solving partial differential equations the subject of partial differential equations (pdesis enormous at the same timeit is very importantsince 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 thisnowhere near full justice to the subject can be made howeverwe still find it valuable to give the reader glimpse of the topic by presenting few basic and general methods that we will apply to very common type of pde we shall focus on one of the most widely encountered partial differential equationsthe diffusion equationwhich in one dimension looks like @ @ dv cg@ @ the multi-dimensional counterpart is often written as @ vr @ we shall restrict the attention here to the one-dimensional case the unknown in the diffusion equation is function xtof space and time the physical significance of depends on what type of process that is described by the diffusion equation for exampleu is the concentration of substance if the diffusion equation models transport of this substance by diffusion diffusion processes are of particular relevance at the microscopic level in biologye diffusive transport of certain ion types in cell caused by molecular collisions there is also diffusion of atoms in solidfor instanceand diffusion of ink in glass of water one very popular application of the diffusion equation is for heat transport in solid bodies then is the temperatureand the equation predicts how the temperature evolves in space and time within the solid body for such applicationsthe equation is known as the heat equation we remark that the temperature in fluid is influenced not only by diffusionbut also by the flow of the liquid if presentthe latter effect requires an extra term in the equation (known as an advection or convection termthe term is known as the source term and represents generationor lossof heat (by some mechanismwithin the body for diffusive transportg models injection or extraction of the substance (cthe author( lingeh langtangenprogramming for computations pythontexts in computational science and engineering doi --