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- W4238795874 abstract "The subject of partial differential equations (PDEs) is enormous. At the same time, it is very important, since so many phenomena in nature and technology find their mathematical formulation through such equations. Knowing how to solve at least some PDEs is therefore of great importance to engineers. In an introductory book like this, nowhere near full justice to the subject can be made. However, we still find it valuable to give the reader a glimpse of the topic by presenting a few basic and general methods that we will apply to a very common type of PDE. We shall focus on one of the most widely encountered partial differential equations: the diffusion equation, which in one dimension looks like $$frac{partial u}{partial t}=betafrac{partial^{2}u}{partial x^{2}}+gthinspace.$$ The multi-dimensional counterpart is often written as $$frac{partial u}{partial t}=betanabla^{2}u+gthinspace.$$ We shall restrict the attention here to the one-dimensional case. The unknown in the diffusion equation is a function $$u(x,t)$$ of space and time. The physical significance of u depends on what type of process that is described by the diffusion equation. For example, u is the concentration of a substance if the diffusion equation models transport of this substance by diffusion. Diffusion processes are of particular relevance at the microscopic level in biology, e.g., diffusive transport of certain ion types in a cell caused by molecular collisions. There is also diffusion of atoms in a solid, for instance, and diffusion of ink in a glass of water." @default.
- W4238795874 created "2022-05-12" @default.
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- W4238795874 date "2016-01-01" @default.
- W4238795874 modified "2023-09-27" @default.
- W4238795874 title "Solving Partial Differential Equations" @default.
- W4238795874 doi "https://doi.org/10.1007/978-3-319-32452-4_5" @default.
- W4238795874 hasPublicationYear "2016" @default.
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