Abstract
Many of the PDEs used in Engineering and Physics are the result of applying physical laws of conservation or balance to systems involving fields, that is, quantities defined over a continuous background of two or more dimensions, such as space and time.
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Notes
- 1.
This point of view is advocated in [4] with particular force by Stephen Wolfram, a physicist and the creator of the Mathematica code (which, ironically, is one of the best tools in the market for the solution of differential equations).
- 2.
On the other hand, recalling the equality of mixed partial derivatives (under assumptions that we assume to be fulfilled), the number of independent entries of this matrix is actually only \(n(n+1)/2\).
- 3.
Vector quantities, such as linear and angular momentum, can be treated in a similar way by identifying U alternatively with each of the components in a global Cartesian frame of reference.
- 4.
Consequently, we will not strictly adhere to the notational convention (2.2).
- 5.
This common policy is adopted in [3]. This is an excellent introductory text, which is highly recommended for its clarity and wealth of examples.
- 6.
The adjective non-homogeneous, in this case, refers to the fact that there are sources or sinks, that is, p does not vanish identically. Material inhomogeneity, on the other hand, would be reflected in a variation of the value of the diffusivity D throughout the tube.
- 7.
Neglecting convective terms.
- 8.
See Sect. 8.1.
- 9.
This is the case of the conservation of mass in conventional Continuum Mechanics. In the context of growing bodies (such as is the case in some biological materials) mass is not necessarily conserved.
References
Chadwick P (1999) Continuum mechanics: Concise theory and problems. Dover, New York
Chorin AJ, Marsden JE (1993) A mathematical introduction to fluid mechanics. Springer, Berlin
Knobel R (2000) An introduction to the mathematical theory of waves. American Mathematical Society, Providence
Wolfram S (2002) A new kind of science. Wolfram Media, Champaign
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Epstein, M. (2017). Partial Differential Equations in Engineering. In: Partial Differential Equations. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-55212-5_2
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