Salsa Sandro - A Primer on PDEs Models, Methods, Simulations

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Part 1
Introduction

Sandro Salsa , Federico M. G. Vegni , Anna Zaretti and Paolo Zunino UNITEXT A Primer on PDEs Models, Methods, Simulations 10.1007/978-88-470-2862-3_1 Springer-Verlag Italia 2013

1. Introduction

Sandro Salsa 1, Federico M. G. Vegni 1, Anna Zaretti 1 and Paolo Zunino 2, 3

(1)

Department of Mathematics, Politecnico di Milano, Italy

(2)

MOX Department of Mathematics, Politecnico di Milano, Italy

(3)

Department of Mechanical Engineering and Materials Science, University of Pittsburgh, USA

Abstract

Mathematical modelling plays a big role in the description of a large part of phenomena in the applied sciences and in several aspects of technical and industrial activity.

1.1 1.1 Mathematical Modelling

Mathematical modelling plays a big role in the description of a large part of phenomena in the applied sciences and in several aspects of technical and industrial activity.

By a mathematical model we mean a set of equations and/or other mathematical relations capable of capturing the essential features of a complex natural or artificial system, in order to describe, forecast and control its evolution. The applied sciences are not confined to the classical ones; in addition to physics and chemistry , the practice of mathematical modelling heavily affects disciplines like finance, biology, ecology, medicine, sociology.

In the industrial activity (e.g. for aerospace or naval projects, nuclear reactors, combustion problems, production and distribution of electricity, traffic control, etc) the mathematical modelling, involving first the analysis and the numerical simulation and followed by experimental tests, has become a common procedure, necessary for innovation, and also motivated by economic factors. It is clear that all of this is made possible by the enormous computational power now available.

In general, the construction of a mathematical model is based on two main ingredients: general laws and constitutive relations. In this book we shall deal with general laws coming from continuum mechanics and appearing as conservation or balance laws (e.g. of mass, energy, linear momentum, etc).

The constitutive relations are of an experimental nature and strongly depend on the features of the phenomena under examination. Examples are the Fourier law of heat conduction, the Fick law for the diffusion of a substance or the way the speed of a driver depends on the density of cars ahead.

The outcome of the combination of the two ingredients is usually a partial differential equation or a system of them.

1.2 1.2 Partial Differential Equations

A partial differential equation is a relation of the following type:

(1.1)

where the unknown u = u (x , ,x n ) is a function of n variables and are its partial derivatives. The highest order of differentiation occurring in the equation is the order of the equation.

A first important distinction is between linear and nonlinear equations. Equation () is linear if F is linear with respect to u and all its derivatives, otherwise it is nonlinear.

A second distinction concerns the types of nonlinearity. We distinguish:

• Semilinear equations where F is nonlinear only with respect to u but is linear with respect to all its derivatives.
• Quasi-linear equations where F is linear with respect to the highest order derivatives of u.
• Fully nonlinear equations where F is nonlinear with respect to the highest order derivatives of u.

The theory of linear equations can be considered sufficiently well developed and consolidated, at least for what concerns the most important questions. On the contrary, the non linearities present such a rich variety of aspects and complications that a general theory does not appear to be conceivable. The existing results and the new investigations focus on more or less specific cases, especially interesting in the applied sciences.

To give the reader an idea of the wide range of applications we present a series of examples, suggesting one of the possible interpretations. Most of them are considered at various level of deepness in this book. In the examples, x represents a space variable (usually in dimension n =1, 2,3) and t is a time variable.

We start with linear equations. In particular, equations () are fundamental and their theory constitutes a starting point for many other equations.

Transport equation (first order):

(1.2)

It describes for instance the transport of a solid polluting substance along a channel; here u is the concentration of the substance and v is the stream speed. We consider the one-dimensional version of (.

Diffusion or heat equation (second order):

(1.3)

where is the Laplace operator. It describes the conduction of heat through a homogeneous and isotropic medium; u is the temperature and D encodes the thermal properties of the material. is devoted to the heat equation and its variants.

Wave equation (second order):

(1.4)

It describes for instance the propagation of transversal waves of small amplitude in a perfectly elastic chord (e.g. of a violin) if n =1, or membrane (e.g. of a drum) if n =2. If n =3 it governs the propagation of electromagnetic waves in vacuum or of small amplitude sound waves ( Section 6.6.4 ). Here u may represent the wave amplitude and c is the propagation speed.

Laplaces or potential equation (second order):

(1.5)

where u = u (x). The diffusion and the wave equations model evolution phenomena. The Laplace equation describes the corresponding steady state , in which the solution does not depend on time anymore. Together with its nonhomogeneous version

called Poissons equation, it plays an important role in electrostatics as well. is devoted to these equations.

Black-Scholes equation (second order):

Here u = u (x,t), x 0, t 0. Fundamental in mathematical finance, this equation governs the evolution of the price u of a so called derivative (e.g. an European option ), based on an underlying asset (a stock, a currency, etc.) whose price is x.

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