Page - 287 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

9SolvingPartialDifferentialEquations We now turn to the solving of differential equations in which the solution is a functionthatdependsonseveral independentvariables.Onesuchequation iscalled apartialdifferentialequation (PDE,plural:PDEs). The subject of 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 readeraglimpseof the topicbypresentinga fewbasicandgeneralmethodsthatwe will apply toa verycommontypeofPDE. We shall focus on one of the most widely encountered partial differential equations: thediffusionequation,which in onedimension looks like ∂u ∂t =β∂ 2u ∂x2 +g. Themulti-dimensionalcounterpart isoftenwrittenas ∂u ∂t =β∇2u+g. We shall restrict theattentionhere to the one-dimensionalcase. The unknown in the diffusion equation is a functionu(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 glassofwater. One very popular application of the diffusion equation is for heat transport in solid bodies. Then u is the temperature, and the equation predicts how the temperatureevolves in space and time within thesolid body.For suchapplications, ©The Author(s) 2020 S.Linge, H.P. Langtangen, Programming forComputations -Python, Texts in Computational Science and Engineering 15, https://doi.org/10.1007/978-3-030-16877-3_9 287