Numerical Methods for Partial Differential Equations

Numerical Methods for Partial Differential Equations
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Introduction to Numerical Methods for Partial Differential Equations PDEs

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Numerical Solution of Partial Differential Equations

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Open Mobile Search. Partial differential equations were not consciously created as a subject but emerged in the 18th century as ordinary differential equations failed to describe the physical principles being studied. The subject was originally developed by the major names of mathematics, in particular, Leonard Euler and Joseph-Louis Lagrange who studied waves on strings; Daniel Bernoulli and Euler who considered potential theory, with later developments by Adrien-Marie Legendre and Pierre-Simon Laplace; and Joseph Fourier's famous work on series expansions for the heat equation.

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Many of the greatest advances in modern science have been based on discovering the underlying partial differential equation for the process in question. Schrodinger's equation for quantum mechanical processes at the atomic level leads to experimentally verifiable results which have changed the face of atomic physics and chemistry in the 20th century. Inevitably the study of partial differential equations is a large undertaking, and falls into several areas of mathematics. JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser.

For instance, we need to estimate probabilities of undesirable events that are critical to public safety such as a chemical being transported to a particular location in groundwater, or the fracture of a stressed metal plate.

Numerical Methods for Partial Differential Equations

Numerical Methods for Partial Differential Equations supports Engineering Reports, a new Wiley Open Access journal dedicated to all areas of engineering and. View the homepage for Numerical Linear Algebra with Applications. Issue Volume 26, Issue 5. October View the homepage for Engineering Reports.

In short, it is imperative to incorporate uncertainty into mathematical models of physical processes so that risk assessments can be performed. We can easily represent the unknown inputs in PDEs as random quantities; we are then faced with solving so-called stochastic PDEs. For material parameters such as permeability coefficients or the modulus of elasticity of an elastic body, it is usually the case that their values, at two distinct spatial locations, are associated and so it is appropriate to talk about correlated random data rather than white noise.

In the past, solving PDEs with correlated random data has been avoided due to limitations in computing resources or else hampered by very primitive approximation schemes. Simply averaging multiple solutions that correspond to particular realisations of the inputs can result in a huge amount of wasted computation time. Recently, more sophisticated numerical methods for approximating solutions to PDEs with correlated random data have been proposed. Unfortunately, this work has been restricted to scalar, elliptic PDEs and the question of efficient linear algebra for the resulting linear systems of equations has been largely overlooked.

So-called stochastic Galerkin methods, in particular, have attractive approximation properties but have been somewhat ignored due to a lack of robust solvers. More simplistic schemes which require less user know-how but ultimately more computing time to implement, have been popularised. The aim of this project is to investigate approximation schemes for quantifying uncertainty in more complex engineering problems modelled by systems of PDEs with two output variables e.