional numerical calculations showing ledge shape, temperature and velocity fields as well as electrical potential for an operating cell are obtained. The effect of thermal field on the electrical current and the effect of fluid motions on the ledge shape in the aluminium cells are presented.

In this thesis, we study the stochastic heat equation (SHE) on bounded domains and on the whole Euclidean space $\R^d.$ We confirm the intuition that as the bounded domain increases to the whole space, both solutions become arbitrarily close to one another. Both vanishing Dirichlet and Neumann boundary conditions are considered.We first study the nonlinear SHE in any space dimension with multiplicative correlated noise and bounded initial data. We prove that the solutions to SHE on an increasing sequence of domains converge exponentially fast to the solution to SHE on $\R^d.$ Uniform convergence on compact set is obtained for all $p$-moments. The conditions that need to be imposed on the noise are the same as those required to ensure existence of a random field solution. A Gronwall-type iteration argument is used together with uniform bounds on the solutions, which are surprisingly valid for the entire sequence of increasing domains.We then study SHE in space dimension $d\ge 2$ with additive white noise and bounded initial data. Even though both solutions need to be considered as distributions, their difference is proved to be smooth. If fact, the order of smoothness depends only on the regularity of the boundary of the increasing sequence of domains. We prove that the Fourier transform, in the sense of distributions, of the solution to SHE on $\R^d$ do not have any locally mean-square integrable representative. Therefore, convergence is studied in local versions of Sobolev spaces. Again, exponential rate is obtained.Finally, we study the Anderson model for SHE with correlated noise and initial data given by a measure. We obtain a special expression for the second moment of the difference of the solution on $\R^d$ with that on a bounded domain. The contribution of the initial condition is made explicit. For example, exponentially fast convergence on compact sets is obtained for any initial condition with polynomial growth. More interestingly, from a given convergence rate, we can decide whether some initial data is admissible.

The aim of the project is double: to understand the flexibility of the Isogeometric Analysis tools through the solution of some PDEs problems; to test the improvement in the computational time given by a partial loops vectorization at compile-time of the LifeV IGA code. Three different appli- cations have been selected: the potential flow problem around an airfoil profile, the heat equation problem in a bent cylinder and the Laplace problem in a multi-patches geometry representing a blood vessels bifurcation. The geometries used are built through the NURBS package available with the software GeoPDEs. The numerical analysis of the first application is performed with both GeoPDEs and LifeV IGA code. The comparison between different implementations shows that the degrees of freedom loop vectorization at compile-time is able to reduce the matrix assembling time of around 20%. The automatic vectorization at compile-time of the loop on the elements requires too much computational effort without having a reasonable improvement in the running time performances. Unsteady problems and multi-patches geometry have not been tested with LifeV IGA code, but GeoPDEs results show the expected solutions.

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