Abstract
In this chapter, we consider several concrete situations stemming from various areas of applications, the mathematical modeling of which involves partial differential equation problems. We will not be rigorous mathematically speaking. There will be quite a few rather brutal approximations, not always convincingly justified. This is however the price to be paid if we want to be able to derive mathematical models that aim to describe the complex phenomena we are dealing with in a way that remains manageable. At a later stage, we will study some of these models with all required mathematical rigor. The simplest examples arise in mechanics. Let us start with the simplest example of all.
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Notes
- 1.
As a general rule, we neglect all terms of order strictly higher than one with respect to \(u'(x)\). This leads to a simplified linearized model. A model that would take into account such higher order terms would be by nature nonlinear, and thus a lot more difficult to study from the point of view of mathematics.
- 2.
The coefficient E is called the Young modulus of the material. It is measured in units of pressure. The higher the coefficient, the more rigid the material.
- 3.
I is an inertia momentum of the cross-section.
- 4.
This can of course be proved with a little more work.
- 5.
The minus sign is due to the physical convention that goes contrary to the mathematical convention in this case.
- 6.
Explicit solutions are very rare in PDE problems.
- 7.
In which, not only is the heat equation derived and solved in special cases, but Fourier series are invented, the heat kernel appears, etc.
- 8.
e stands for épaisseur in French, i.e. thickness.
- 9.
The heat flux is the quantity of heat going through a unit area in the plane during a unit of time.
- 10.
Here \(i^2=-1\).
- 11.
Think Avogadro’s number.
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© 2016 Springer International Publishing Switzerland
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Le Dret, H., Lucquin, B. (2016). Mathematical Modeling and PDEs. In: Partial Differential Equations: Modeling, Analysis and Numerical Approximation. International Series of Numerical Mathematics, vol 168. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-27067-8_1
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DOI: https://doi.org/10.1007/978-3-319-27067-8_1
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