Page - 161 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

5SolvingPartialDifferentialEquations Thesubjectofpartial differentialequations (PDEs) is enormous.At thesame time, it is very important, since somanyphenomena innature and technologyfind their mathematical formulation through such equations. Knowinghow to solve at least somePDEs is therefore of great importance to engineers. In an introductorybook like this, nowhere near full justice to the subject can bemade. However, we still find it valuable to give the reader a glimpse of the topic bypresenting a fewbasic andgeneralmethods thatwewill apply toaverycommontypeofPDE. Weshall focusononeof themostwidely encounteredpartial differential equa- tions: thediffusionequation,which inonedimension looks like @u @t Dˇ@ 2u @x2 Cg: Themulti-dimensionalcounterpart is oftenwrittenas @u @t Dˇr2uCg: Weshall restrict theattentionhere to theone-dimensionalcase. The unknown in the diffusion equation is a functionu.x;t/ of space and time. The physical significance ofu depends onwhat type of process that is described by thediffusionequation. For example,u is the concentrationof a substance if the diffusion equationmodels transport of this substance by diffusion. Diffusion pro- cesses areofparticular relevanceat themicroscopic level inbiology,e.g., diffusive transportof certain ion types inacell causedbymolecularcollisions. There is also diffusionofatoms ina solid, for instance, anddiffusionof ink inaglassofwater. One very popular application of the diffusion equation is for heat transport in solidbodies. Thenu is the temperature, and theequationpredictshow the temper- ature evolves in space and timewithin the solid body. For such applications, the equation is knownas theheat equation. We remark that the temperature in afluid is influenced not only by diffusion, but also by the flowof the liquid. If present, the latter effect requires an extra term in the equation (known as an advection or convection term). Thetermg isknownasthesourcetermandrepresentsgeneration,orloss,ofheat (by somemechanism)within thebody. For diffusive transport,gmodels injection orextractionof thesubstance. 161©TheAuthor(s)2016 S.Linge,H.P.Langtangen,Programming for Computations –Python, Texts inComputational Science andEngineering15,DOI10.1007/978-3-319-32428-9_5