we remark that separate ode for the (knownboundary condition tis not strictly needed we can just work with the ode system for un and in the ode for replace tby thoweverthese authors prefer to have an ode for every point value ui which requires formulating the known boundary at as an ode the reason for including the boundary values in the ode system is that the solution of the system is then the complete solution at all mesh pointswhich is convenientsince special treatment of the boundary values is then avoided the condition @ =@ at is bit more complicatedbut we can approximate the spatial derivative by centered finite differencev @ vv un un @ dn this approximation involves fictitious point xn outside the domain common trick is to use ( for and eliminate un by use of the discrete boundary condition (un un )dun un un tc gn tdv dt ( that iswe have special version of ( at the boundary what about simpler finite differences at the boundarysome reader may think that smarter trick is to approximate the boundary condition @ =@ at by one-sided differencev @ vv un un @ vi dn this gives simple equation un un for the boundary valueand corresponding ode howeverthis approximation has an error of order xwhile the centered approximation we used above has an error of order the finite difference approximation we used for the second-order derivative in the diffusion equation also has an error of order thusif we use the simpler one-sided difference aboveit turns out that we reduce the overall accuracy of the method we are now in position to summarize how we can approximate the partial differential equation problem ( )-( by system of ordinary differential equationsdu /dt dui ui ui tc ui / gi / dt dun un un / gn td dt ( ( (